Posted October 1, 2003

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 243
Patrick Gilmer, Mathematics Department, LSU

Integrality for TQFTs

Posted October 21, 2003

Last modified January 10, 2022

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 243
Richard A. Litherland, Mathematics Department, LSU

On the Ozsváth-Szabó homology theory

Posted October 14, 2003

Geometry and Topology Seminar Seminar website

3:40 pm – 4:40 pm 243 Lockett Hall
Charles Frohman, University of Iowa

Symplectic measure, Reidemeister torsion and the Jones polynomial

Posted January 27, 2004

Last modified January 30, 2004

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 285
Tara Brendle, Department of Mathematics, LSU

On finite order generators of the mapping class group

Visit supported by Visiting Experts Program in Mathematics, Louisiana Board of Regents LEQSF(2002-04)-ENH-TR-13

Posted February 12, 2004

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 285
Ian Agol, University of Illinois, Chicago

Tameness of hyperbolic 3-manifolds

Visit supported by Visiting Experts Program in Mathematics, Louisiana Board of Regents LEQSF(2002-04)-ENH-TR-13

Posted March 8, 2004

Geometry and Topology Seminar Seminar website

2:40 pm – 3:30 pm Lockett 381
Graham Denham, University of Western Ontario

The Homotopy Lie Algebra of an Arrangement

Posted March 3, 2004

Last modified March 12, 2004

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm tba
Neal Stoltzfus, Mathematics Department, LSU

Diagonalization of the Lickorish Form on Non-crossing Chord Diagrams

Posted March 22, 2004

Last modified March 25, 2004

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm 285 Lockett
Thomas Kerler, Ohio State University

Mapping Class Group Representations from TQFT

Abstract: The TQFTs of Witten Reshetikhin Turaev imply representations of the mapping class

groups over the cyclotomic integers Z[\\zeta] for \\zeta a prime root of unity. These

representations are highly structured and allow \"perturbative\" filtrations due to the

rich ideal structure of Z[\\zeta]. It is not too surprising that they are related to

well known filtrations of the mapping class groups, given, for example, by the

Johnson subgroups. We will describe such explicit relations in \"low order\" examples.

Visit supported by Visiting Experts Program in Mathematics, Louisiana Board of Regents.

LEQSF(2002-04)-ENH-TR-13

Posted September 8, 2004

Geometry and Topology Seminar Seminar website

4:10 pm – 5:00 pm Lockett 284
Scott Baldridge, Louisiana State University

Introduction to 4-Manifold Theory, I

Posted September 20, 2004

Geometry and Topology Seminar Seminar website

4:10 pm – 5:00 pm Lockett 284
Scott Baldridge, Louisiana State University

Introduction to 4-manifold theory, II

Posted September 21, 2004

Geometry and Topology Seminar Seminar website

4:10 pm – 5:00 pm Lockett 284
Scott Baldridge, Louisiana State University

Introduction to 4-Manifolds III

Posted September 24, 2004

Last modified October 1, 2004

Geometry and Topology Seminar Seminar website

4:00 pm – 5:00 pm Lockett 284
Gregor Masbaum, University Paris 7

Integral lattices in TQFT

Visit supported by Visiting Experts Program in Mathematics, Louisiana Board of Regents LEQSF(2002-04)-ENH-TR-13

Posted February 10, 2005

Last modified February 15, 2005

Geometry and Topology Seminar Seminar website

1:00 pm – 2:00 pm Life Sciences A 663 Access Grid Video Conference Room
Daniel C. Cohen, Mathematics Department, LSU

Topology and Combinatorics of boundary manifolds of arrangements

Joint Virtual Seminar with the University of Iowa

Posted February 20, 2005

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 285
William Schellhorn, LSU

Virtual Strings for Closed Curves with Multiple Components

Abstract: A Gauss paragraph is a combinatorial formulation of a generic closed curve with multiple components on some surface. A virtual string is a collection of circles with arrows that represent the crossings of such a curve. Every closed curve has an underlying virtual string and every virtual string has an underlying Gauss paragraph. A word-wise partition is a partition of the alphabet set of a Gauss paragraph that satisfies certain conditions with respect to the Gauss paragraph. This talk will discuss how the theory of virtual strings can be used to obtain necessary and sufficient conditions for a Gauss paragraph and word-wise partition to represent a closed curve in the 2-sphere.

Posted March 1, 2005

Last modified October 1, 2021

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 239
Kee Lam, University of British Columbia

Low dimensional spinor bundles over projective spaces

Given a k-dimensional vector bundle E over a real projective space, the "geometric dimension problem" asks for the maximal s such that E contains an s-dimensional trivial sub-bundle. This problem originates from the study of immersions of projective spaces into Euclidean space, and has been much pursued by topologists over the last 40 years. As a general phenomenon, k-s will be smaller when k is divisible by a higher power of 2. In this talk we shall examine such a phenomenon from the view point of spinor representations, and obtain some partial results. Some of these results turn out to be best possible.

Posted March 9, 2005

Last modified March 11, 2005

Geometry and Topology Seminar Seminar website

4:00 pm – 5:00 pm Lockett 285
Patrick Gilmer, Mathematics Department, LSU

Integral Lattices in TQFT

Posted March 29, 2005

Geometry and Topology Seminar Seminar website

1:30 pm – 2:30 pm
Xiao-Song Lin, University of California Riverside

Representations of Braid Groups and Colored Homfly Polynomials

Virtual Seminar together with U Iowa

Posted March 29, 2005

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 285
Khaled Qazaqzeh, LSU

Integral Bases for the SU(2)-TQFT-modules in genus one

Posted March 15, 2005

Last modified October 1, 2021

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Life Science A663
Dror Bar-Natan, University of Toronto

I don't understand Khovanov-Rozansky homology

Visit supported in part by Visiting Experts Program in Mathematics, Louisiana, Board of Regents LEQSF(2002-04)-ENH-TR-13.

Posted April 6, 2005

Last modified April 19, 2005

Geometry and Topology Seminar Seminar website

1:40 pm – 2:30 pm Life Science A 663
Cameron Gordon, University of Texas, Austin

Knots with Unknotting Number 1 and Conway Spheres

Virtual Seminar with U Iowa.

Cameron Gordon is visiting U Iowa.

Posted April 19, 2005

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 285
Ambar Sengupta, Mathematics Department, LSU

Quantum Physics from Pure Logic

Posted March 27, 2005

Last modified April 27, 2005

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 285
Abhijit Champanerkar, University of South Alabama

Scissors congruence and Bloch invariants of hyperbolic 3-manifolds.

Abstract: I will give a background of scissors congruence in various geometries. The complexified Dehn invariant for scissors congruence in hyperbolic 3-space gives rise to invariants of hyperbolic 3-manifolds called Bloch invariants introduced by Neumann and Yang. I will talk about the variation of the Bloch invariant and its relation to the PSL A-polynomial.

Posted April 27, 2005

Last modified January 10, 2022

Geometry and Topology Seminar Seminar website

4:00 pm – 5:00 pm Lockett 232
Tom Mark, Southeastern Louisiana University

Heegaard Floer invariants for fibered manifolds.

Heegaard Floer invariants, introduced by Ozsváth and Szabó several years ago, are proving to be valuable tools in low-dimensional topology: in particular the theory reproduces and extends many results obtained previously using Seiberg-Witten and/or Donaldson gauge theory, as well as yielding novel results. I will discuss an ongoing project, joint with Slaven Jabuka, whose goal is to understand the Ozsváth-Szabó invariants of Lefschetz fibered 4-manifolds. A natural place to start is to study the Heegaard Floer homology groups of 3-manifolds that fiber over the circle, particularly in terms of the expression of their monodromy as a product of Dehn twists. We give some preliminary results in this area and indicate some directions for future work.

Posted August 16, 2005

Last modified September 6, 2005

Geometry and Topology Seminar Seminar website

4:10 pm – 5:00 pm Lockett 285
Nathan Broaddus, Cornell University

Non-cyclic covers of knot complements

Posted September 8, 2005

Last modified September 13, 2005

Geometry and Topology Seminar Seminar website

4:10 pm – 5:00 pm Lockett 285
Brendan Owens, LSU

Floer homology of double branched covers

Posted September 9, 2005

Geometry and Topology Seminar Seminar website

4:10 pm – 5:00 pm Lockett 285
Brendan Owens, LSU

Floer homology of double branched covers, Part II

Posted September 13, 2005

Geometry and Topology Seminar Seminar website

4:10 pm – 5:00 pm Lockett 285
Brendan Owens, LSU

Floer homology of double branched covers, Part III

Posted September 19, 2005

Geometry and Topology Seminar Seminar website

4:10 pm – 5:00 pm Lockett 285
Scott Baldridge, Louisiana State University

Symplectic 4-manifolds with prescribed fundamental group

Posted October 4, 2005

Geometry and Topology Seminar Seminar website

4:10 pm – 5:00 pm Lockett 285
Scott Baldridge, Louisiana State University

Symplectic 4-manifolds with prescribed fundamental group, Part II

Posted October 12, 2005

Last modified October 15, 2005

Geometry and Topology Seminar Seminar website

4:10 pm – 5:00 pm Lockett 285
Tara Brendle, Department of Mathematics, LSU

The Birman-Craggs-Johnson homomorphism and the homology of the Johnson Kernel

Posted October 12, 2005

Last modified October 14, 2005

Geometry and Topology Seminar Seminar website

4:10 pm – 5:00 pm 285 Lockett
Tara Brendle, Department of Mathematics, LSU

The Birman-Craggs-Johnson homomorphism and the homology of the Johnson Kernel, Part II

Posted November 1, 2005

Geometry and Topology Seminar Seminar website

4:10 pm – 5:00 pm Lockett 285
Tara Brendle, Department of Mathematics, LSU

The Birman-Craggs-Johnson homomorphism and the homology of the Johnson Kernel, Part III

Posted November 14, 2005

Geometry and Topology Seminar Seminar website

4:10 pm – 5:00 pm Lockett 285
Khaled Qazaqzeh, LSU

Integral Bases for Certain TQFT-Modules of the Torus

Posted November 20, 2005

Geometry and Topology Seminar Seminar website

4:10 pm – 5:00 pm 285 Lockett
Atle Hahn, University of Bonn and LSU

Towards a path integral derivation of the Reshetikhin-Turaev invariants

Posted January 24, 2006

Last modified January 26, 2006

Geometry and Topology Seminar Seminar website

4:30 pm – 5:30 pm Lockett 284
Daniel C. Cohen, Mathematics Department, LSU

tba

Posted January 26, 2006

Last modified February 1, 2006

Geometry and Topology Seminar Seminar website

4:40 pm – 5:30 pm Lockett 284
Neal Stoltzfus, Mathematics Department, LSU

Root Posets and Temperley-Lieb Algebras

Posted February 8, 2006

Geometry and Topology Seminar Seminar website

4:40 pm – 5:30 pm Lockett 284
Ben McReynolds, UT Austin

Separable subgroups of mapping class groups

Posted February 23, 2006

Geometry and Topology Seminar Seminar website

4:40 pm – 5:30 pm Lockett 284
Matilde Lalin, University of British Columbia

Some aspects of the Multivariable Mahler Measure

Posted March 9, 2006

Geometry and Topology Seminar Seminar website

4:40 pm – 5:30 pm Lockett 284
Patrick Gilmer, Mathematics Department, LSU

Lollipop trees in TQFT

Posted April 19, 2006

Last modified March 2, 2021

Geometry and Topology Seminar Seminar website

1:30 pm – 2:30 pm Life Science A663
David Futer, Michigan State University

Geometry and combinatorics of arborescent link complements

Virtual Seminar together with U Iowa

Posted April 16, 2006

Geometry and Topology Seminar Seminar website

4:40 pm – 5:30 pm Lockett 284
Alissa Crans, University of Chicago/Loyola Marymount University

Self-Distributivity in Coalgebras

Abstract: Self-distributive binary operations have appeared extensively in knot theory in recent years, specifically in algebraic structures called `quandles.\' A quandle is a set equipped with two binary operations satisfying axioms that capture the essential properties of the operations of conjugation in a group. The self-distributive axioms of a quandle correspond to the third Reidemeister move in knot theory. Thus, quandles give a solution to the Yang-Baxter equation, which is an algebraic distillation of the third Reidemeister move. We formulate analogues of self-distributivity in the categories of coalgebras and Hopf algebras and use these to construct additional solutions to the Yang-Baxter equation.

Posted May 1, 2006

Last modified October 1, 2021

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Lockett 284
Abhijit Champanerkar, University of South Alabama

On the Mahler measure of Jones polynomials

We show that the Mahler measure of the Jones polynomial and of the colored Jones polynomials converges under twisting for any link. In terms of Mahler measure convergence, the Jones polynomial behaves like hyperbolic volume under Dehn surgery. We also show that after sufficiently many twists, the coefficient vector of the Jones polynomial and of any colored Jones polynomial decomposes into fixed blocks according to the number of strands twisted. We will also discuss recent results about links with cyclotomic Jones polynomials.

Posted September 5, 2006

Last modified September 19, 2006

Geometry and Topology Seminar Seminar website

4:40 pm – 5:30 pm Lockett 285
Patrick Gilmer, Mathematics Department, LSU

Surgery of type-p and quantum invariants of 3-manifolds

Posted September 6, 2006

Last modified September 19, 2006

Geometry and Topology Seminar Seminar website

4:40 pm – 5:30 pm Lockett 284
Neal Stoltzfus, Mathematics Department, LSU

Dessins in Knot Theory

Posted September 19, 2006

Last modified January 10, 2022

Geometry and Topology Seminar Seminar website

4:40 pm – 5:30 pm Lockett 284
David Cimasoni, UC Berkeley

Generalized Seifert surfaces and signatures of colored links

The Seifert surface is a well-known and very useful tool in link theory. For instance, it permits to study the Alexander invariants, the Conway polynomial, and the signature of an oriented link. In this talk, we shall introduce 'generalized Seifert surfaces' for colored links. They provide a geometric interpretation of the multivariable Alexander invariants and of the Conway potential function. They also make it possible to define (and compute easily) a multivariable signature that generalizes the Levine-Tristram signature. This multivariable signature turns out to be a slight generalization of invariants introduced by P. Gilmer and L. Smolinsky.

Posted October 4, 2006

Geometry and Topology Seminar Seminar website

4:40 pm – 5:30 pm Lockett 284
Neal Stoltzfus, Mathematics Department, LSU

Skein Modules of Cylinders and Quantum Cluster Algebras

Posted October 4, 2006

Last modified October 10, 2006

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Lockett 284
Indira Lara Chatterji, Ohio State University

A characterization of hyperbolicity.

Posted October 17, 2006

Geometry and Topology Seminar Seminar website

4:40 pm – 5:30 pm Lockett 284
Brendan Owens, LSU

Knot surgeries and negative definite four manifolds

Posted October 25, 2006

Geometry and Topology Seminar Seminar website

4:40 pm – 5:30 pm Lockett 284
Matilde Lalin, University of British Columbia

Functional equations for the Mahler measure of genus 1 curves

Posted November 3, 2006

Geometry and Topology Seminar Seminar website

11:00 am – 12:00 pm Johnston 338
Ronald Fintushel, Michigan State University

Surgery on Nullhomologous Tori

Virtual Seminar together with Rice University

Posted November 3, 2006

Last modified January 10, 2022

Geometry and Topology Seminar Seminar website

4:40 pm – 5:30 pm Lockett 284
Matthew Hedden, Michigan State University

The meaning and comparison of smooth concordance invariants

In the past three years, several new invariants of smooth knot concordance have been discovered. This lecture will focus on two of these invariants, denoted $\tau(K)$ and $s(K)$, respectively. Here $K$ denotes a knot in the three-sphere. The former invariant was discovered by Ozsváth and Szabó and independently by Rasmussen and is defined using the Floer homology theory for knots introduced by the aforementioned authors. $s(K)$ was introduced by Rasmussen and is defined in the context of Khovanov knot homology. The invariants share several formal properties and agree for many knots. In particular, each invariant is a homomorphism from the smooth knot concordance group to the integers, and each bounds the smooth four-genus, $g_4(K)$. Moreover, each invariant can be used to determine the smooth four-genera of torus knots and provide new proofs of Milnor's famous conjecture on the four-genera and unknotting numbers of these knots. It was conjectured by Rasmussen that $2\tau$ and $s$ agree for all knots. If confirmed, this conjecture would point to a surprising connection between the analytically defined Ozsváth-Szabó homology theory and the combinatorially defined Khovanov homology. Moreover, it would seem to indicate a relationship between the gauge theory of three and four-manifolds and the quantum framework underlying the Jones polynomial.

This lecture will explore Rasmussen's conjecture by discussing evidence for its validity and families of knots for which the conjecture holds. In this pursuit, it will be appropriate to briefly comment on the geometry contained by the $\tau$ invariant—in particular I'll discuss a theorem which indicates that $tau$ can be used to detect when a knot arises as the intersection of a complex curve in $C^2$ with the three-sphere. This connection partially arises with the $s$ invariant. The main purpose, however, will be to present the first counterexamples to Rasmussen's conjecture, discovered last year by myself and Philip Ording. The examples come from the Whitehead double construction. I will try to say some words about how Rasmussen's conjecture, though false, could be interpreted in the context of a larger conjecture connecting Floer homology to Khovanov homology, also due to Rasmussen.

Posted January 10, 2007

Last modified January 21, 2007

Geometry and Topology Seminar Seminar website

4:40 pm – 5:30 pm Lockett 276
Emille K. Davie, University of Georgia

Characterizing Right-Veering Surface Diffeomorphisms Via the Burau Representation

Posted March 7, 2007

Geometry and Topology Seminar Seminar website

4:40 pm – 5:00 pm Lockett 276
Kathy Zhong, Cal State Sacramento

Calculate Kauffman Polynomials of some Knots Using Kauffman Skeins

Posted March 21, 2007

Last modified March 28, 2007

Geometry and Topology Seminar Seminar website

11:00 am Lockett 381(Originally scheduled for Wednesday, March 28, 2007, 4:40 pm)

Stephen Bigelow, UC Santa Barbara

Representations of Planar Algebras

Time/Date Changed

Posted August 27, 2007

Last modified August 31, 2007

Geometry and Topology Seminar Seminar website

5:10 pm – 6:00 pm Lockett 276
Adam Lowrance, Department of Mathematics, Vassar College

On Knot Floer Width and Turaev Genus

Posted September 5, 2007

Geometry and Topology Seminar Seminar website

4:30 pm – 5:30 pm Lockett 276
Steve Wallace, LSU

Surgery untying of knots

Posted September 7, 2007

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Biological Sciences Annex Building - A663
Moshe Cohen, Department of Mathematics, Bar-Ilan University, Israel

Introductory remarks on Khovanov homology

This is a virtual topology seminar together with U Iowa

Posted September 7, 2007

Geometry and Topology Seminar Seminar website

4:40 pm – 5:30 pm Lockett 276
Hee Jung Kim, Department of Mathematics, LSU

Topological triviality of smoothly knotted surfaces in 4-manifolds

Posted September 13, 2007

Last modified September 18, 2007

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Biological Sciences Annex Building - A663
Neal Stoltzfus, Mathematics Department, LSU

Quasi-Trees and Khovanov homology

Virtual Seminar together with U Iowa

Posted September 19, 2007

Geometry and Topology Seminar Seminar website

4:40 pm – 5:30 pm Lockett 276
Scott Baldridge, Louisiana State University

A symplectically aspherical manifold with b_1=1

Posted September 24, 2007

Last modified October 1, 2021

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Biological Sciences Annex Building - A663
Adam Lowrance, Department of Mathematics, Vassar College

On knot Floer width and Turaev genus, Part I

On knot Floer width and Turaev genus

Posted September 24, 2007

Geometry and Topology Seminar Seminar website

4:40 pm – 5:30 pm Lockett 276
Scott Baldridge, Louisiana State University

A symplectically aspherical manifold with b_1=1, Part II

Posted September 24, 2007

Last modified October 1, 2021

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Biological Sciences Annex Building - A663
Adam Lowrance, Department of Mathematics, Vassar College

On knot Floer width and Turaev genus, Part II

On knot Floer width and Turaev genus

Posted October 1, 2007

Last modified October 1, 2021

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Biological Sciences Annex Building - A663
Moshe Cohen, Department of Mathematics, Bar-Ilan University, Israel

On a result of Ozsvath and Manolescu

Virtual Seminar together with U Iowa

Posted October 2, 2007

Last modified January 10, 2022

Geometry and Topology Seminar Seminar website

4:40 pm – 5:30 pm Lockett 276
Neal Stoltzfus, Mathematics Department, LSU

The Bollobás-Riordan-Tutte polynomial as a tri-graded Poincaré-polynomial (due to N. Forman)

Posted October 3, 2007

Last modified October 11, 2007

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Biological Sciences Annex Building - A663
John Etnyre, Georgia Institute of Technology

A geometric reason for the non-sharpness of Bennequin's inequality for some fibered knots

Virtual Seminar together with U Iowa

Posted October 22, 2007

Last modified January 10, 2022

Geometry and Topology Seminar Seminar website

4:40 pm – 5:30 pm Lockett 276
Neal Stoltzfus, Mathematics Department, LSU

The Bollobás-Riordan-Tutte polynomial as a tri-graded Poincaré-polynomial (due to N. Forman), Part II

Posted October 22, 2007

Last modified March 2, 2021

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Biological Sciences Annex Building - A663
Charles Frohman, University of Iowa

On Bar-Natan's skein module

Virtual Seminar together with U Iowa (talk this week is from Iowa)

Posted October 24, 2007

Geometry and Topology Seminar Seminar website

4:40 pm – 5:30 pm Lockett 276
Ambar Sengupta, Mathematics Department, LSU

Gaussian Matrix Integrals

Abstract: The talk of the same title given in the probability seminar concluded with a definition of a who a topologist is. In this talk we will strive to define a probabilist. Along the way we shall examine the representation of Gaussian integrals of matrix-trace functions in terms of sums over surfaces of varying genus. This is an illustration of a broader phenomenon of integrals arising from physical theories having topological interpretations.

Posted October 25, 2007

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Biological Sciences Annex Building - A663Jeff Boerner (U Iowa): On the Asaeda-Przytycki-Sikora homology

Virtual Seminar together with U Iowa (talk this week is from Iowa)

Posted October 25, 2007

Last modified October 1, 2021

Geometry and Topology Seminar Seminar website

12:40 pm – 1:30 pm tbaJunior Topology Seminar

This is a reading seminar.

Posted November 4, 2007

Last modified March 2, 2021

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Biological Sciences Annex Building - A663
Heather Russell, University of Iowa

Embedded Khovanov homology of $S^1\times D^2$ and the homology of the $(n,n)$-Springer Fiber

Virtual Seminar together with U Iowa (talk this week is from Iowa)

Posted November 6, 2007

Last modified September 17, 2021

Geometry and Topology Seminar Seminar website

12:40 pm – 1:30 pm Lockett 119Junior Topology Seminar

Reading seminar.

Posted November 6, 2007

Geometry and Topology Seminar Seminar website

4:40 pm – 5:30 pm Lockett 276
Patrick Gilmer, Mathematics Department, LSU

Congruence and quantum invariants

Posted November 6, 2007

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Biological Sciences Annex Building - A663
Steve Wallace, LSU

Surgery equivalence invariants of colored knots

Virtual Seminar together with U Iowa

Posted November 6, 2007

Last modified September 17, 2021

Geometry and Topology Seminar Seminar website

12:40 pm – 1:30 pm Lockett 119Junior Topology Seminar

Reading Seminar.

Posted November 6, 2007

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pmNO VIRTUAL SEMINAR (THANKSGIVING)

Posted November 6, 2007

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Biological Sciences Annex Building - A663
Adam Lowrance, Department of Mathematics, Vassar College

On a paper by Ozsvath, Rasmussen and Szabo on the odd Khovanov homology

Virtual Seminar together with U Iowa

Posted January 21, 2008

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Bio. Sciences Annex, A663
Charles Frohman, University of Iowa

An introduction to Frobenius extensions and TQFT over rings

Virtual Seminar together with UIowa

Posted January 22, 2008

Last modified January 28, 2008

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Bio. Sciences Annex, A663
Charles Frohman, University of Iowa

sl_3 Topological Quantum Field Theory after Khovanov

Virtual Seminar together with UIowa

Posted January 24, 2008

Last modified February 14, 2008

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Bio. Sciences Annex, A663A diagramless link homology

Adam McDougall (Virtual Seminar together with UIowa)

Posted January 24, 2008

Last modified February 15, 2008

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Bio. Sciences Annex, A663
Hee Jung Kim, Department of Mathematics, LSU

Embeddings of Surfaces in 4-manifolds

(Virtual Seminar together with UIowa)

Posted January 30, 2008

Last modified February 25, 2008

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Lockett 276
Gregor Masbaum, University Paris 7

TQFT and the Nielsen-Thurston classification of surface homeomorphisms

Posted February 20, 2008

Last modified February 27, 2008

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Bio. Sciences Annex, A663
Alissa Crans, University of Chicago/Loyola Marymount University

2-groups: Categorified groups

(Virtual Seminar together with UIowa; the talk is broadcasted from Iowa)

Posted February 20, 2008

Last modified March 20, 2008

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Bio. Sciences Annex, A663
Hee Jung Kim, Department of Mathematics, LSU

Knotting Surfaces in 4-manifolds, Part II

(Virtual Seminar together with UIowa)

Posted January 24, 2008

Last modified March 31, 2008

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Lockett 276
Charles Livingston, Indiana University

Twisted Alexander polynomials, metabelian representations, and the knot slicing problem

Posted February 20, 2008

Last modified April 1, 2008

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Bio. Sciences Annex, A663Models for evaluating the Homfly polynomial

Anna Meyers (UIowa) (Virtual Seminar together with UIowa)

Posted April 7, 2008

Last modified April 8, 2008

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Bio. Sciences Annex, A663
Cody Armond, Department of Mathematics, LSU

On the Huynh-Le Quantum Determinant for the Colored Jones Polynomial

(Virtual Seminar together with UIowa)

Posted February 18, 2008

Last modified April 4, 2008

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Lockett 276
Oleg Viro, SUNY Stony Brook

Twisted acyclicity of circle and link signatures

Posted February 27, 2008

Last modified April 21, 2008

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Lockett 276
Joan Birman, Barnard College, Columbia University
Recipient of the Chauvenet Prize

Twisted torus knots and Lorenz knots

Posted August 22, 2008

Geometry and Topology Seminar Seminar website

3:40 pm X-labTest run for Virtual Seminar

Participating Universities: LSU, U Iowa, GWU, U Miami

Posted September 10, 2008

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm X-lab: Lockett 233
Heather Russell, USC

Virtual Seminar

Live from Iowa City

Posted September 23, 2008

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm X-lab: Lockett 233
Hee Jung Kim, Department of Mathematics, LSU

tba

Virtual Seminar together with UIowa and the University of Miami

Posted October 1, 2008

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm X-lab: Lockett 233Virtual Seminar with UIowa/UMiami

Adam McDougall (U Iowa): On the diagramless link homology

Talk is broadcasted from Iowa

Posted October 1, 2008

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm X-lab: Lockett 233Virtual Seminar with UIowa/UMiami

Ken Baker (University of Miami)

(talk is broadcasted from Miami)

Posted November 17, 2008

Last modified October 1, 2021

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm X-lab: Lockett 233
Lawrence Roberts, Michigan State University

Knot Floer homology for some fibered knots

I will talk about a computing the knot Floer homology of a class of fibered knots in rational homology spheres, for which the computation is particularly simple.

Joint virtual seminar with UIowa, Rice, UMiami, Boise State, GWU .

Posted December 2, 2008

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm X-lab: Lockett 233
Leah Childers, LSU

Birman-Craggs-Johnson Homomorphism of the Torelli Group

Virtual Seminar together with UIowa, Rice University, UMiami, Boise State University, George Washington University

Posted February 10, 2009

Last modified October 1, 2021

Geometry and Topology Seminar Seminar website

3:40 pm – 4:40 pm Lockett 233
Chad Giusti, University of Oregon

Virtual Seminar: Unstable Vassiliev Theory

This week's AccessGrid virtual seminar will be presented locally by Chad Giusti.

Posted February 17, 2009

Last modified October 1, 2021

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm 233 Lockett
Scott Baldridge, Louisiana State University

Virtual Seminar: Cube knots and knot Floer homology from cube diagrams

This week's AccessGrid virtual seminar will be presented locally by Scott Baldridge.

Posted March 11, 2010

Last modified October 1, 2021

Geometry and Topology Seminar Seminar website

3:40 pm – 5:00 pm Lockett 276
Gregor Masbaum, University Paris 7

The Arf-invariant formula for graphs on surfaces

Kasteleyn showed how to count dimer coverings (= perfect matchings) on a planar bipartite graph by evaluating the determinant of a certain matrix. The method works for non-bipartite graphs as well, upon replacing the determinant with a Pfaffian. If the graph is not planar, but embedded in a surface of genus g, Kasteleyn stated and Gallucio-Loebl proved a formula expressing the number of dimer coverings as a linear combination of 4^g Pfaffians. The main aim of the talk is to explain a new proof of this formula based on the theory of Arf invariants of quadratic forms on the mod 2 homology of the surface. I will then discuss the question of whether the minimal number of Pfaffians needed to count dimer coverings is always a power of 4. If time remains, I will explain a recent result of Loebl and myself which gives an affirmative answer to the analogous question for the Ising model on a graph.

Posted June 28, 2010

Geometry and Topology Seminar Seminar website

10:30 am – 11:30 am Lockett 233
Paul Kirk, Indiana University

Untwisted Whitehead doubles of $(2, 2^k-1)$ torus knots are linearly independent in the smooth knot concordance group.

Abstract: We revisit an argument of Furuta, using SO(3) instanton moduli spaces on 4-manifolds with boundary and estimates of Chern-Simons invariants of flat SO(3) connections on 3-manifolds, to prove that the infinite family of untwisted positive clasped Whitehead doubles of the $(2, 2^k-1)$ torus knots are linearly independent in the smooth knot concordance group. (joint work with Matt Hedden)

Posted March 18, 2011

Last modified March 29, 2011

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm 276 Lockett
Kate Kearney, Indiana University

An Obstruction to Knots Bounding Moebius Bands

Posted August 18, 2011

Last modified March 3, 2022

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Locket 233
Charles Frohman, University of Iowa

Virtual Seminar: Projective Representations of the Mapping Class group of a surface with boundary coming from TQFT

(Joint with Joanna Kania-Bartoszynska and Mike Fitzpatrick)

For each odd prime p, and primitive 2pth root unity, there is a projective representation of the mapping class group of a torus of dimension 2, that comes from the projective action of the mapping class group of a one punctured torus ( aka the modular group) on a portion of the state space assigned to a once punctured torus. I will prove up to conjugacy, this family extends to a continuous family of representations of the modular group defined on the unit circle. This family includes a twisted version of the canonical representation of the modular group. This means that the dilation coefficient of pseudo-anosov mapping classes can be computed as a limit of quantum invariants of mapping tori. It also means that the hyperbolic volume should also be computable, though the connection is less direct.

Posted August 29, 2011

Last modified October 1, 2021

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Locket 233
Greg Muller, Department of Mathematics, LSU

Virtual Seminar: Skein algebras of marked surfaces

Given a surface with boundary and a collection of marked points on the boundary, one may consider all curves in the surface which end at the marked points. One may define the Kauffman skein algebra (at q=1) generated by these curves; this generalizes the `unmarked' definition where only loops are allowed. Generalizing results of Bullock, Barrett and Przytycki-Sikora, this algebra can be realized as the algebra of functions on a space of (twisted) SL_2(C) local systems with extra data at the marked points. Additionally, new phenomena arise in the marked case which do not generalize any unmarked results. When there are enough marked points for the surface to admit a triangulation, then each triangulation gives an embedding of the skein algebra into a ring of Laurent polynomials. Through these embeddings, it can be shown that the skein algebra coincides with the `upper cluster algebra' of the marked surface, an algebra with significance in combinatorics, Lie theory and Teichmüller theory. Part of this work is joint with Peter Samuelson.

Posted September 9, 2011

Last modified March 3, 2022

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Lockett 233
Shin Satoh, Kobe University

Virtual Seminar: Quandle cocycle invariants of roll-spun knots

We have two fundamental families in 2-knot theory; one is a ribbon 2-knot and the other is a deform-spun knot. Since any ribbon 2-knot is represented by a diagram with no triple point, the quandle cocycle invariant is always trivial. As special families of deform-spun knots, we have twist-spun knots and roll-spun knots. The invariant of a twist-spun knot have been studied in many papers. The aim of this talk is to explain how to calculate the quandle cocycle invariant of a roll-spun knot and give several properties.

Posted August 29, 2011

Last modified September 26, 2011

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Locket 233
Trenton Schirmer, Department of Mathematics, University of Iowa

Virtual Seminar: The degeneration ratio of tunnel number under connect sum

The tunnel number $t(L)$ of a link $L$ in $S^3$ is the minimal number of arcs $\{t_1, .... , t_n\}$ that can be embedded in the closure of $S^3-N(L)$ so that S^3-N(L \cup t_1 \cup ... \cup t_n is a handlebody. When $L$ is a knot $t(K)+1$ is just the Heegaard genus of its complement. The ``degeneration ratio'' of a connect sum $L =L_1$ # $L_2$ is defined as t L/(t(L_1)+t(L_2)). We give some new examples of links for which the degeneration ratio becomes low.

Posted September 11, 2011

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Lockett 233
Eamonn Tweedy, Department of Mathematics, Rice University

Virtual Seminar: A filtration on the Heegaard Floer chain complex of a double branched cover

Abstract: Seidel and Smith defined their fixed-point symplectic Khovanov cohomology theory for links in the 3-sphere. For the case of a knot K, they described how to define a particular filtration on their complex. Via an observation of Manolescu, this filtration induces a spectral sequence from the Seidel-Smith theory to the Heegaard Floer hat theory for the double cover of the 3-sphere branched along K. This spectral sequence is itself a knot invariant, and has some nice properties. We also discuss how the construction leads to a family of rational-valued knot invariants.

Posted September 8, 2011

Last modified September 22, 2011

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Lockett 233
Effie Kalfagianni, Michigan State University

Virtual Seminar: Polyhedral decompositions, essential surfaces and colored Jones polynomials.

Abstract: We generalize the checkerboard decompositions of alternating knots and links: For A- or B-adequate diagrams, we show that the checkerboard knot surfaces are incompressible, and we obtain an ideal polyhedral decomposition of their complement. In the talk I will describe these decompositions and give some of the applications, which include fibering knot criteria and relations between hyperbolic volume and colored Jones polynomials. The talk will be based on joint work with Dave Futer (Temple) and Jessica Purcell (BYU).

Posted September 23, 2011

Last modified October 25, 2011

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Lockett 233
Adam Lowrance, Department of Mathematics, Vassar College

Virtual Seminar: "A categorification of the Tutte polynomial"

Abstract: The Tutte polynomial is a graph and matroid polynomial which has a close relationship with the Jones polynomial. We construct a categorification of the Tutte polynomial for graphs and matroids. Our construction is modeled after the construction of odd Khovanov homology, which is a categorification of the Jones polynomial developed by Ozsvath, Rasmussen, and Szabo. Many properties of the Tutte polynomial lift to expected properties of our categorification. The deletion-contraction relation of the Tutte polynomial becomes an exact triangle in the categorification, and the formula for the Tutte polynomial of the dual matroid has an analog for our categorification. We will also present examples and an application that leads to an invariant of (mutation classes of) alternating links.

Posted November 4, 2011

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm In R^4
Scott Baldridge, Louisiana State University

Knotted embedded tori in R^4

Abstract: One of the barriers to studying knotted surfaces in R^4 has been that there are few ways to represent them that lead to powerful yet easy-to-compute invariants. In this talk we will describe a new way to represent 2-dimensional knotted tori in R^4 using 4n half-integer valued points in the cube [0,n]^4. We will illustrate why the construction represents knotted spun tori and discuss the ramifications of the representation to topics such as Heegaard Floer Homology and Contact Homology.

Posted September 23, 2011

Last modified November 3, 2011

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Lockett 233
Heather Russell, USC

Virtual Seminar: Springer varieties and spider webs

Abstract: Springer varieties are certain flag varieties classically studied because their cohomology rings are Weyl group representations. Khovanov, Stroppel-Webster, Cautis-Kamnitzer, Seidel-Smith and others have studied the connections of Springer varieties to knot theory. In past work we built on ideas of Khovanov and Stroppel-Webster to give a diagrammatic framework enabling the study of Springer representations as well as the topology of certain Springer varieties via sl_2 webs. We will discuss recent work extending some of these results to other classes of Springer varieties using sl_3 webs.

Posted October 31, 2011

Last modified November 4, 2011

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Lockett 233
Ben McCarty, LSU

Virtual Seminar: "On the rotation class of knotted Legendrian tori in R^5"

Abstract Legendrian knots in R^3 have been studied extensively in recent years. However, much less is known about Legendrian knots in higher dimensions. We present Lagrangian hypercube diagrams as a convenient tool to study knotted Legendrian tori in R^5 with the standard contact structure. In particular, we describe an easy way to compute a Legendrian invariant, the rotation class, from a Lagrangian hypercube diagram, and discuss applications to contact homology. (Joint work with S. Baldridge)

Posted December 21, 2011

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Lockett 233
Benjamin Himpel, Centre for Quantum Geometry of Moduli Spaces, Aarhus, Denmark

tba

Posted January 11, 2012

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Lockett 233
Yeonhee Jang, Hiroshima University

Virtual Seminar: Bridge presentations of links

Abstract: This talk will be devoted to introduce the speaker\'s works related to bridge presentations of links. In the first half of this talk, we introduce results on the classification or characterization of certain 3-bridge links and their 3-bridge presentations. In the last half, we introduce results on Cappell-Shaneson\'s question which asks whether the bridge numbers of links are equal to the minimal numbers of meridian generators of link groups.

Posted January 28, 2012

Last modified January 31, 2012

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Lockett 233
Cody Armond, Department of Mathematics, LSU

"The colored Jones polynomial and adequate links"

Posted February 6, 2012

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Lockett 233
Dan Rutherford, University of Arkansas

A combinatorial Legendrian knot DGA from generating families

Abstract: This is joint work with Brad Henry. A generating family for a Legendrian knot L in standard contact R^3 is a family of functions f_x whose critical values coincide with the front projection of L. Pushkar introduced combinatorial analogs of generating families which have become known as Morse complex sequences. In this talk, I will describe how to associate a differential graded algebra (DGA) to a Legendrian knot with chosen Morse complex sequence. In addition, I will discuss the geometric motivation from generating families and the relationship with the Chekanov-Eliashberg invariant.

Posted February 26, 2012

Last modified March 14, 2012

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Lockett 233
John Etnyre, Georgia Institute of Technology

Virtual Seminar: Open books decompositions and the geometry of contact structures

Abstract: Giroux's correspondence between open books decompositions and contact structures on 3-manifolds has been key to many advances in contact geometry and its application to topology. In this talk I will discuss several recent advances that describe how properties of a contact structure, such as tightness and fillability, are reflected in its associated open book decompositions and vice vera. In addition I will discuss how some operations on open books decompositions, such as cabling a binding component, affect the associated contact structure.

Posted March 7, 2012

Last modified March 14, 2012

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Lockett 233
Matt Clay, Allegheny College

Virtual Seminar: The geometry of right-angled Artin subgroups of mapping class groups

Abstract: We describe sufficient conditions which guarantee that a finite set of mapping classes generate a right-angled Artin group quasi-isometrically embedded in the mapping class group. This is joint work with Chris Leininger and Johanna Mangahas.

Posted April 9, 2012

Last modified October 1, 2021

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Lockett 233
Rafal Komendarczyk, Tulane University

Virtual Seminar: "Towards the $\kappa$-invariant conjecture"

A parametrization of an $n$-component link in $R^3$, produces a natural evaluation map from the $n$-torus to the configuration space of $n$ distinct points in $R^3$. Denote by $\kappa$ the map from homotopy links to the set of homotopy classes of evaluation maps. A natural conjecture arises, that $\kappa$ classifies homotopy links. Koschorke first proved that $\kappa$ has this property for homotopy Brunnian links. In this talk, I will show how to recast Koschorke's correspondence in the language of torus homotopy groups, which reveals an interesting algebraic structure. Further, time permitting, I will describe progress towards extending the result beyond the Brunnian case. This is joint work with Frederick Cohen at Rochester and Clayton Shonkwiler at UGA.

Posted April 3, 2012

Last modified October 1, 2021

Geometry and Topology Seminar Seminar website

3:40 pm – 4:30 pm Lockett 233(Originally scheduled for Wednesday, April 4, 2012)

Dennis Roseman

Small Lattice Surfaces in Four Dimensions

A lattice point is a point with integer coordinates. The standard p x q x r x s lattice box is all lattice points in R^4 within [0,p-1] x [0,q-1] x [0, r-1] x [0, s-1]. A lattice square in R^4 is a unit square whose vertices are lattice points. A lattice surface or lattice surface link is a finite union of lattice squares which is topologically is a closed two-dimensional manifold (perhaps not connected, perhaps not orientable).

We focus on the question: which surface link types can be represented as lattice surfaces in a given small lattice box? We show that any orientable surface link in a 3x3x3x3 lattice box is a pseudo-ribbon link,
and discuss a new surface link invariant that can detect non-pseudo-ribbon links.
We give a table of surface links that lie in a 3x3x3x2 lattice box and develop notations, terminology, mathematical strategies and visualization tools for investigating these and surface links in slightly larger boxes.

Posted August 26, 2012

Last modified September 3, 2012

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm
Anastasiia Tsvietkova, LSU

Virtual Seminar: "Hyperbolic structures from link diagrams"

W. Thurston demonstrated that every link in $S^3$ is a torus link, a satellite link or a hyperbolic link and these three categories are mutually exclusive. It also follows from work of W. Menasco that an alternating link represented by a prime diagram is either hyperbolic or a $(2,n)$--torus link. The talk will introduce an alternative method for computing the hyperbolic structure of the complement of a hyperbolic link. It allows computing the structure directly from the link diagram. Some of its consequences will be discussed, including a surprising rigidity property of certain tangles, and the formulas that allow one to calculate the exact hyperbolic volume, as well as complex volume, of hyperbolic 2--bridge links. This is joint work with M. Thistlethwaite.

Posted August 21, 2012

Last modified September 3, 2012

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm Lockett 233(Originally scheduled for Sunday, September 2, 2012)

Shea Vela-Vick, Louisiana State University

Virtual Seminar: "The equivalence of transverse link invariants in knot Floer homology"

Abstract: The Heegaard Floer package provides a robust tool for studying contact 3-manifolds and their subspaces. Within the sphere of Heegaard Floer homology, several invariants of Legendrian and transverse knots have been defined. The first such invariant, constructed by Ozsvath, Szabo and Thurston, was defined combinatorially using grid diagrams. The second invariant was obtained by geometric means using open book decompositions by Lisca, Ozsvath, Stipsicz and Szabo. We show that these two previously defined invariant agree. Along the way, we define a third, equivalent Legendrian/transverse invariant which arises naturally when studying transverse knots which are braided with respect to an open book decomposition.

Posted September 1, 2012

Last modified September 14, 2012

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm
Ken Baker, University of Miami

Virtual Seminar: "Annular twists and Bridge numbers of knots"

Abstract: Performing +1/n and -1/n Dehn surgery on the boundary components of an annulus A in a 3-manifold M provides a homeomorphism of M similar to a Dehn twist. If a knot intersects the interior of A in an essential manner, then this twisting produces an infinite family of knots. In joint work with Gordon and Luecke, we show (under certain hypotheses) that if the bridge numbers of this family with respect to a given Heegaard surface of M are bounded, then the annulus may be isotoped to embed in the Heegaard surface. With this we construct genus 2 manifolds that each contain a family of knots with longitudinal surgeries to S^3 and unbounded genus 2 bridge number. In contrast, our earlier work gives an a priori upper bound on the bridge number of a knot in a genus g manifold with a non-longitudinal S^3 surgery.

Posted September 14, 2012

Last modified March 2, 2021

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm
Moshe Cohen, Department of Mathematics, Bar-Ilan University, Israel

Virtual Seminar: "Kauffman's clock lattice as a graph of perfect matchings: a formula for its height"

Kauffman gives a state sum formula for the Alexander polynomial of a knot using states in a lattice that are connected by his clock moves. We show that this lattice is more familiarly the graph of perfect matchings of a bipartite graph obtained from the knot diagram by overlaying the two dual Tait graphs of the knot diagram. Using a partition of the vertices of the bipartite graph, we give a direct computation for the height of Kauffman\'s clock lattice obtained from a knot diagram with two adjacent regions starred and without crossing information specified. We prove structural properties of the bipartite graph in general and mention applications to Chebyshev or harmonic knots (obtaining the popular grid graph) and to discrete Morse functions.

Posted September 14, 2012

Last modified October 16, 2012

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm
Adam Lowrance, Department of Mathematics, Vassar College

Virtual Seminar: "Khovanov homology and oriented ribbon graphs"

Abstract: We define Khovanov homology of ribbon graphs and discuss how it ties together the Khovanov homology of both classical and virtual links. The spanning tree complex of Khovanov homology generalizes in the ribbon graph setting to a quasi-tree complex, which shows a relation between the Khovanov homology (of both classical and virtual links) and Turaev genus. We also discuss ribbon graph Reidemeister moves and discuss how they may be used to give distinct virtual links with isomorphic Khovanov homology.

Posted September 19, 2012

Last modified January 10, 2022

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm Lockett 233
Bulent Tosun, CIRGET (Montreal)

Virtual Seminar: "Cabling and Legendrian simplicity"

This talk will be about Legendrian and transverse knots in cabled knot types in standard contact three sphere and their classifications up to contact isotopy. We will be able to give structural theorems that ensure when cables a of a Legendrian simple knot type are Legendrian simple. We will then give complete classification in case of cables of positive torus knots. These results exhibits many new phenomena about structural understanding of Legendrian and transverse knot theory. The key ingredient of the proofs will be understanding of certain quantities associated to contact solid tori representing positive torus knots in standard contact three sphere. Part of the results are joint work with John Etnyre and Douglas LaFountain.

Posted December 3, 2012

Last modified January 22, 2013

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm Lockett 233
Doug LaFountain, Western Illinois University

Virtual Seminar: "Links and doubling branched surfaces"

Abstract: We consider oriented links in the 3-sphere which are braided positively with respect to two different braid fibrations, and hence represent two different braid conjugacy classes for the link type. Following work of Morton, we show that these two braid fibrations may be assumed to be mutually braided with respect to each other; furthermore, after isotopies of the link which are non-increasing on braid index, the link projects onto one of a family of well-defined branched surfaces in the complement of the braid axes. Time permitting we discuss potential applications; this is joint work with Bill Menasco and Hiroshi Matsuda.

Posted December 3, 2012

Last modified January 25, 2013

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm Lockett 233
Steven Sivek, Harvard

Virtual Seminar: "Donaldson invariants of symplectic manifolds"

Abstract: Donaldson proved in the late 80s that his polynomial invariants of smooth 4-manifolds are nonzero for Kaehler surfaces, and this was only recently extended to symplectic manifolds by Kronheimer and Mrowka. In this talk, we will give a new proof that symplectic 4-manifolds have nonzero Donaldson invariants. Our proof will rely on Kronheimer and Mrowka's structure theorem for manifolds of "simple type" together with some known cases of Witten's conjecture relating the Donaldson and Seiberg-Witten invariants.

Posted December 3, 2012

Last modified January 28, 2013

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm Lockett 233
Tye Lidman, UT Austin

Virtual Seminar: "Left-orderability and Floer homology"

Abstract: We will study the seemingly unnatural question of when the fundamental group of a three-manifold can be given a left-invariant order. This is related to the existence of taut foliations on the manifold as well as the structure of its Heegaard Floer homology groups.

Posted January 22, 2013

Last modified March 2, 2021

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm Lockett 233
Anne Thomas, University of Sydney

Virtual Seminar: "Infinite reduced words and the Tits boundary of a Coxeter group"

Let (W,S) be a Coxeter system with W infinite. An infinite reduced word of W is an infinite sequence of elements of S such that each finite subsequence is a reduced word. We prove that the limit weak order on the blocks of infinite reduced words of W is encoded by the topology of the Tits boundary of the Davis complex of W. We consider many special cases, including W word hyperbolic and X with isolated flats. This is joint work with Thomas Lam.

Posted January 22, 2013

Last modified March 5, 2013

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm Lockett 233
Clayton Shonkwiler, University of Georgia

Virtual Seminar: "The geometry and topology of random polygons"

Abstract: What is the expected shape of a random closed curve in space? For example, what is the expected radius of gyration or expected total curvature? What is the likelihood that the curve is knotted? As a first step, what are the corresponding answers when I restrict to closed n-gons in space? Aside from purely mathematical interest, such questions are natural in the context of statistical physics since n-gons in space are simple models for ring polymers with n monomers in solution. When we restrict attention to equilateral n-gons such questions become quite challenging, even numerically: current algorithms for sampling equilateral n-gons use a Markov process which "folds" polygons while preserving closure and edgelengths and are only expected to converge in O(n^3) time. The main point of this talk is that a much better sampling algorithm and indeed much better answers are available if we widen our view to the space of n-gons in three dimensional space of fixed total length (rather than with fixed edgelengths). I will describe a natural probability measure on n-gons of total length 2 which is pushed forward from the standard measure on the Stiefel manifold of 2-frames in complex n-space using methods from algebraic geometry. We can directly sample the Stiefel manifold in O(n) time, which gives us a fast, direct sampling algorithm for closed n-gons via the pushforward map. We can also explicitly compute the expected radius of gyration and expected total curvature and even recover some topological information. This talk describes joint work primarily with Jason Cantarella (University of Georgia) and Tetsuo Deguchi (Ochanomizu University).

Posted January 29, 2013

Last modified March 21, 2013

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm Lockett 233
Margaret Doig, Indiana University

Virtual Seminar: "Obstructing finite surgery"

Abstract: We will discuss using Heegaard Floer invariants towards a partial classification of Dehn surgery on knots $K$ in $S^3$ which give elliptic manifolds $Y$ other than the lens spaces, sometimes called \emph{finite, non-cyclic surgeries}. Recent results using these techniques include: if $p<10$ and $K$ is hyperbolic, there are no such $Y$; for any fixed $p$, there are at most finitely many $Y$ given by any $p/q$-surgery; if $p\leq4$, there is a unique $p/q$-surgery (up to orientation) that gives an elliptic manifold, other than a lens space (for $g=1$, this was previously proved by Ghiggini: the surgery is $+1$ on the right-handed trefoil, and the manifold is the Poincar\'e homology sphere).

Posted April 8, 2013

Last modified March 2, 2021

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm Lockett 233
Sam Nelson, Claremont McKenna College

Virtual Seminar: "Rack and Birack Module Invariants"

In 2002, Andruskiewitsch and Graña defined an algebra $R[x]$ associated to a rack $X$ and used it to generalize rack homology. In recent work we have extended the rack algebra to the cases of biracks and twisted virtual biracks. In this talk we will see new invariants of knots and links defined from modules over these algebras.

Posted October 1, 2013

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm Lockett 233
Juanita Pinzon-Caicedo, Indiana University

Traceless SU(2) representations of 2-stranded tangles

Abstract: Given a codimension 2 submanifold A⊂X define R(X,A) as the space of traceless SU(2) representations of π_1(X\\A) modulo conjugation. For Y a 3-manifold and K⊂Y a knot, Kronheimer-Mrowka defined the Instanton Knot Homology of (Y,K) as the homology of a chain complex whose groups are generated by the elements of R(Y,K). In the talk we describe a method to determine R(S^3,K) whenever K is a torus or pretzel knot.

Posted October 9, 2013

Last modified October 10, 2013

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm Lockett 233
Susan Abernathy, Louisiana State University

Virtual Seminar: "The Kauffman bracket ideal for genus-1 tangles"

Abstract: A genus-1 tangle is a 1-manifold with two boundary components properly embedded in the solid torus. A genus-1 tangle G embeds in a link L if we can complete G to L via a 1-manifold in the complement of the solid torus containing G. A natural question to ask is: given a tangle G and a link L, how can we tell if G embeds in L? We define the Kauffman bracket ideal, which gives an obstruction to tangle embedding, and outline a method for computing a finite list of generators for this ideal. We also give an example of a genus-1 tangle with non-trivial Kauffman bracket ideal and discuss how the concept of partial closures relates to this ideal.

Posted April 3, 2013

Last modified June 8, 2020

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm Lockett 233
Peter Horn, Syracuse University

Virtual Seminar: "Computing higher-order Alexander polynomials of knots"

Abstract: The classical Alexander polynomial of a knot can be defined in several ways, one of which is via covering spaces. Using higher covering spaces, Cochran defined the higher-order Alexander polynomials. It is known that the degree of the classical Alexander polynomial gives a lower bound for the genus of a knot, and so do the degrees of the higher-order Alexander polynomials. These higher-order bounds are known to be stronger than the classical bound for satellite knots, but little is known about low crossing knots. We will present an algorithm to compute the degree of the first higher-order Alexander polynomial of any knot, and we will discuss some interesting computations.

Posted August 29, 2013

Last modified November 5, 2013

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Ina Petkova, Rice University

Virtual Seminar: "Bordered Floer homology and decategorification"

Abstract: Bordered Floer homology is a TQFT-type generalization of Heegaard Floer homology to 3-manifolds with boundary, which satisfies a nice gluing formula. I will give a brief description of this generalized theory, and discuss some applications to topology. For example, bordered Floer homology categorifies the kernel of the homology map induced by the inclusion of the boundary into the 3-manifold.

Posted August 29, 2013

Last modified January 10, 2022

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm Lockett 233
Eamonn Tweedy, Department of Mathematics, Rice University

Virtual Seminar: "Positive Links"

Cochran and Gompf defined a notion of positivity for concordance classes of knots that simultaneously generalizes the usual notions of sliceness and positivity of knots. Their positivity essentially amounts to the knot being slice in a positive-definite simply-connected four manifold. I'll discuss an analogous property for links, describe a concrete characterization of positivity up to concordance, and give some obstructions to positivity.

Posted August 29, 2013

Last modified November 14, 2013

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm Lockett 233
Eamonn Tweedy, Department of Mathematics, Rice University

Virtual Seminar: tba

Posted March 13, 2014

Last modified March 2, 2021

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm Lockett 233
Olga Plamenevskaya, SUNY Stony Brook

Virtual Topology Seminar: "Looking for flexibility in higher-dimensional contact manifolds"

Contact manifolds are odd-dimensional cousins of symplectic manifolds; a contact structure on a smooth manifold is a hyperplane field given as a kernel of a "non-degenerate" 1-form. Locally, all contact structures look the same, but globally, a lot of interesting topological phenomena arise. By a classical result of Eliashberg, contact manifolds in dimension 3 come in two flavors: tight (rigid) and overtwisted (flexible). While the tight ones are quite subtle, overtwisted contact structures are completely described by their algebraic topology. In higher dimensions, a class of flexible contact structures is yet to be found. We will describe some conjectural "overtwisted pieces" (due to Niederkruger et al.) and an important flexibility principle for certain Legendrian knots discovered by Murphy. Then, we will present some results (joint with E. Murphy, K. Niederkruger, and A. Stipsicz) showing that in the presence of an "overtwisted piece", all Legendrian knots are "flexible", and demonstrating some flexibility phenomena for contact manifolds in higher dimensions.

Posted March 14, 2014

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Arkady Berenstein, University of Oregon

Equivariant Littlewood-Richardson coefficients

ABSTRACT. The goal of my talk (based on joint work with Edward Richmond) is to compute all equivariant Littlewood-Richardson (LR) coefficients for semisimple and Kac-Moody groups G, that is, the structure constants of the equivariant cohomology algebra H_B(G/B), where B is the Borel subgroup of G. These coefficients are of importance in enumerative geometry, algebraic combinatorics and representation theory. Our formula for the LR coefficients is purely combinatorial and is given in terms of the Cartan matrix and the Weyl group of G. In particular, our formula gives a combinatorial proof of positivity of the equivariant LR coefficients in the cases when all off-diagonal Cartan matrix entries are less than or equal to -2.

Posted March 30, 2014

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Chris Cornwell, Duke University

Virtual Seminar: Knot contact homology, knot group representations, and the A-polynomial

Abstract: In the knot contact homology of a knot K there are augmentations that may be associated to a flat connection on the complement of K. We show that all augmentations arise this way. As a consequence, a polynomial invariant of K called the augmentation polynomial represents a generalization of the classical A-polynomial. A recent conjecture, similar to the AJ conjecture concerning the A-polynomial, relates a 3-variable augmentation polynomial to colored HOMFLY-PT polynomials. Our results can be seen as motivation for this conjecture having an augmentation polynomial in place of the A-polynomial.

Posted April 21, 2014

Last modified March 2, 2021

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm Lockett 233
Mustafa Hajij, Department of Mathematics, LSU
Graduate Student

Virtual Topology Seminar: "Skein Theory and q-Series"

We study the tail a q-power series invariant of a sequence of admissible trivalent graphs with edges colored n or 2n. We use local skein relations to understand and compute the tail of these graphs. This allows us to understand the tail of the colored Jones polynomial for a large class of knots and links. For many quantum spin networks they turn out to be interesting number-theoretic q-series. In particular, certain quantum spin networks give a skein theoretic proof for the Andrews-Gordon identities for the two variable Ramanujan theta function as well to corresponding identities for the false theta function. Finally, we give product formula that the tail of such graphs satisfies.

Posted August 12, 2014

Last modified March 2, 2021

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm Lockett 233
Kristen Hendriks, UCLA

Virtual Seminar: "Localization and the link Floer homology of doubly-periodic knots"

A knot K in S^3 is said to be q-periodic if there is an orientation-preserving action of Z_q on S^3 which preserves K and has fixed set an unknot disjoint from K. There are many classical obstructions to the possible periods of a knot, including Edmonds's condition on the genus and Murasugi's conditions on the Alexander polynomial. We construct localization spectral sequences on the link Floer homology of 2-periodic knots, and show that they give a simultaneous generalization of Edmonds's condition and one of Murasugi's conditions. We conclude with an example in which our spectral sequences give a stronger obstruction than these (although not all) classical conditions.

Posted September 22, 2014

Last modified March 2, 2021

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm 233 Lockett Hall
Robert Lipshitz, Columbia University

Virtual Seminar: "A Khovanov stable homotopy type"

Khovanov homology is a knot invariant which refines (categorifies) the Jones polynomial. After recalling the definition of Khovanov homology we will introduce a space-level version, and sketch some computations and (modest) applications. This is joint work with Sucharit Sarkar and Tyler Lawson.

Posted October 7, 2014

Last modified October 23, 2014

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm Lockett 233
James Conway, Georgia Tech

Virtual Seminar: "Transverse Surgery in Contact 3-Manifolds"

Abstract: Much ink has been spilled on surgery on Legendrian knots; much less well studied is surgery on transverse knots. We will investigate transverse surgery, and study its effect on open books, the Heegaard Floer contact invariant, and tightness. We show that surgery on the connected binding of a genus g open book that supports a tight contact structure preserves tightness if the surgery coefficient is greater than 2g-1. In a complementary direction, we give criteria for when positive contact surgery on Legendrian knots will result in an overtwisted manifold.

Posted February 2, 2015

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm 233 Lockett Hall
Emily Stark, Tufts University

Abstract commensurability and quasi-isometric classification in dimension two

Two foundational questions in geometric group theory are to characterize the abstract commensurability and quasi-isometry classes within a class of groups, and to understand for which classes of groups the classifications coincide. In this talk, I will present a solution within the class of groups isomorphic to the fundamental group of two closed hyperbolic surfaces identified along an essential simple closed curve in each. I will discuss current work, joint with Pallavi Dani and Anne Thomas, for right-angled Coxeter groups.

Posted March 3, 2015

Last modified March 16, 2015

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm
Jeremy Van Horn-Morris, University of Arkansas

Virtual Seminar: On the coarse classification of Stein fillings

Abstract: In the '90s, Donaldson showed that every symplectic 4-manifold can be equipped with the structure of a Lefschetz pencil, a kind of singular surface bundle over CP^1. This pencil can be (non-uniquely) encoded as a relation in the mapping class group of a punctured surface, and while this factorization completely determines the manifold, it is in general very complicated. One might hope that some simpler shadow of the pencil might give useful information about the topology of the symplectic manifold. For example, what information does the genus of the pencil tell you about the symplectic manifold? Many of the initial conjectures about this relationship, as well as its generalization to open book decompositions, have been shown to be false. But, it turns out that in certain cases, there is very useful information available. We'll discuss the examples and the constraints. This is joint work with Inanc Baykur and Naoyuki Monden.

Posted October 7, 2014

Last modified January 10, 2022

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm tba
Tullia Dymarz, University of Wisconsin, Madison

Virtual Seminar: Non-rectifiable Delone sets in amenable groups

In 1998 Burago-Kleiner and McMullen constructed the first examples of coarsely dense and uniformly discrete subsets of R^n that are not bi-Lipschitz equivalent to the standard lattice Z^n. Similarly we find subsets inside the three dimensional solvable Lie group SOL that are not bi-Lipschitz equivalent to any lattice in SOL. The techniques involve combining ideas from Burago-Kleiner with quasi-isometric rigidity results from geometric group theory.

Posted June 3, 2015

Last modified September 9, 2015

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Adam Saltz, Boston College

Virtual Seminar: A transverse invariant from annular Khovanov homology

Abstract: Annular Khovanov homology is a refinement of Khovanov homology for links embedded in an annulus. Braid closures are natural examples of such links, and there is a well-known correspondence between braids and transverse links. Expanding on work of Plamenevskaya, I will present a computable conjugacy class invariant whose minimum we hope to be an effective transverse invariant. The invariant has applications to the word problem, the lengths of certain spectral sequences, and some classical questions about braids. This is joint work with Diana Hubbard.

Posted October 19, 2015

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm 233 Lockett Hall
Nicholas Owad, University of Nebraska, Lincoln

Virtual Seminar: Recent results concerning bridge spectra

Abstract: The bridge spectrum of a knot is a generalization of the classic invariant defined by Schubert, the bridge number of a knot. We will introduce the relevant background and some known results. Then we will give a short sketch of a proof of our main result, and end with open questions.

Posted June 17, 2015

Last modified November 4, 2015

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm 233 Lockett Hall
Clayton Shonkwiler, Colorado State University

Virtual Seminar: "The Symplectic Geometry of Polygon Space and How to Use It"

Abstract: In statistical physics, the basic (and highly idealized) model of a ring polymer is a closed random walk in 3-space with equal-length steps, often called a random equilateral polygon. In this talk, I will describe the moduli space of random equilateral polygons, giving a sense of how this fits into a larger symplectic and algebraic geometric story. In particular, the space of equilateral n-gons turns out to be a toric symplectic manifold, yielding a (nearly) global coordinate system. These coordinates are powerful tools both for proving theorems and for developing numerical techniques, some of which I will describe, including a very fast algorithm for directly sampling random polygons recently developed with Jason Cantarella (University of Georgia), Bertrand Duplantier (CEA/Saclay), and Erica Uehara (Ochanomizu University).

Posted September 24, 2015

Last modified November 30, 2015

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm
Chris Hruska, UW Milwaukee

Virtual Seminar: Distortion of surfaces in 3-dimensional graph manifolds

Abstract: (Joint with Hoang Thanh Nguyen) In geometric group theory, one often studies a finitely generated group as a geometric object, by equipping the group with a word metric. Using the word metric, Milnor observed that the fundamental group of any compact manifold closely resembles the universal cover of the manifold. If H is a finitely generated subgroup of G, then the inclusion of H into G may distort the geometry of H. In other words, distances between elements of H may be quite different when measured in the metrics of G and of H. We examine the large scale geometry of immersed horizontal surfaces in 3-dimensional graph manifolds. An immersed surface in a 3-manifold is said to be virtually embedded if the immersion lifts to an embedding into a finite sheeted cover of the manifold. We prove that the distortion of a horizontal surface is quadratic if the surface is virtually embedded, and is exponential otherwise. The proof depends on a combinatorial characterization of horizontal surfaces that virtually embed, due to Rubinstein-Wang. I will not assume any familiarity with geometric group theory or 3-dimensional manifolds in this talk.

Posted January 7, 2016

Last modified February 8, 2016

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm
Abhijit Champanerkar, CSI NY/CUNY

Virtual Seminar: "Densities and semi-regular tilings"

Abstract: For a hyperbolic knot or link $K$ the volume density is a ratio of hyperbolic volume to crossing number, and the determinant density is the ratio of 2\pi\log(det(K)) to the crossing number. We explore limit points of both densities for families of links approaching semi-regular biperiodic alternating links. We explicitly realize and relate the limits for both using techniques from geometry, topology, graph theory, dimer models, and Mahler measure of two-variable polynomials. This is joint work with Ilya Kofman and Jessica Purcell.

Posted November 21, 2015

Last modified February 25, 2016

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm 233 Lockett
Max Forester, University of Oklahoma

Virtual Seminar: "The geometry of Stallings-Bieri groups"

Abstract: The Stallings-Bieri groups are a family of finitely presented groups that have exotic homological finiteness properties, while also being quite easy to define and describe. They occur naturally as subgroups of non-positively curved groups (products of free groups, in fact). They are not non-positively curved themselves, however, and their large-scale geometry is quite interesting. I will discuss recent work with Will Carter in which we determine the large-scale isoperimetric behavior of these groups.

Posted March 9, 2016

Last modified March 2, 2021

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm
Roland van der Veen, Universiteit Leiden

Virtual Seminar: "Shadows, spines and gluing equations"

Ideal triangulations were applied very effectively to understand 3-manifolds. For example Thurston set up a system of gluing equations to produce hyperbolic structures from the ideal triangulation. I will argue that their dual 2-complexes, known as spines, are both easier to visualize and more flexible than ideal triangulations. We will reformulate Thurston's construction in terms of spines and show how one proves their symplectic properties first found by Neumann and Zagier. Time permitting we will also mention relations to four-manifolds and the Andrews-Curtis conjecture that become apparent in terms of spines.

Posted September 22, 2016

Last modified October 5, 2016

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm Lockett 233
Fang Sun, Tulane University

Topological Symmetries of R^3

The absence of geometric rigidity regarding topological actions of finite group on R^3 drives us into looking for possible algebraic rigidity. The outcome is positive: If a finite group G acts topologically and faithfully on R^3, then G is isomorphic to a subgroup of O(3).

Posted September 22, 2016

Last modified October 11, 2016

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm Lockett 233
Rafal Komendarczyk, Tulane University

Ropelength, crossing number and finite-type invariants

Ropelength and embedding thickness are related measures of geometric complexity of classical knots and links in Euclidean space. In their recent work, Freedman and Krushkal posed a question regarding lower bounds for embedding thickness of n-component links in terms of Milnor linking numbers (mu-invariants). In this talk we will show how to obtain such estimates, generalizing the known linking number bound. In the process, we generalize the results of Kravchenko and Polyak on the arrow polynomial formulas of mu-invariants of string links. We also collect several facts about finite type invariants and ropelength/crossing number of knots giving examples of families of knots, where estimates via the finite type invariants outperform the well-known knot--genus estimate. This is joint work with Andreas Michaelides.

Posted November 1, 2016

Last modified January 3, 2017

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Sam Nelson, Claremont McKenna College

Biquasiles and Dual Graph Diagrams

Dual graph diagrams are an alternate way to present oriented knots and links in R^3. In this talk we will see how to turn dual graph Reidemeister moves into an algebraic structure known as biquasiles and use this structure to define new integer-valued counting invariants of oriented knots and links.

Posted February 27, 2017

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm Lockett 233
Christine Lee, University of Texas at Austin

Jones slopes and Murasugi sums of links

Abstract: A Jones surface for a knot in the three-sphere is an essential surface whose boundary slopes, Euler characteristic, and number of sheets correspond to quantities defined from the asymptotics of the degrees of colored Jones polynomial. The Strong Slope Conjecture by Garoufalidis and Kalfagianni-Tran predicts that there are Jones surfaces for every knot.

A link diagram D is said to be a Murasugi sum of two links D' and D'' if a state graph of D has a cut vertex, which separates the graph into two state graphs of D' and D'', respectively. We may obtain a state surface in the complement of the link K represented by D by gluing the state surface for D and the state surface for D' along the disk filling the circle represented by the cut vertex in the state graph. The resulting surface is called the Murasugi sum of the two state surfaces.

We consider near-adequate links which are certain Murasugi sums of near-alternating link diagrams with an adequate link diagram along their all-A state graphs with an additional graphical constraint. For a near-adequate knot, the Murasugi sum of the corresponding state surface is a Jones surface by the work of Ozawa. We discuss how this proves the Strong Slope Conjecture for this class of knots and we will also discuss the stability properties of their colored Jones polynomial.

Posted March 8, 2017

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm Lockett 233
Gregor Masbaum, CNRS, Institut de Mathematiques de Jussieu, Paris, France

An application of TQFT to modular representation theory

Posted February 3, 2017

Geometry and Topology Seminar Seminar website

3:30 pm – 4:20 pm Lockett 233
Jose Ceniceros, Louisiana State University

TBD

Posted August 14, 2017

Last modified August 15, 2017

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Peter Lambert-Cole, Georgia Institute of Technology

Conway mutation and knot Floer homology

Abstract: Mutant knots are notoriously hard to distinguish. Many, but not all, knot invariants take the same value on mutant pairs. Khovanov homology with coefficients in Z/2Z is known to be mutation-invariant, while the bigraded knot Floer homology groups can distinguish mutants such as the famous Kinoshita-Terasaka and Conway pair. However, Baldwin and Levine conjectured that delta-graded knot Floer homology, a singly-graded reduction of the full invariant, is preserved by mutation. In this talk, I will give a new proof that Khovanov homology mod 2 is mutation-invariant. The same strategy can be applied to delta-graded knot Floer homology and proves the Baldwin-Levine conjecture for mutations on a large class of tangles.

Posted August 24, 2017

Last modified September 12, 2017

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Mike Wong, Louisiana State University

An unoriented skein exact triangle for grid homology

Abstract: Like the Jones and Alexander polynomials, Khovanov and knot Floer homology (HFK) both satisfy an oriented and an unoriented skein exact triangle. Manolescu (2007) proved the unoriented triangle for HFK over Z/2Z. In this talk, we will give a combinatorial proof of the same using grid homology (GH), which is isomorphic to knot Floer homology. This gives rise to a cube-of-resolutions complex that calculates GH-tilde. If time permits, we will outline the generalisation to the case over Z, and an application to quasi-alternating links. No prior experience with the subject is needed, as a brief introduction to grid homology will be given.

Posted August 24, 2017

Last modified September 20, 2017

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
John Etnyre, Georgia Institute of Technology

Contact surgeries and symplectic fillings

Abstract: It is well known that all contact manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. What is not so well understood is what properties of a contact structure are preserved by positive contact surgeries (the case for negative contact surgeries is fairly well understood now). In this talk we will discuss some new results about positive contact surgeries and in particular completely characterize when contact r surgery is symplectically fillable when r is in (0,1].

Posted August 24, 2017

Last modified September 19, 2017

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Mike Wong, Louisiana State University

TBD

Posted August 27, 2017

Last modified October 16, 2017

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Robin Koytcheff, University of Louisiana, Lafayette

Homotopy string links, configuration spaces, and the kappa invariant

Abstract: A link is an embedding of disjoint circles in space. A link homotopy is a path between two links where distinct components may not pass through each other, but where a component may pass through itself. In the 1990s, Koschorke conjectured that link homotopy classes of n-component links are distinguished by the kappa invariant. This invariant is essentially the map that a link induces on configuration spaces of n points. In joint work with F. Cohen, Komendarczyk, and Shonkwiler, we proved an analogue of this conjecture for string links. A key ingredient is a multiplication on maps of configuration spaces, akin to concatenation of loops in a space. This approach is related to recent joint work with Budney, Conant, and Sinha on finite-type knot invariants and the Taylor tower for the space of knots.

Posted September 13, 2017

Last modified September 20, 2017

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Yilong Wang, The Ohio State University

Integrality for SO(p)_2-TQFTs

Abstract: Representation theory of quantum groups at roots of unity give rise to modular tensor categories hence TQFTs, and the 3-manifold invariants from such constructions are known to be algebraic integers. In this talk, I will introduce the SO(p)_2-TQFT as an example of the above construction, and I will present our results on the integral lattices of the SO(p)_2-TQFT in genus 1 and one-punctured torus.

Posted November 7, 2017

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Mike Wong, Louisiana State University

Ends of moduli spaces in bordered Floer homology I

Abstract: Bordered Floer homology is an invariant associated to 3-manifolds with parametrized boundary, created by Lipshitz, Ozsvath, and Thurston as an extension of Heegaard Floer homology. In this framework, we associate a differential graded algebra A(F) to each surface, and an A-infinity module CF^(Y) to each bordered 3-manifold Y. The module CF^(Y) satisfies a structural equation that should be thought of as the analogue of the condition d^2=0 for chain complexes, obtained by considering ends of moduli spaces that appear in the definition of CF^(Y). In two consecutive expository talks, we will discuss specific examples that illustrate how these ends of moduli spaces match up in pairs. As a starting point, in this talk, we will first focus on the case of grid homology, a specialization of Heegaard Floer homology. No prior knowledge is necessary, as a brief introduction to grid homology will be given.

Posted November 15, 2017

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Mike Wong, Louisiana State University

Ends of moduli spaces in bordered Floer homology II

Abstract: This is the second in two consecutive talks about the ends of moduli spaces in Bordered Floer homology. Bordered Floer homology is an invariant associated to 3-manifolds with parametrized boundary, created by Lipshitz, Ozsvath, and Thurston as an extension of Heegaard Floer homology. In this framework, we associate a differential graded algebra A(F) to each surface, and an A-infinity module CF^(Y) to each bordered 3-manifold Y. The module CF^(Y) satisfies a structural equation that should be thought of as the analogue of the condition d^2=0 for chain complexes, obtained by considering ends of moduli spaces that appear in the definition of CF^(Y). In the talk last week, we discussed how these ends of moduli spaces match up in pairs in grid homology. In this talk, we will focus on the situation in bordered Floer homology, for both type A and type D structures.

Posted January 30, 2018

Last modified January 10, 2022

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Shea Vela-Vick, Louisiana State University

Knot Floer homology and fibered knots

We prove that the knot Floer homology of a fibered knot is nontrivial in its next-to-top Alexander grading. Immediate applications include a new proof that L-space knots prime and a classification of knots 3-manifolds with rank 3 knot Floer homology. We will also discuss a numerical refinement of the Ozsváth-Szabó contact invariant. This is joint work with John Baldwin.

Posted November 13, 2017

Last modified February 26, 2018

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Ivan Levcovitz, CUNY Graduate Center

Coarse geometry of right-angled Coxeter groups

Abstract: A main goal of geometric group theory is to understand finitely generated groups up to a coarse equivalence (quasi-isometry) of their Cayley graphs. Right-angled Coxeter groups (RACGs for short), in particular, are important classical objects that have been unexpectedly linked to the theory of hyperbolic 3-manifolds through recent results, including those of Agol and Wise. I will give a background on the relevant geometric group theory, RACGs and what is currently known regarding the quasi-isometric classification of RACGs. I will then describe a new computable quasi-isometry invariant, the hypergraph index, and its relation to other invariants such as divergence and thickness.

Posted October 18, 2017

Last modified February 27, 2018

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Bulent Tosun, University of Alabama

Obstructing Stein structures on contractible 4-manifolds

Abstract: A Stein manifold is a complex manifold with particularly nice convexity properties. In real dimensions above 4, existence of a Stein structure is essentially a homotopical question, but for 4-manifolds the situation is more subtle. An important question that has been circulating among contact and symplectic topologist for some time asks: whether every contractible smooth 4-manifold admits a Stein structure? In this talk we will provide examples that answer this question negatively. Moreover, along the way we will provide new evidence to a closely related fascinating conjecture of Gompf, which asserts that a nontrivial Brieskorn homology sphere, with either orientation, cannot be embedded in complex 2-space as the boundary of a Stein submanifold. This is a joint work with Tom Mark.

Posted October 18, 2017

Last modified March 13, 2018

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm
Ina Petkova, Dartmouth College

Knot Floer homology and the gl(1|1) link invariant

Abstract: The Reshetikhin-Turaev construction for the standard representation of the quantum group gl(1|1) sends tangles to C(q)-linear maps in such a way that a knot is sent to its Alexander polynomial. After a brief review of this construction, I will give an introduction to tangle Floer homology - a combinatorial generalization of knot Floer homology which sends tangles to (homotopy equivalence classes of) bigraded dg bimodules. Finally, I will discuss how to see tangle Floer homology as a categorification of the Reshetikhin-Turaev invariant. This is joint work with Alexander Ellis and Vera Vertesi.

Posted January 10, 2018

Last modified March 20, 2018

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Adam Levine, Duke University

Piecewise-linear disks and spheres in 4-manifolds

Abstract: We discuss a variety of problems related to the existence of piecewise-linear (PL) embedded surfaces in smooth 4-manifolds. We give the first known example of a smooth, compact 4-manifold which is homotopy equivalent to the 2-sphere but for which the homotopy equivalence cannot be realized by a PL embedding. We also show that the PL concordance group of knots in homology 3-spheres is infinitely generated and contains elements of infinite order. This is joint work with Jen Hom and Tye Lidman.

Posted November 12, 2017

Last modified November 5, 2021

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Miriam Kuzbary, Rice University

Perspectives on Link Concordance Groups

The knot concordance group has been the subject of much study since its introduction by Ralph Fox and John Milnor in 1966. One might hope to generalize the notion of a concordance group to links; however, the immediate generalization to the set of links up to concordance does not form a group since connected sum of links is not well-defined. In this talk, I will discuss two notions of a link concordance group: the string link concordance group due to Le Dimet in 1988 and one due to Matthew Hedden and myself based on the knotification construction of Peter Ozsvath and Zoltan Szabo. I will present invariants for studying these groups coming from Heegaard Floer homology and a new group theoretic invariant for studying concordance of knots inside in certain types of 3-manifold, as well a preliminary result involving more classical link concordance invariants.

Posted August 14, 2018

Last modified August 27, 2018

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Ignat Soroko, Louisiana State University

Dehn functions of subgroups of right-angled Artin groups

The question of what is a possible range for the Dehn functions (a.k.a. isoperimetric profile) for certain classes of groups is a natural and interesting one. Due to works of many authors starting with Gromov, we know a lot about the isoperimetric profile for the class of all finitely presented groups. Much less is known for many natural subclasses of groups, such as subgroups of right-angled Artin groups. We prove that polynomials of arbitrary degree are realizable as Dehn functions of subgroups of right-angled Artin groups. The key step is to construct for each natural k a free-by-cyclic group with the monodromy automorphism growing as n^k, which is virtually special in the sense of Haglund and Wise. Then its double will have Dehn function growing as n^{k+2}. This is a joint work with Noel Brady.

Posted August 16, 2018

Last modified September 3, 2018

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Andrew Zimmer, Louisiana State University

Limit sets of discrete subgroups

Given a discrete group of matrices one can define an associated limit set in projective space. In this talk I'll describe some results concerning the regularity of this limit set when the discrete group satisfies certain geometric properties.

Posted August 16, 2018

Last modified September 24, 2018

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Yilong Wang, Louisiana State University

Modular tensor categories and Reshetikhin-Turaev TQFTs

Abstract: In this talk, we give a detailed introduction to modular tensor categories and the Reshetikhin-Turaev TQFT associated to them. Time permitted, I will talk about algebraic properties of the RT-TQFTs.

Posted October 1, 2018

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Yilong Wang, Louisiana State University

Modular categories and RT-TQFTs: part II

Abstract: In this talk, I will define ribbon and modular categories, and show how modular categories give rise representations of the modular group SL(2,Z) using the graphical calculus introduced last time. Time permitted, I will explain how to generalize the construction to obtain a TQFT for closed surfaces.

Posted August 14, 2018

Last modified September 17, 2018

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Joshua Sabloff, Haverford College

Length and Width of Lagrangian Cobordisms

Abstract: In this talk, I will discuss two measurements of Lagrangian cobordisms between Legendrian submanifolds in symplectizations: their length and their relative Gromov width. The Gromov width, in particular, is a fundamental global invariant of symplectic manifolds, and a relative version of that width helps understand the geometry of Lagrangian submanifolds of a symplectic manifold. Lower bounds on both the length and the width may be produced by explicit constructions; this talk will concentrate on upper bounds that arise from a filtered version of Legendrian contact homology, a Floer-type invariant. This is joint work with Lisa Traynor.

Posted September 14, 2018

Last modified October 18, 2018

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Scott Baldridge, Louisiana State University

A new cohomology for planar trivalent graphs with perfect matchings

Abstract: In this lecture, I will describe a simple-to-compute polynomial invariant of a planar trivalent graph with a perfect matching (think: Jones polynomial for graphs). This polynomial is interesting because of what it detects: If the polynomial is non-zero when evaluated at one, then the perfect matching is even. Such a perfect matching implies that the graph can be 4-colored. I will then show how to categorify this polynomial to get a Khovanov-like cohomology theory for planar trivalent graphs and compute a couple of simple examples. If time, I will talk about some consequences of the cohomology theory.

Posted August 14, 2018

Last modified October 26, 2018

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Matthew Haulmark, Vanderbilt

Non-hyperbolic groups with Menger curve boundary

Abstract: In the setting of hyperbolic groups groups with Menger curve boundary are known to be abundant. Given the prevalence of negatively curved groups, it is was a surprising observation of Ruane that there were no known examples of non-hyperbolic groups with Menger curve boundary found in the literature. Thus Ruane posed the problem (early 2000's) of finding examples (alt. interesting classes) of non-hyperbolic groups with Menger curve boundary. In this talk I will discuss the first class of such examples. This is joint work with Chris Hruska and Bakul Sathaye.

Posted October 15, 2018

Last modified October 23, 2018

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Andrew McCullough, Georgia Institute of Technology

Legendrian Large Cables and Non-uniformly Thick Knots

Abstract: We will define the notion of a knot type having Legendrian large cables, and discuss the fact that having this property implies that the knot type is not uniformly thick. In this case, there are solid tori in this knot type that do not thicken to a solid torus with integer slope boundary torus, and that exhibit new phenomena; specifically, they have virtually overtwisted contact structures. We will give an example of an infinite family of ribbon knots that have Legendrian large cables which fail to be uniformly thick in several ways not previously seen.

Posted October 15, 2018

Last modified January 10, 2022

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Marco Marengon, UCLA

Strands algebras and Ozsváth-Szabó's Kauffman states functor

Ozsváth and Szabó introduced in 2016 a knot invariant, which they announced to be isomorphic to the usual knot Floer homology. Their construction is reminiscent of bordered Floer homology: for example, their invariant is defined by tensoring bimodules over certain algebras. During the talk I will introduce a more geometric construction, closer in spirit to bordered sutured Floer homology, based on strands on a particular class of generalized arc diagrams. The resulting strands algebras are quasi-isomorphic to the Ozsváth-Szabó's algebras, suggesting that Ozsváth and Szabó's theory may be part of a hypothetical generalization of bordered sutured Floer homology. This is a joint work with Mike Willis and Andy Manion.

Posted September 7, 2018

Last modified January 6, 2019

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Genevieve Walsh, Tufts University

Relatively hyperbolic groups with planar boundary

Abstract: I will describe what a relatively hyperbolic group is, and give a lot of examples where the boundary is planar. Furthermore, I will explore some of the interesting phenomena that can occur and explain the significance of cut points in the boundary. Lastly I will discuss restrictions on the peripheral groups when the boundary is planar and without cut points.

Posted August 28, 2018

Last modified January 14, 2019

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Francesco Lin, Princeton University

Monopole Floer homology and spectral geometry

Abstract: By studying the Seiberg-Witten equations, Kronheimer and Mrowka defined a package of invariants of three-manifolds called monopole Floer homology. In this talk, we discuss some interactions between this topological invariant and the spectral geometry of the Laplacian on the underlying Riemannian manifold, with the goal of understanding concrete examples of hyperbolic and Solv manifolds.

Posted October 22, 2018

Last modified March 11, 2019

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Caitlin Leverson, Georgia Institute of Technology

Representations, Ruling Polynomials, and the Colored HOMFLY-PT Polynomial

Abstract: Given a pattern braid beta in J1(S1), to any Legendrian knot Lambda in R3 with the standard contact structure, we can associate the Legendrian satellite knot S(Lambda,beta). We will discuss the relationship between counts of augmentations of the Chekanov-Eliashberg differential graded algebra of S(Lambda,beta) and counts of certain representations of the algebra of Lambda. We will then define an m-graded n-colored ruling polynomial from the m-graded ruling polynomial, analogously to how the n-colored HOMFLY-PT polynomial is defined from the HOMFLY-PT polynomial, and extend results of the second author, to show that the 2-graded n-colored ruling polynomial appears as a specialization of the n-colored HOMFLY-PT polynomial. (Joint work with Dan Rutherford.)

Posted August 15, 2018

Last modified March 26, 2019

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Tye Lidman, North Carolina State University

Splices, Heegaard Floer homology, and Seifert manifolds

Abstract: A natural way to construct three-manifolds is to glue two knot exteriors together. We will study properties of the Heegaard Floer homology of such manifolds. We then use this to characterize homeomorphisms between a special class of three-manifolds. This is joint work with Cagri Karakurt and Eamonn Tweedy.

Posted March 20, 2019

Last modified March 21, 2019

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Linh Truong, Columbia University

An infinite rank summand of the homology cobordism group

Abstract: We show that the homology cobordism group of integer homology three-spheres contains an infinite rank summand. The proof uses an algebraic modification of the involutive Heegaard Floer package of Hendricks-Manolescu and Hendricks-Manolescu-Zemke. This is inspired by Hom's techniques in the setting of knot concordance. This is joint work with Irving Dai, Jen Hom and Matt Stoffregen.

Posted January 25, 2019

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Peter Lambert-Cole, Georgia Institute of Technology

TBA

Posted September 11, 2019

Last modified October 22, 2019

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Hung Cong Tran, University of Oklahoma

The local-to-global property for Morse quasi-geodesics

Abstract: We show the mapping class group, CAT(0) groups, the fundamental groups of compact 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. As a consequence, we generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgroups of these groups. In the case of the mapping class group, this gives a combination theorem for convex cocompact subgroups. This is a joint work with Jacob Russell and Davide Spriano.

Posted September 9, 2019

Last modified October 25, 2019

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Viet Dung Nguyen, Vietnam Academy of Science and Technology Institute of Mathematics

The higher topological complexity of the complement of fiber type arrangements and related topics

Abstract: In the talk we present our method to compute the higher topological of the complement of fiber type arrangements. The same method will be applied to compute the higher topological complexity for some other spaces.

Posted September 16, 2019

Last modified October 1, 2021

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Jason Behrstock, CUNY Graduate Center and Lehman College

Hierarchically hyperbolic groups: an introduction

Hierarchical hyperbolicity provides a uniform framework for working with many important examples, including mapping class groups, right angled Artin groups, Teichmüller space, and others. In this talk I'll provide an introduction to studying groups and spaces from this point of view. This talk will include joint work with M. Hagen and A. Sisto.

Posted February 7, 2020

Last modified February 17, 2020

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Mike Wong, Louisiana State University

Ribbon Homology Cobordisms

Abstract: A cobordism between 3-manifolds is ribbon if it has no 3-handles. Such cobordisms arise naturally from several different topological and geometric contexts. In this talk, we describe a few obstructions to their existence, from Thurston geometries, character varieties, and instanton and Heegaard Floer homologies, and some applications. This is joint work with Aliakbar Daemi, Tye Lidman, and Shea Vela-Vick.

Posted January 30, 2020

Last modified March 3, 2020

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Eva Elduque, University of Michigan

Mixed Hodge structures on Alexander modules

Abstract: Given an epimorphism from the fundamental group of a smooth complex algebraic variety U onto the integers Z, one naturally obtains an infinite cyclic cover of the variety. In analogy with knot theory, the homology groups of this infinite cyclic cover, which are endowed with Z-actions by deck transformations, determine the family of Alexander modules associated to the epimorphism. In this talk, we will talk about how to equip the torsion part of the Alexander modules (with respect to the Z-actions) with canonical mixed Hodge structures in the case when the epimorphism is the induced map on fundamental groups of an algebraic map f from U into the punctured complex plane. Furthermore, we will compare the resulting mixed Hodge structure to other well studied mixed Hodge structures in the literature, including the limit mixed Hodge structure on the generic fiber of f. The relevant concepts will be introduced during the talk. Joint work with C. Geske, L. Maxim, and B. Wang.

Posted November 11, 2019

Last modified March 8, 2020

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Zhenkun Li, MIT

Decomposing sutured Instanton Floer homology

Abstract: Sutured Instanton Floer homology was introduced by Kronheimer and Mrowka. In this talk I will explain how to decompose sutured Instanton Floer homology with respect to a properly embedded surfaces inside the sutured manifold, and explain how this decomposition could be used to study the topological complexities of sutured manifolds and taut foliations. This work is partially joint with Sudipta Ghosh.

Posted August 25, 2021

Last modified September 17, 2021

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm
Kevin Schreve, Louisiana State University

TBD

Posted November 30, 2021

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Sam Shepherd, Vanderbilt University

TBA

Posted September 6, 2021

Geometry and Topology Seminar Seminar website

Posted September 6, 2021

Last modified September 7, 2021

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Kevin Schreve, Louisiana State University

Homological growth of groups

Lück's approximation theorem says that the L^2-Betti numbers of a residually finite group measure the rational homological growth of residual sequences of finite index normal subgroups. One can then ask about mod p homology growth or growth of torsion in integral homology. I will calculate these for right-angled Artin groups, and mention some consequences.

Posted August 25, 2021

Last modified September 8, 2021

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Roland van der Veen, Bernoulli Institute, University of Groningen

Generating functions and quantum knot invariants

Abstract: Calculations of quantum knot invariants often get complicated quickly. Be it skein relations or representations or quantum groups there always seems to be a huge number of unruly terms even for moderately simple knots. The goal of this talk is to show how generating functions can improve the situation, with a focus on the case of the sl_2 invariant (colored Jones polynomial). The main idea is to place the entire multiplication table of the relevant algebra in a generating function. Instead multiplying terms directly we will compose the generating functions in a way that is reminiscent of Feynman diagram calculus. The result is a strong yet computable knot invariant that shares many properties with the Alexander polynomial. This is joint work with Dror Bar-Natan and I will discuss (part of) sections 2,6 and 8 of our recent preprint: https://arxiv.org/abs/2109.02057

Posted September 6, 2021

Geometry and Topology Seminar Seminar website

3:30 pm
Angela Wu, Louisiana State University

Obstructing Lagrangian concordance for closures of 3-braids

Two knots are said to be concordant if they jointly form the boundary of a cylinder in four-dimensional Euclidean space. In the symplectic setting, we say they are Lagrangian concordant if the knots are Legendrian and the cylinder is Lagrangian. In this talk I'll show that no Legendrian knot which is both concordant to and from the unstabilized Legendrian unknot can be the closure of an index 3 braid except the unknot itself.

Posted September 6, 2021

Last modified October 1, 2021

Geometry and Topology Seminar Seminar website

9:30 am Lockett 233
Michael Farber, Queen Mary University of London

Ample simplicial complexes

I will first describe a remarkable simplicial complex X which can be uniquely characterized by its universality and homogeneity. It contains an isomorphic copy of any simplicial complex with countably many vertexes as an induced subcomplex. A random simplicial complex on countably many vertexes is isomorphic to X with probability 1. The main focus of the talk will be on r-ample simplicial complexes which are finite approximations to X and possess many striking properties. The r-ample complexes can potentially be used for designing stable and resilient networks. The talk is based on joint work with C. Even-Zohar, L. Mead and L. Strauss.

Posted September 20, 2021

Last modified October 4, 2021

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Christin Bibby, Louisiana State University

Homology representations of compactified configurations on graphs applied to tropical moduli spaces

The homology of a compactified configuration space of a graph is equipped with commuting actions of a symmetric group and the outer automorphism group of a free group. We construct an efficient free resolution for these homology representations. Using the Peter-Weyl Theorem for symmetric groups, we consider irreducible representations individually, vastly simplifying the calculation of these homology representations from the free resolution. As our main application, we obtain computer calculations of the top weight rational cohomology of the moduli space of genus 2 curves with n marked points, equivalently the rational homology of tropical moduli space, as a representation of the symmetric group acting by permuting point labels for all n≤10. We further give new multiplicity calculations for specific irreducible representations of the symmetric group appearing in cohomology for n≤17. Our approach produces information about these homology groups in a range well beyond what was feasible with previous techniques. This is joint work with Melody Chan, Nir Gadish, and Claudia He Yun.

Posted September 7, 2021

Last modified October 4, 2021

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Matt Clay, University of Arkansas

Chain flaring and L^2–torsion of free-by-cyclic groups

We introduce a condition on the monodromy of a free-by-cyclic group, G_φ, called the chain flare condition, that implies that the L^2–torsion, ρ^(2)(G_φ), is non-zero. We conjecture that this condition holds whenever the monodromy is exponentially growing.

Posted September 8, 2021

Last modified October 18, 2021

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Nikita Nikolaev, University of Sheffield

Abelianisation of Filtered Local Systems on Surfaces

I will describe an approach to analysing local systems on surfaces which is a functorial correspondence between local systems of rank n with C*-local systems an n-fold covering surface. Such an approach, called abelianisation, emerged in the last decade in the work of Gaiotto, Moore, Neitzke on spectral networks that arise in the context of supersymmetric gauge theories. It can be seen as a generalisation of the abelianisation of Higgs bundles (a.k.a., the spectral correspondence, a key step in the analysis of Hitchin integrable systems) to flat bundles. I will explain my point of view on the mathematical theory behind abelianisation (which involves the deformation theory of the direct image functor) and give an outlook of what kind of things I hope to prove in the near future.

Posted September 20, 2021

Last modified October 27, 2021

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Orsola Capovilla-Searle, University of California, Davis

Infinitely many Lagrangian Tori in Milnor fibers constructed via Lagrangian Fillings of Legendrian links

One approach to studying symplectic manifolds with contact boundary is to consider Lagrangian submanifolds with Legendrian boundary; in particular, one can study exact Lagrangian fillings of Legendrian links. There are still many open questions on the spaces of exact Lagrangian fillings of Legendrian links in the standard contact 3-sphere, and one can use Floer theoretic invariants to study such fillings. We show that a family of oriented Legendrian links has infinitely many distinct exact orientable Lagrangian fillings up to Hamiltonian isotopy. Within this family, we provide some of the first examples of a Legendrian link that admits infinitely many planar exact Lagrangian fillings. Weinstein domains are examples of symplectic manifolds with contact boundary that have a handle decomposition compatible with the symplectic structure of the manifold. Weinstein handlebody diagrams are given by projections of Legendrian submanifolds. We provide Weinstein handlebody diagrams of the 4-dimensional Milnor fibers of T_{p,q,r} singularities, which we then use to construct infinitely many Lagrangian tori and spheres in these spaces.

Posted September 9, 2021

Last modified October 28, 2021

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Yvon Verberne, Georgia Tech

Automorphisms of the fine curve graph

The fine curve graph of a surface was introduced by Bowden, Hensel and Webb. It is defined as the simplicial complex where vertices are essential simple closed curves in the surface and the edges are pairs of disjoint curves. We show that the group of automorphisms of the fine curve graph is isomorphic to the group of homeomorphisms of the surface, which shows that the fine curve graph is a combinatorial tool for studying the group of homeomorphisms of a surface. This work is joint with Adele Long, Dan Margalit, Anna Pham, and Claudia Yao.

Posted September 18, 2021

Last modified November 29, 2021

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Assaf Bar-Natan, University of Toronto

Geodesics in the Thurston Metric on Teichmüller Space

For a given surface S, Teichmüller space is the space of all hyperbolic metrics on S up to isotopy homotopic to the identity. In the 70s, Thurston defined his eponymous metric by considering lipschitz constants between maps from one hyperbolic structure to another. The Thurston metric is asymmetric, but that's okay! Geodesics work as you'd expect, but there are forward and backwards geodesics, and they are also sometimes unique! In this talk, I will describe the structure of geodesics in the Thurston metric, and explain where non-uniqueness of geodesics comes from, and how we can fight it! This talk will include a brief introduction to Teichmüller spaces.

Posted September 9, 2021

Last modified November 5, 2021

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Daniel Platt, Imperial College London

G2-instantons on resolutions of G2-orbifolds

In dimensions 3 and 4, useful invariants of smooth manifolds can be defined by counting certain principal bundle connections, namely the Casson invariant and Donaldson theory. There is a big research programme trying to define analogues in dimension 7 for manifolds with holonomy G2. However, in this dimension, there are some problems that don’t appear in lower dimensions, and not many examples are known. In this talk, I will explain a new construction for G2-instantons. This is intimately related to gauge theory in four dimensions and the Fueter equation in three dimensions. In the beginning I will briefly explain holonomy and the holonomy group G2, no previous knowledge of G2 required!

Posted October 1, 2021

Last modified November 29, 2021

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Dahye Cho, Stony Brook University

Symplectic Criteria on Stratified Uniruledness of Affine Varieties and Applications

We explain about Hamiltonian Floer cohomology of certain open symplectic manifolds including affine varieties, that is Morse cohomology of the space of loops on a symplectic manifold. Using the long exact sequences of symplectic cohomology, we develop criteria for affine varieties to admit uniruled subvarieties of certain dimensions. If time permits, we provide applications of the criteria in birational geometry of log pairs in the direction of the Minimal Model Program.

Posted December 13, 2021

Last modified January 19, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Gage Martin, Boston College

Annular links, double branched covers, and annular Khovanov homology

Given a link in the thickened annulus, you can construct an associated link in a closed 3-manifold through a double branched cover construction. In this talk we will see that perspective on annular links can be applied to show annular Khovanov homology detects certain braid closures. Unfortunately, this perspective does not capture all information about annular links. We will see a shortcoming of this perspective inspired by the wrapping conjecture of Hoste-Przytycki. This is partially joint work with Fraser Binns.

Posted December 10, 2021

Last modified February 11, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Shelly Harvey, Rice University

Knotting and linking in 4-dimensions

Knots are circles embedded into Euclidean space. Links are knots with multiple components. The classification of links is essential for understanding the fundamental objects in low-dimensional topology: 3- and 4-dimensional manifolds since every 3- and 4-manifold can be represented by a weighted link. When studying 3-manifolds, one considers isotopy as the relevant equivalence relation whereas when studying 4-manifolds, the relevant condition becomes knot and link concordance. In some sense, the nicest class of links are the ones called boundary links; like a knot, they bound disjointly embedded orientable surfaces in Euclidean space, called a multi-Seifert surface. The strategy to understand link concordance, starting with Levine in the 60s, was to first understand link concordance for boundary links and then to try to relate other links to boundary links. However, this point of view was foiled in the 90's when Tim Cochran and Kent Orr showed that there were links (with all known obstructions vanishing, i.e., Milnor's invariants) that were not concordant to any boundary link. In this work, Chris Davis, Jung Hwan Park, and I consider weaker equivalence relations on links filtering the notion of concordance, called n-solvable equivalence. We will show that most links are 0- and 0.5-solvably equivalent but that for larger n, that there are links not n-solvably equivalent to any boundary link (thus cannot be concordant to a boundary link). This is joint work with C. Davis and J. H. Park.

Posted December 17, 2021

Last modified February 9, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Fraser Binns, Boston College

Links with Knot Floer homology of Low Rank

Knot Floer homology is a powerful link invariant due to Ozsváth–Szabó and J. Rasmussen. In this talk I will discuss a rank bound in knot Floer homology for fibered links coming from Baldwin–Vela-Vick–Vértesi's BRAID invariant and explain how it can be used to classify links with knot Floer homology of low rank. This talk is based on joint work with Subhankar Dey.

Posted February 1, 2022

Last modified March 3, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Zsuzsanna Dancso, University of Sydney

Welded Tangles and the Kashiwara–Vergne Groups

I'll explain a general method of “translating” between a certain type of problem in quantum topology, and solving equations in graded spaces in (quantum) algebra. I'll go through old and new applications of this principle: Drinfel'd associators and parenthesised braids, Grothendieck–Teichmüller groups, welded tangles and the Kashiwara–Vergne equations, and a topological description of the Kashiwara–Vergne groups. The “recent” portion of the talk is based on joint work with Iva Halacheva and Marcy Robertson (arXiv: 2106.02373), with Dror Bar-Natan (arXiv: 1405.1955), and work-in-progress with Marcy Robertson and Tamara Hogan.

Posted December 10, 2021

Last modified February 24, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Sam Shepherd, Vanderbilt University

Commensurability of lattices in right-angled buildings

Given compact length spaces $X_1$ and $X_2$ with a common universal cover, it is natural to ask whether $X_1$ and $X_2$ have a common finite cover. In particular, are there properties of $X_1$ and $X_2$, or of their fundamental groups, that guarantee the existence of a common finite cover? We will discuss several examples, as well as my new result which concerns the case where the common universal cover is a right-angled building. Examples of right-angled buildings include products of trees and Davis complexes of right-angled Coxeter groups. My new result will be stated in terms of (weak) commensurability of lattices in the automorphism group of the building.

Posted January 14, 2022

Last modified February 2, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Sudipta Ghosh, Louisiana State University

Connected sums and directed systems in knot Floer homologies

Knot Floer homology is an invariant of knots which was first introduced in the context of Heegaard Floer homology and later extended to other Floer theories. In this talk, we discuss a new approach to the connected sum formula using direct limits. Our methods apply to versions of knot Floer homology arising in the context of Heegaard, instanton and monopole Floer homology. The same argument we use to prove the connected sum formula also generalizes Kronheimer-Mrowka's oriented skein exact triangle from the hat version of instanton knot homology to the minus version of instanton knot homology. This is joint work with Ian Zemke.

Posted March 6, 2022

Last modified March 30, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Nurdin Takenov, Louisiana State University

On a relation between ADO and Links-Gould invariants

Akutsu-Deguchi-Ohtsuki (ADO) and Links-Gould are two link invariants. Both of them are connected to Alexander polynomial and can be considered its generalizations. In the talk I will define them, describe some of their properties and state a conjecture about a relation between them. Then I will sketch the proof of the conjecture for some classes of links and some thoughts about perspectives of a full proof of a conjecture.

Posted January 19, 2022

Last modified March 31, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Dan Rutherford, Ball State University

Normal rulings, augmentations, and the colored Kauffman polynomial

Normal rulings are certain decompositions of front diagrams of Legendrian links in $R^3$ that were discovered independently by Chekanov-Pushkar and Fuchs in the context of generating families and augmentations of the Legendrian DG-algebra respectively. They can be used to define combinatorial invariants of Legendrian links called ruling polynomials. In this talk, I will survey some results connecting normal rulings with augmentations and with 2-variable topological knot polynomials (HOMFLY-PT and Kauffman). In particular, I will discuss joint work with C. Leverson and J. Murray relating counts of higher rank augmentations to the n- colored HOMFLY-PT and Kauffman polynomials.

Posted March 25, 2022

Last modified March 28, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Rachael Boyd, University of Cambridge

A Deligne complex for Artin monoids

This talk is based on joint work with Charney and Morris-Wright (https://arxiv.org/abs/2007.12156). I will motivate the study of Artin monoids before outlining our construction of the monoid Deligne complex, which generalises the notion of Deligne complexes for Artin groups. I will discuss geometric properties of this complex, including the fact that the monoid Deligne complex is always contractible, with a locally isometric embedding into the Deligne complex of the corresponding Artin group.

Posted January 26, 2022

Last modified April 14, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Locket 233
Sümeyra Sakallı , University of Arkansas

Complex Ball Quotients and New Symplectic 4-Manifolds with Nonnegative Signatures

We first construct a complex surface with positive signature, which is a ball quotient. We obtain it as an abelian Galois cover of CP^2 branched over the Hesse arrangement. Then we analyze its fibration structure, and by using it we build new symplectic and also non-symplectic exotic 4-manifolds with positive signatures. In the second part of the talk, we discuss Cartwright-Steger surfaces, which are also ball quotients. Next, we present our constructions of new symplectic and non-symplectic exotic 4-manifolds with non-negative signatures that have the smallest Euler characteristics in the so-called ‘arctic region’ on the geography chart. More precisely, we prove that there exist infinite families of irreducible symplectic and infinite families of irreducible non-symplectic, exotic 4-manifolds that have the smallest Euler characteristics among the all known simply connected 4-manifolds with nonnegative signatures and with more than one smooth structures. This is a joint work with A. Akhmedov and S.-K. Yeung.

Posted March 10, 2022

Last modified April 22, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Alan Logan, University of St Andrews

The Surface Group Conjectures

The Surface Group Conjectures are statements about recognising surface groups among one-relator groups. In this talk, I will motivate these conjectures and add a new conjecture to this family. I will then explain the proofs of all these conjectures in the case of both two-generator one-relator groups and one-relator groups with torsion. Based on joint work with Giles Gardam and Dawid Kielak.

Posted March 6, 2022

Last modified April 26, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Rob Quarles, Louisiana State University

A new perspective on a polynomial time knot polynomial

In this talk we consider the Z_1(K) polynomial time knot polynomial defined and described by Dror Bar-Natan and Roland van der Veen in their 2018 paper "A polynomial time knot polynomial". We first look at some of the basic properties of Z_1(K) and develop an invariant of diagrams \Psi_m(D) related to this polynomial. We use this invariant as a model to prove how Z_1(K) acts under the connected sum operation. We then discuss the effect of mirroring the knot on Z_1(K) and describe a geometric interpretation of some of the building blocks of the invariant. Finally, we describe a base set of knots which can be used to build the Z_1(K), or rather its normalization \rho_1(K), showcasing some of its symmetry properties, and we use this idea to give an explicit expansion of \rho_1(K) for the family of T(2,2p+1) torus knots in terms of this base set of knot invariants.

Posted April 26, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Nurdin Takenov, Louisiana State University

On a relation between ADO and Links-Gould invariants

Akutsu-Deguchi-Ohtsuki(ADO) and Links-Gould are two link invariants. Both of them are connected to Alexander polynomial and can be considered its generalizations. In the talk I will define them, describe some of their properties and state a conjecture about a relation between them. Then I will sketch the proof of the conjecture for some classes of links and some thoughts about perspectives of a full proof of a conjecture.

Posted July 27, 2022

Last modified August 18, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Jean Pierre Mutanguha, Institute for Advanced Study

Canonical forms for free group automorphisms

The Nielsen–Thurston theory of surface homeomorphisms can be thought of as a surface analogue to the Jordan Canonical Form. I will discuss my progress in developing a similar canonical form for free group automorphisms. (Un)Fortunately, free group automorphisms can have arbitrarily complicated behaviour. This is a significant barrier to translating arguments that worked for surfaces into the free group setting; nevertheless, the overall ideas/strategies do translate!

Posted August 17, 2022

Last modified August 21, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Daniel C. Cohen, Mathematics Department, LSU

Topological complexity of motion planning

Motivated by the motion planning problem from robotics, topological complexity is a mathematical invariant of the space of all configurations of a mechanical system. This invariant provides a measure of the complexity of designing a motion planning algorithm for the mechanical system, that is, the complexity of navigation in the configuration space. I will introduce this notion (as well as a recent generalization if time permits), and provide illustrations of its determination. Examples of configuration spaces considered will include familiar topological spaces such as spheres and surfaces, and possibly some classical configuration spaces of points in Euclidean spaces.

Posted August 29, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Mike Wong, Louisiana State University

Annular link Floer homology and gl(1|1)

In earlier work by Ellis, Petkova, and Vertesi, tangle Floer bimodules (a combinatorial generalization of link Floer homology) are shown to decategorify to the Reshetikhin–Turaev invariants arising in the representation theory of gl(1|1). In this talk, we describe how this can give rise to a gl(1|1) action on annular link Floer homology, viewed as the Hochschild homology—or horizontal trace—of a tangle Floer bimodule. The gl(1|1) action turns out to be closely related to a known basepoint action in Floer theory. This is based on joint work in progress with Andy Manion and Ina Petkova.

Posted July 21, 2022

Last modified August 29, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Jeff Hicks, University of Edinburgh

Cute constructions from Lagrangian cobordisms

The symplectic squeezing problem asks: What kinds of subsets "fit" inside a symplectic manifold X. The most famous example comes from Gromov's non-squeezing theorem, which shows that the symplectic 4-ball of radius R cannot be symplectically embedded inside the product of two balls of radius A,B whenever A is less than R. In this talk, we will look at squeezing problems related to Lagrangian submanifolds. Let X be the product of two disks of radius A and B. A Lagrangian submanifold L of X is called integral if the symplectic form takes integer values on H2(X,L). The Lagrangian packing problem asks how many disjoint integral Lagrangian tori can we fit inside X. It is easy to construct (⌊A⌋-1)(⌊B⌋-1) such tori, by taking the products of circles bounding integer areas. When A,B are less than 2 of Richard Hind shows that you can do no better. In this talk, I will discuss how to fit an additional integral Lagrangian torus into this space whenever A is greater than 2. The main insight is to treat Lagrangian submanifolds in X as Lagrangian cobordisms (with Lagrangian ends) and employ tools used to construct Lagrangian cobordisms. Time permitting, I will discuss 2 other examples where techniques from Lagrangian cobordisms can be used to improve known constructions for squeezing problems. This is joint work with Cheuk Yu Mak.

Posted August 17, 2022

Last modified September 19, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Akram Alishahi, University of Georgia

Khovanov homology and involutive Heegaard Floer homology

Heegaard Floer theory is a package is invariants for 3- and 4-manifolds, knots, links, contact structure, etc. Khovanov homology is a knot (link) invariant defined around the same time as Heegaard Floer homology. Studying the interaction between these invariants has been the subject of many research works over the past two decades. In 2003, Ozsváth and Szabó construct a spectral sequence from Khovanov homology to the Heegaard Floer homology of the branched double cover of the knot. In 2017, Hendricks and Manolescu, incorporate the conjugation action on Heegaard Floer homology to produce a richer 3-manifold invariant, called involutive Heegaard Floer homology. In this talk, we will discuss an involutive version of Ozsváth-Szabó’s spectral sequence that converges to the involutive Heegaard Floer homology of the branched double cover of the knot. This is a work in progress, joint with Linh Truong and Melissa Zhang.

Posted July 21, 2022

Last modified September 15, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Ipsita Datta, Institute for Advanced Study

Lagrangian cobordisms, enriched knot diagrams, and algebraic invariants

We introduce new invariants to the existence of Lagrangian cobordisms in R^4. These are obtained by studying holomorphic disks with corners on Lagrangian tangles, which are Lagrangian cobordisms with flat, immersed boundaries.

Posted July 23, 2022

Last modified September 30, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Emily Stark, Wesleyan University

Graphically discrete groups and rigidity

Rigidity theorems in geometric group theory prove that a group’s geometric type determines its algebraic type, typically up to virtual isomorphism. We study graphically discrete groups, which impose a discreteness criterion on the automorphism groups of graphs the group acts on and are well suited to studying rigidity problems. Classic examples of graphically discrete groups include virtually nilpotent groups and fundamental groups of closed hyperbolic manifolds; nonabelian free groups are non-examples. We will present new families of graphically discrete groups and demonstrate this property is not a quasi-isometry invariant. We will discuss rigidity phenomena for free products of graphically discrete groups. This is joint work with Alex Margolis, Sam Shepherd, and Daniel Woodhouse.

Posted September 30, 2022

Last modified October 3, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Marco Marengon, Alfréd Rényi Institute of Mathematics

Sliceness and the rank of some knot homologies

A popular notion in knot theory is that of “sliceness”: a knot in S^3 is called slice if it bounds a smooth disc in B^4. There are various reasons why this concept is so fundamental: for example, sliceness is at the core of a trendy strategy proposed to disprove the smooth 4-dimensional Poincaré conjecture, and it has recently been shown that a generalisation of this concept to 4-manifolds other than B^4 can detect exotic pairs, i.e. 4-manifolds that are homeomorphic but not diffeomorphic to each other. We study whether sliceness poses any restrictions on the rank of certain homology theories associated with knots. We prove some results and formulate some curious conjectures. This is joint work with Hockenhull and Willis, and partly also with Dunfield and Gong.

Posted July 29, 2022

Last modified October 5, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Caitlin Leverson, Bard College

Lagrangian Realizations of Ribbon Cobordisms

Similarly to how every smooth knot has a Legendrian representative (in fact, infinitely many different representatives), in this talk we will discuss why every ribbon cobordism has a Legendrian representative. Meaning, if $C$ is a ribbon cobordism in $[0,1]\times S^3$ from the link $K_0$ to $K_1$, then there are Legendrian realizations $\Lambda_0$ and $\Lambda_1$ of $K_0$ and $K_1$, respectively, such that $C$ may be isotoped to a decomposable Lagrangian cobordism from $\Lambda_0$ to $\Lambda_1$. We will also give examples of some interesting constructions of such decomposable Lagrangian cobordisms. This is joint work with John Etnyre.

Posted August 18, 2022

Last modified October 14, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Pallavi Dani, Department of Mathematics, LSU

Divergence, thickness, and hypergraph index for Coxeter groups

Divergence and thickness are geometric properties of finitely generated groups that are invariant under quasi-isometry. In general, they can be quite difficult to compute. In the case of right-angled Coxeter groups, Levcovitz introduced the notion of hypergraph index, which can be algorithmically computed from the defining graph, and proved that it determines the thickness and divergence of the group. After introducing the basics, I will talk about joint work with Yusra Naqvi, Ignat Soroko, and Anne Thomas, in which we propose a definition of hypergraph index for general Coxeter groups. We show that it determines the divergence and thickness in an infinite family of non-right-angled Coxeter groups.

Posted August 11, 2022

Last modified October 19, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Miriam Kuzbary, Georgia Institute of Technology

Asymptotic bounds on the d-invariant

As shown by Morita, every integral homology 3-sphere Y has a Heegaard decomposition into two handlebodies where the gluing map along the boundary is an element of the Torelli subgroup of the mapping class group of the boundary composed with the standard gluing map for the 3-sphere. In work in progress with Santana Afton and Tye Lidman, we show that the d-invariant of Y, a homology cobordism invariant of homology spheres defined using Heegaard Floer homology, is bounded above by a linear function of the word length of a corresponding gluing element in the Torelli group for any fixed, finite generating set when the genus is larger than 2. Moreover, we show the d-invariant is bounded for homology spheres corresponding to various large families of mapping classes.

Posted July 28, 2022

Last modified November 7, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Alex Margolis, Vanderbilt University

Model geometries of groups and discretisability

The central theme of geometric group theory is to study groups via their actions on metric spaces. A model geometry of a finitely generated group is a proper geodesic metric space admitting a geometric group action. Every finitely generated group has a model geometry that is a locally finite graph, namely its Cayley graph with respect to a finite generating set. In this talk, I investigate which finitely generated groups G have the property that all model geometries of G are (essentially) locally finite graphs. I introduce the notion of domination of metric spaces and give necessary and sufficient conditions for all model geometries of a finitely generated group to be dominated by a locally finite graph. Among groups of cohomological dimension two, the only such groups are surface groups and generalised Baumslag-Solitar groups. Time permitting, I will discuss applications to quasi-isometric rigidity.

Posted July 23, 2022

Last modified November 11, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Hannah Turner , Georgia Institute of Technology

Generalizing the (fractional) Dehn twist coefficient

The fractional Dehn twist coefficient (FDTC) is a rational number associated to a mapping class on a (finite-type) surface with boundary. This 2-dimensional invariant has many applications to 3-manifold topology and contact geometry. One way to think of the FDTC is as a real-valued function on the mapping class group of a surface with many nice properties. In this talk, we will give sufficient conditions on a more general group to admit a function which behaves like the FDTC. In particular, we use this to generalize the FDTC to infinite-type surfaces (with boundary); in this setting, we show that the "fractional" Dehn twist coefficient need not be rational. This is joint work with Peter Feller and Diana Hubbard.

Posted September 6, 2022

Last modified November 28, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Agniva Roy, Georgia Institute of Technology

On the doubling construction for Legendrian submanifolds

In high dimensional contact and symplectic topology, finding interesting constructions for Legendrian submanifolds is an active area of research. Further, it is desirable that the constructions lend themselves nicely to computation of invariants. The doubling construction was defined by Ekholm, which uses Lagrangian fillings of a Legendrian knot in R^{2n-1} to produce a Legendrian in R^{2n+1}. Later Courte-Ekholm showed that symmetric doubles of embedded fillings are "uninteresting". In recent work the symmetric doubling construction was generalised to any contact manifold, giving two isotopic constructions related to open book decompositions of the ambient manifold. In a separate joint work with James Hughes, we explore the asymmetric doubling construction through Legendrian weaves.

Posted December 1, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Katherine Goldman, Ohio State University

TBA

Posted December 1, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Katherine Goldman, Ohio State University

TBA

Posted February 21, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Nir Gadish, University of Michigan

(Lots of) unstable cohomology of moduli spaces of curves with marked points

The moduli spaces of smooth projective curves with marked points have cohomology that attaches characteristic classes to surface bundles with disjoint sections. As such, this cohomology plays a fundamental role in algebraic geometry and topology. However, only a tiny fraction of the cohomology is understood. I will present joint works with Bibby, Chan and Yun, and with Hainaut, in which we gain access to the least algebraic part of the cohomology for curves of genus 2, using tropical geometry and configuration spaces on graphs. In particular we find the first examples of families of cohomology classes in the top cohomological dimension, which seem to tell a geometric story that is yet to be understood.

Posted January 10, 2023

Last modified August 7, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Jingyin Huang, Ohio State University

Integral measure equivalence versus quasi-isometry for some right-angled Artin groups

Recall that two finitely generated groups G and H are quasi-isometric, if they admit a topological coupling, i.e. an action of G times H on a locally compact topological space such that each factor acts properly and cocompactly. This topological definition of quasi-isometry was given by Gromov, and at the same time he proposed a measure theoretic analogue of this definition, called the measure equivalence, which is closely related to the notion of orbit equivalence in ergodic theory. Despite the similarity in the definition of measure equivalence and quasi-isometry, their relationship is rather mysterious and not well-understood. We study the relation between these two notions in the class of right-angled Artin groups. In this talk, we show if H is a countable group with bounded torsion which is integrable measure equivalence to a right-angled Artin group G with finite outer automorphism group, then H is finitely generated, and H and G are quasi-isometric. This allows us to deduce integrable measure equivalence rigidity results from the relevant quasi-isometric rigidity results for a large class of right-angled Artin groups. Interestingly, this class of groups are rigid for a reason which is quite different from other cases of measure equivalence rigidity. We will also do a quick survey of relevant measure equivalence rigidity and quasi-isometric rigidity results of other classes of groups, motivating our choice of right-angled Artin groups as a playground. This is joint work with Camille Horbez.

Posted January 12, 2023

Last modified March 1, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Catherine Cannizzo, University of California, Riverside

Homological Mirror Symmetry for Theta Divisors

Symplectic geometry is a relatively new branch of geometry. However, a string theory-inspired duality known as “mirror symmetry” reveals more about symplectic geometry from its mirror counterparts in complex geometry. M. Kontsevich conjectured an algebraic version of mirror symmetry called “homological mirror symmetry” (HMS) in his 1994 ICM address. HMS results were then proved for symplectic mirrors to Calabi-Yau and Fano manifolds. Those mirror to general type manifolds have been studied in more recent years, including my research. In this talk, we will introduce HMS through the example of the 2-torus T^2. We will then outline how it relates to HMS for a hypersurface of a 4-torus T^4, in joint work with Haniya Azam, Heather Lee, and Chiu-Chu Melissa Liu. From there, we generalize to hypersurfaces of higher dimensional tori, otherwise known as “theta divisors.” This is also joint with Azam, Lee, and Liu.

Posted November 29, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Katherine Raoux, University of Arkansas

TBA

Posted January 10, 2023

Last modified March 14, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Grigori Avramidi, Max Planck Institute for Mathematics

Division, group rings, and negative curvature

In 1997 Delzant observed that fundamental groups of hyperbolic manifolds with large injectivity radius have nicely behaved group rings. In particular, these rings have no zero divisors and only the trivial units. In this talk I will discuss an extension of this observation showing that such rings have a division algorithm (generalizing the division algorithm for group rings of free groups discovered by Cohn) and ``freedom theorems’’ saying ideals generated by two elements are free (which can be viewed as generalizations from subgroups to ideals of some freedom theorems of Delzant and Gromov). This has geometric consequences to the homotopy classification of 2-complexes with surface fundamental groups and to complexity of cell structures on hyperbolic manifolds.

Posted January 10, 2023

Last modified March 13, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Jonathan Johnson, Oklahoma State University

Non-standard orders on torus bundles with one boundary

Consider a torus bundle over the circle with one boundary. Perron-Rolfsen shows that having an Alexander polynomial with real positive roots is a sufficient condition for a surface bundle with one boundary to have bi-orderable fundamental group. This is done by showing the action induced by the monodromy preserves a "standard" bi-ordering of the fundamental group of the surface. In this talk, we discuss if there are other ways to bi-order the fundamental group of a torus bundle with one boundary component. This work is joint with Henry Segerman. This work is partially funded by NSF grant DMS-2213213.

Posted January 18, 2023

Last modified August 7, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Allison Miller, Swarthmore

Generalizing sliceness

A knot in the 3-sphere is said to be smoothly slice if it bounds a smoothly embedded disc in the 4-ball. Sliceness questions are closely related to interesting phenomena in 4-manifold topology: for example, the existence of a non smoothly slice knot that bounds a flatly embedded disc can be used to give a relatively quick proof of the existence of nonstandard smooth structures on 4-dimensional euclidean space. There are (at least!) two reasonable generalizations of sliceness to arbitrary 4-manifolds: in each of these directions, we will highlight open questions and give some results from joint work with Kjuchukova-Ray-Sakallı and Marengon-Ray-Stipsicz.

Posted January 10, 2023

Last modified April 4, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Giovanni Paolini, Caltech

The K(π,1) conjecture

Artin groups are a generalization of braid groups, and arise as the fundamental groups of configuration spaces associated with Coxeter groups. A long-standing open problem, called the K(π,1) conjecture, states that these configuration spaces are classifying spaces for the corresponding Artin groups. In the case of finite Coxeter groups, this was proved by Deligne in 1972. In the first part of this talk I will introduce Coxeter groups, Artin groups, and the K(π,1) conjecture. Then I will outline a recent proof of the K(π,1) conjecture in the affine case and further developments in the hyperbolic case. This is joint work with Mario Salvetti and Emanuele Delucchi.

Posted January 10, 2023

Last modified August 7, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Chindu Mohanakumar, Duke University

DGA maps Induced by Decomposable Fillings with Z-coefficients

To every Legendrian link in R3, we can assign a differential graded algebra (dga) called the Chekanov-Eliashberg DGA. An exact Lagrangian cobordism between two Legendrian links induces a DGA map between the corresponding Chekanov-Eliashberg DGAs, and this association is functorial. This DGA map was written down explicitly for exact, decomposable Lagrangian fillings as Z2-count of certain pseudoholomorphic disks by Ekholm, Honda, and Kálmán, and this was combinatorially upgraded to an integral count by Casals and Ng. However, this upgrade only assigned an automorphism class of DGA maps. We approach the same problem of integral lifts by a different strategy, first done for the differential in the Chekanov-Eliashberg DGA by Ekholm, Etnyre, and Sullivan. Here, we find the precise DGA maps for all exact, decomposable Lagrangian cobordisms through this more analytic method.

Posted January 10, 2023

Last modified April 19, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Ying Hu, University of Nebraska Omaha

Left-orderability of branched covers and link orientations

A non-trivial group G is called left-orderable (LO) if G admits a total linear order invariant under left-multiplication. We call a 3-manifold is LO if its fundamental group is. In this talk, we will discuss the (in)dependence of the left-orderability of cyclic branched covers of a link on the orientation of the link. We show that for certain links, such as fibered strongly quasi-positive hyperbolic links, changing the link's orientation does not affect the LO of the cyclic branched covers. Our proof involves a construction that mutates the Homeo(S^1) representations obtained from "pseudo-Anosov" flows on $3$-orbifolds. We will discuss additional applications of this construction. This is joint work with Steve Boyer and Cameron Gordon.

Posted January 11, 2023

Last modified April 29, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Ben Knudsen, Northeastern University

Farber's conjecture and beyond

Topological complexity is a numerical invariant quantifying the difficulty of motion planning; applied to configuration spaces, it measures the difficulty of collision-free motion planning. In many situations of practical interest, the environment is reasonably modeled as a graph, and the topological complexity of configuration spaces of graphs has received significant attention for this reason. This talk will discuss a proof of a conjecture of Farber, which asserts that this invariant is as large as possible in the stable range, and of an analogue of this result in the setting of unordered configuration spaces.

Posted April 3, 2023

Last modified May 8, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Oleg Lazarev, University of Massachusetts Boston

Localization and flexibilization in symplectic geometry

Localization is an important construction in algebra and topology that allows one to study global phenomena a single prime at a time. Flexibilization is an operation in symplectic topology introduced by Cieliebak and Eliashberg that makes any two symplectic manifolds that are diffeomorphic (plus a bit of tangent bundle data) become symplectomorphic. In this talk, I will explain joint work with Sylvan and Tanaka that shows that flexibilization is a localization functor of a certain category of symplectic manifolds and also constructs new localization functors of symplectic manifolds associated to primes P. These P-flexibilization functors interpolate between rigidity and flexibility and are a symplectic analog of topological localization of Sullivan, Quillen, and Bousfield. I will also describe work joint work with Datta, Mohanakumar, and Wu that gives explicit descriptions of Legendrians used to create P-flexible symplectic manifolds.

Posted July 7, 2023

Last modified August 23, 2023

Geometry and Topology Seminar Seminar website

1:30 pm Lockett 233
Rima Chatterjee, University of Cologne

Knots in overtwisted manifolds

Knots in contact manifolds are interesting objects to study. In this talk, I'll focus on knots in overtwisted manifolds with tight complements also known as non-loose knots. These knots play an important role in contact topology as one can get interesting tight contact manifolds by doing surgery on them, though they appear to be rare. I'll discuss some of their classification results and how they differ from knots in tight manifolds. Next I’ll show that the cabling construction of these knots will give us a family of non-loose knots but only under certain conditions. Part of this is joint work with Geiges-Onaran and with Etnyre, Min and Mukherjee.

Posted July 7, 2023

Last modified August 23, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Yu Chan Chang, Wesleyan University

The RAAG Recognition Problem for Bestvina–Brady Groups

Right-angled Artin groups (RAAGs) have been a central object of study in geometric group theory because they contain many interesting subgroups. The RAAG Recognition Problem seeks to determine whether a given group is isomorphic to a RAAG. In this talk, we will discuss this recognition problem for Bestvina–Brady groups (BBGs). I will describe a graph-theoretic condition that certifies a BBG as a RAAG. On the other hand, I will use the BNS-invariants of BBGs to demonstrate that certain BBGs are not RAAGs. Finally, we will see a complete solution to the RAAG Recognition Problem for the BBGs defined on 2-dimensional flag complexes. This is joint work with Lorenzo Ruffoni.

Posted August 15, 2023

Last modified August 31, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
David Sheard, King's College London

Reflection and Nielsen equivalence in Coxeter groups

Nielsen equivalence – the natural notion of equivalence between generating sets of finitely generated groups – has been studied for the last century. Early techniques were often combinatorial, however more modern approaches use algebra and geometric/topological methods. In the last decade significant progress has been made studying it in surface and Fuchsian groups. In this talk I will introduce Nielsen equivalence in Coxeter groups, a class of groups with very rich geometry, and a related notion called reflection equivalence which is specialised for reflections. I will prove that any reflect generating set of a Coxeter group is equivalent to a “geometrically simple” generating set and provide a complete classification in some classes of Coxeter groups. Time permitting, I will also mention one approach to Nielsen equivalence in the right-angled case based on recent work generalising Stallings’ folds to RACGs.

Posted September 6, 2023

Last modified September 11, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Agniva Roy, Louisiana State University

Symplectic fillings of lens spaces and torus bundles

Classifying minimal strong symplectic fillings of contact manifolds is a problem with a rich history. The first result was by Eliashberg who showed that the 4-ball is the unique such filling of the standard tight contact 3-sphere. Since then, various techniques have been developed, by Eliashberg, McDuff, Wendl, Lisca, Christian-Menke, and Lisi-van Horn-Morris-Wendl, among others, to study this problem using holomorphic curves, and the literature has seen complete classifications for certain families of 3-manifolds -- notably lens spaces. I will discuss the primary techniques that are used to study these problems and talk about some of my own work in classifying fillings of lens spaces and developing techniques to understand fillings of torus bundles -- the former joint with Etnyre, and the latter joint with Min and Wang.

Posted August 1, 2023

Last modified September 12, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
James Hughes, Duke University

Legendrian loops and cluster modular groups

Given a Legendrian link L in the contact 3-sphere, one can hope to classify the set of exact Lagrangian fillings of L, i.e. exact Lagrangian surfaces in the symplectic 4-ball with boundary equal to L. Much of the recent progress towards this classification relies on the theory of cluster algebras. In this talk, I will describe a cluster structure on the augmentation variety of Legendrian positive braid closures. I will then discuss how Legendrian loops -- Legendrian isotopies fixing L setwise -- yield cluster automorphisms of the augmentation variety. These automorphism groups behave very similarly to mapping class groups and I will use this connection to describe generating sets, fixed point properties, and some applications to the mapping tori of Legendrian loops.

Posted September 6, 2023

Last modified September 22, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Ana Bălibanu, Louisiana State University

Moment maps and multiplicative reduction

Symplectic reduction is a process that eliminates the symmetries of a Poisson manifold equipped with a Hamiltonian group action. Many algebraic varieties which are of interest to representation theory arise as reductions of symplectic spaces associated to algebraic groups. We introduce several new reduction procedures, some of which are multiplicative analogues of ”classical” examples of symplectic reduction. This is joint work with Maxence Mayrand.

Posted August 23, 2023

Last modified September 27, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Yuan Yao, Sorbonne University

Morse-Bott theory and embedded contact homology

Embedded contact homology (ECH) is a Floer theory associated to a contact 3-manifold $(Y,\lambda)$. Its generators are orbits of the Reeb vector field and its differential counts pseudo-holomorphic curves. It has been shown to be isomorphic to versions of Heegaard Floer homology and monopole Floer homology. It has wide ranging applications in the study of symplectic topology and dynamics, for example in finding obstructions to symplectic embeddings and studying periodic orbits of Reeb vector fields. In this talk I will first give an overview of ECH, then explain how to compute ECH in certain Morse-Bott settings via enumerations of J-holomorphic cascades. One of the main technical tools will be how to do Morse-Bott theory in the pseudo-holomorphic curves setting.

Posted August 20, 2023

Last modified September 28, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Wenyuan Li, Northwestern University

Conjugate fillings and Legendrian weaves

In this talk, we compare two different approaches to constructing Lagrangian fillings of Legendrian knots. The first one is conjugate Lagrangian fillings of alternating Legendrians, introduced by Shende-Treumann-Williams-Zaslow, which are characterized using bipartite graphs, and the second one is Lagrangian projections of Legendrian weaves, introduced by Casals-Zaslow, which are depicted by planar graphs encoding their wavefronts. We will develop a diagrammatic calculus to show that conjugate Lagrangian fillings are Hamiltonian isotopic to certain Lagrangian projections of Legendrian weaves. The result includes Legendrian positive braid closures and ideal triangulations on punctured surfaces. We will then explain some implications on Lagrangian mutations and cluster theory. This is joint work with Roger Casals.

Posted August 17, 2023

Last modified October 15, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Laura Wakelin, Imperial College London

Characterising slopes for knots of hyperbolic type

A slope p/q is characterising for a knot K in the 3-sphere if the oriented homeomorphism type of the manifold obtained by performing Dehn surgery of slope p/q on K uniquely determines the knot K. Sorya showed that for any knot K, there exists a constant C(K) such that any slope p/q with |q|≥C(K) is characterising for K. However, the proof of the existence of C(K) in the general case is non-constructive, which naturally evokes the question of how to compute explicit values for C(K). In this talk, I will explore methods for finding C(K) in the case where K is a knot of hyperbolic type (meaning that the JSJ decomposition of its complement has a hyperbolic outermost JSJ piece). This is ongoing joint work with Patricia Sorya.

Posted August 20, 2023

Last modified October 12, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Amanda Wilkens, University of Texas

Poisson-Voronoi tessellations and fixed price in higher rank

We overview the cost of a group action and the ideal Poisson-Voronoi tessellation (IPVT), a new random limit with interesting geometric features. In recent work, we use the IPVT to prove all measure preserving and free actions of a higher rank semisimple Lie group on a standard probability space have cost 1, answering Gaboriau's fixed price question for this class of groups. This further implies results on the rank gradient and growth of first mod-p homology groups. We sketch a proof, which relies on some simple dynamics of the group action and the definition of a Poisson point process. No prior knowledge on cost or IPVTs will be assumed. This is joint work with Mikolaj Fraczyk and Sam Mellick.

Posted August 23, 2023

Last modified October 27, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Mustafa Hajij, University of San Francisco

Topological Deep Learning: Going Beyond Graph Data

Over the past decade, deep learning has been remarkably successful at solving a massive set of problems on datatypes including images and sequential data. This success drove the extension of deep learning to other discrete domains such as sets, point clouds, graphs, 3D shapes, and discrete manifolds. While many of the extended schemes have successfully tackled notable challenges in each domain, the plethora of fragmented frameworks have created or resurfaced many long-standing problems in deep learning such as explainability, expressiveness and generalizability. Moreover, theoretical development proven over one discrete domain does not naturally apply to the other domains. Finally, the lack of a cohesive mathematical framework has created many ad hoc and inorganic implementations and ultimately limited the set of practitioners that can potentially benefit from deep learning technologies. This talk introduces the foundation of topological deep learning, a rapidly growing field that is concerned with the development of deep learning models for data supported on topological domains such as simplicial complexes, cell complexes, and hypergraphs, which generalize many domains encountered in scientific computations including images and sequence data. It introduces the main notions while maintaining intuitive conceptualization, implementation and relevance to a wide range of practical applications. It also demonstrates the practical relevance of this framework with practical applications ranging from drug discovery to mesh and image segmentation.

Posted August 27, 2023

Last modified October 25, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Biji Wong, Duke University

Branched double covers of links in the 3-sphere, involutions, and bordered Floer theory

Branched double covers $\Sigma_2(L)$ of links $L$ in the 3-sphere are a nice tractable class of 3-manifolds that can be studied using Heegaard Floer theory. In this talk, we will discuss recent work to compute the Heegaard Floer d-invariants of $\Sigma_2(L)$ for L a 2-component plumbing link, using involutions and tools from bordered Floer theory. We will also give new families of links L where the d-invariants of $\Sigma_2(L)$ (in the spin structures) are determined by the signatures of L. This project is joint with J. Hanselman and M. Marengon.

Posted September 6, 2023

Last modified November 1, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Federico Salmoiraghi, Queen's University

Surgery on Anosov flows using bi-contact geometry

An example of the beautiful intertwine between hyperbolic dynamics, foliation theory, and contact geometry is given by an Anosov flow. Geometrically an Anosov flow is defined by two transverse invariant foliations with expanding and contracting behaviors. Much of our understanding of the structure of an Anosov flow relies on the study of the leaves space of the invariant foliations. Mitsumatsu first noticed that an Anosov vector field also belongs to the intersection of two transverse contact structures rotating towards each other. After giving the necessary background, I will show how to use this point of view to address questions in the theory of surgery on Anosov flows.

Posted October 12, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Basile Coron, Queen Mary University of London

Operadic structures and matroid invariants

We will begin with a general discussion on operadic structures, and then show that many well-known matroid invariants come naturally equipped with such structures. Applying the general tools of operadic theory such as Grobner bases and Koszul duality will shed new light on those invariants.