Calendar
Calendar
Posted August 29, 2024
Last modified October 7, 2024
Geometry and Topology Seminar Seminar website
3:30 pm
Bin Sun, Michigan State University
$L^2$-Betti numbers of Dehn fillings
I will talk about a recent joint work with Nansen Petrosyan where we obtain conditions under which $L^2$-Betti numbers are preserved by group-theoretic Dehn fillings. As an application, we verify the Singer Conjecture for certain Einstein manifolds and provide new examples of hyperbolic groups with exotic subgroups. We also establish a virtual fibering criterion and obtain bounds on deficiency of Dehn fillings. A key step in our approach of computations of $L^2$-Betti numbers is the construction of a tailored classifying space, which is of independent interest.
Posted October 7, 2024
Last modified October 28, 2024
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Monika Kudlinska, University of Cambridge
Solving equations in free-by-cyclic groups
A group G is said to be free-by-cyclic if it maps onto the infinite cyclic group with free kernel of finite rank. Free-by-cyclic groups form a large and widely-studied class with close links to 3-manifold topology. A group G is said to be equationally Noetherian if any system of equations over G is equivalent to a finite subsystem. In joint work with Motiejus Valiunas we show that all free-by-cyclic groups are equationally Noetherian. As an application, we deduce that the set of exponential growth rates of a free-by-cyclic group is well ordered.
Posted September 17, 2024
Last modified November 11, 2024
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Arka Banerjee, Auburn University
Urysohn 1-width and covers
A metric space has small Urysohn 1-width if it admits a continuous map to a 1-dimensional complex where the preimage of each point has small diameter. An open problem is, if a space's universal cover has small Urysohn 1-width, must the original space also have small Urysohn 1-width? While one might intuitively expect this to be true, there are strange examples that suggest otherwise. In this talk, I will explore the motivations behind this question and discuss some partial progress we have made in understanding it. This is a joint work with H. Alpert and P. Papasoglu.
Posted December 11, 2024
Last modified January 27, 2025
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Akram Alishahi, University of Georgia
Contact invariants in Heegaard Floer homology
Over the past two decades multiple invariants of contact structures have been defined in different variations of Heegaard Floer homology. We will start with an overview of these invariants and their connections. Then, we will discuss one of these invariants that is defined for a contact 3-manifold with a foliated boundary and lives in bordered sutured Floer homology in more details. This is a joint work with Földvári, Hendricks, Licata, Petkova and Vertesi.
Posted January 23, 2025
Last modified January 27, 2025
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Matthew Stoffregen, Michigan State University
Pin(2) Floer homology and the Rokhlin invariant
In this talk, we describe a family of homology cobordism invariants that can be extracted from Pin(2)-equivariant monopole Floer homology (using either Manolescu or Lin's definitions), that have some properties in common with both the epsilon and upsilon invariants in knot Floer homology. We'll show a relationship of this family to questions about torsion in the homology cobordism group, and to triangulation of higher-dimensional manifolds. This is joint work in progress with Irving Dai, Jen Hom, and Linh Truong.
Posted November 12, 2024
Last modified February 10, 2025
Geometry and Topology Seminar Seminar website
2:30 pm
Porter Morgan, University of Massachusetts Amherst
Irreducible 4-manifolds with order two fundamental group
Let R be a closed, smooth, oriented 4–manifold with order two fundamental group. The works of Freedman and Hambleton-Kreck show that R is determined up to homeomorphism by just a few basic properties. That said, there are often many different manifolds that are homeomorphic to R, but not diffeomorphic to it or each other. In this talk, we’ll describe how to construct irreducible copies of R; roughly speaking, these are smooth manifolds that are homeomorphic to R, and don’t decompose into non-trivial connected sums. We’ll show that if R has odd intersection form and non-negative first Chern number, then in all but seven cases, it has an irreducible copy. We’ll describe some of the techniques used to realize these irreducible smooth structures, including torus surgeries, symplectic fiber sums, and a novel approach to constructing Lefschetz fibrations equipped with free involutions. This is joint work with Mihail Arabadji.
Posted February 6, 2025
Last modified February 12, 2025
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Neal Stoltzfus, Mathematics Department, LSU
Discrete Laplacians, Ribbon Graphs, and Link Polynomials
The Whitney homology of the independence lattices of the state space of a ribbon graphs supports three independent anti-commuting discrete Laplacians. They relate to the three fundamental combinatorial invariants of independent subsets: rank, nullify and genus. We explore the combinations that give link invariants.
Posted January 14, 2025
Last modified February 18, 2025
Geometry and Topology Seminar Seminar website
3:30 pm
Dave Auckly, Kansas State University
Restrictions on the genus of trivial families of surfaces in twisted families of 4-manifolds
Several notions of equivalence in topology may be expressed via the existence of families. Thus, asking when an untwisted family of surfaces can be placed in a twisted family of manifolds in a natural question. This talk will describe a generalized adjunction inequality for families.
Posted March 4, 2025
Last modified March 10, 2025
Geometry and Topology Seminar Seminar website
2:30 pm Lockett 233
Maarten Mol, University of Toronto
Constructibility of momentum maps and variation of singular symplectic reduced spaces (Joint with Mathematical Physics and Representation Theory Seminar)
Proper maps in various categories studied in singularity theory (for example, the real analytic category) are known to be constructible, in the sense that the image of the map can be stratified in such a way that the map is a topological fiber bundle over each stratum. Such stratifications provide insight into how the fibers of the map vary. In this talk we will discuss the existence of such a stratification for momentum maps of Hamiltonian Lie group actions (a natural class of maps studied in symplectic/Poisson geometry), which provides insight into how the so-called symplectic reduced spaces of the Hamiltonian action vary. Along the way we will also try to give an overview of some more classical results on the geometry of such maps.
Posted March 10, 2025
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Scott Baldridge, Louisiana State University
A new way to prove the four color theorem using gauge theory
In this talk, I show how ideas coming out of gauge theory can be used to prove that certain configurations in the list of "633 unavoidable's" are reducible. In particular, I show how to prove the most important initial example, the Birkhoff diamond (four “adjacent" pentagons), is reducible using our filtered $3$- and $4$-color homology. In this context reducible means that the Birkhoff diamond cannot show up as a “tangle" in a minimal counterexample to the 4CT. This is a new proof of a 111-year-old result that is a direct consequence of a special (2+1)-dimensional TQFT. I will then indicate how the ideas used in the proof might be used to reduce the unavoidable set of 633 configurations to a much smaller set. This is joint work with Ben McCarty.
Posted March 26, 2025
Geometry and Topology Seminar Seminar website
2:30 pm – 3:30 pm Lockett 233
Scott Baldridge, Louisiana State University
A new way to prove the four color theorem using gauge theory, Part 2
This is a continuation of last week’s talk in which we explain the definition of the homology theory used to prove that Birkhoff’s diamond is reducible. I will quickly summarize last week's discussion before heading into new material, so people can attend this week even if they couldn’t attend last week. This is joint work with Ben McCarty at University of Memphis.
Posted January 23, 2025
Last modified April 29, 2025
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Annette Karrer, The Ohio State University
Connected Components in Morse boundaries of right-angled Coxeter groups
Every finitely generated group G has an associated topological space, called a Morse boundary, that captures the hyperbolic-like behavior of G at infinity. It was introduced by Cordes generalizing the contracting boundary invented by Charney--Sultan. In this talk, we study subgroups arising from connected components in Morse boundaries of right-angled Coxeter groups and of such that are quasi-isom
Posted September 1, 2025
Last modified September 10, 2025
Geometry and Topology Seminar Seminar website
3:30 am Lockett 233
Kyle Binder, Louisiana State University
Cohomology of Toric Varieties Associated with Matroids
The Chow ring of a matroid is an important tool in studying the combinatorics of matroids through geometric techniques, and it played a central role in the Adiprasito, Huh, and Katz proof of the Rota—Heron—Welsh conjecture for matroids. This ring is defined to be the Chow ring of the smooth, quasi-projective toric variety associated with the Bergman fan of the matroid, and, remarkably, it enjoys many of the Hodge-theoretic properties of Chow rings of smooth, projective varieties. In this talk, we will extend the Chow ring of these toric varieties to the larger (singular) cohomology ring, compute the top-graded piece of cohomology in terms of the associated matroid, and describe how to compute all of the Betti numbers in the case of uniform matroids.
Posted September 1, 2025
Last modified September 15, 2025
Geometry and Topology Seminar Seminar website
3:30 am Lockett 233
Kevin Schreve, Louisiana State University
L^2-homology of right-angled Coxeter groups
A flag triangulation of an (n-1)-dimensional sphere determines a right-angled Coxeter group and a closed n-manifold which is a K(G,1) for the commutator subgroup. The Singer Conjecture predicts that the L^2-homology of the universal cover is only nonzero in dimension n/2. We will show the Singer conjecture holds if 1) L is the barycentric subdivision of the boundary of a simplex, 2) L is the barycentric subdivision of a triangulation of an odd-dimensional sphere Based on joint work with Grigori Avramidi and Boris Okun.
Posted September 1, 2025
Last modified September 23, 2025
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Robin Koytcheff, University of Louisiana, Lafayette
Milnor invariants and thickness of spherical links
Various authors have studied the question of how long a rope of a given thickness is needed to tie a given isotopy class of knot or link. In joint work with Rafal Komendarczyk and Fedya Manin, we generalize this work to spherical links in arbitrary dimensions. In more detail, we study their Milnor invariants in terms of Massey products and prove asymptotically optimal upper bounds on Milnor invariants in terms of embedding thickness. Interestingly, there is a dichotomy between polynomial and exponential bounds, depending on the dimensions of the spheres. We apply our results to answer a question of Freedman and Krushkal about exponentially thin 2-complexes in 4-space.
Posted September 29, 2025
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Scott Baldridge, Louisiana State University
What does the Four Color Theorem have to do with the sound a drum makes?
Mark Kac famously asked in 1966, “Can one hear the shape of a drum?” While the answer to this question is now known to not be true in general, it popularized an investigation into eigenvalues of Laplacians that continues to this day. One formulation of it is as follows: two closed Riemannian manifolds are said to {\em isospectral} if the eigenvalues of their Laplace-Beltrami operator, counted with multiplicities, coincide. Modern questions ask to what extent having the same eigenvalues determine the geometry of the two manifolds. In this talk, we introduce a Laplace-de Rham operator on a cochain complex derived from a cellularly embedded graph into a surface. (When the surface is a $2$-sphere, this is simply a plane graph.) In degree zero, the dimension of the subspace of the harmonic solutions to this operator counts the number of $4$-face colorings of the graph. Therefore, there are zero eigenvalue solutions for a plane graph if and only if the graph does not have a bridge (the Four Color Theorem). The nonzero eigenvalues of this operator are also quite interesting, which leads us to pose the following isospectral conjecture by the end of the talk, "Can one hear the shape of the CW structure of a surface?”
Posted October 6, 2025
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Scott Baldridge, Louisiana State University
How to Build a Toy 2+1 ``Theory of Everything’’ model of the universe in 137 Easy Steps
Can a simple 2D combinatorial model already show us how to fuse matter and geometry into one quantum framework? String Theory (with its background-choice and vacuum-multiplicity issues) and Loop Quantum Gravity (with its dynamical ambiguities) both leave gaps. To keep the talk simple, I stay on a closed 2D surface and use metric triangulations to build a refinement-invariant Penrose polynomial (invariant under 1-3 Pachner refinements) that, under resampling, converges to a smooth metric. This polynomial is then an invariant of the triangulation-refinement class of a Riemannian manifold. I next tie the Penrose polynomial to the Regge action to produce a quantum gravity action whose equations of motion match the Einstein equations of general relativity (in 2D), and I use 2-2 Pachner flips as a ``discrete time step’’ in the toy model to illustrate dynamics. The talk focuses on explicit, easy-to-follow graph constructions and computations suitable for graduate students (and advanced undergraduates). If time, I conclude by outlining how the same blueprint extends to 3D, actual spacetime, where the model becomes genuinely dynamical. Note: The 137 steps is obviously a joke! It’s more like 35 steps, but I’ll only show you a few of them to give you the idea of how it works. Also: This talk is NOT a continuation of last week’s talk. However, the full theory does use aspects of it for those who attended.
Posted September 26, 2025
Last modified October 13, 2025
Geometry and Topology Seminar Seminar website
1:30 pm Virtual
Naageswaran Manikandan, Max Planck Institute
Obstructions to positivity notions using Khovanov-type theories.
In this talk, we discuss how Khovanov homology theories can be employed to construct obstructions to various notions of positivity in knot theory. We begin by discussing a result showing that, for a positive link, the first Khovanov homology is supported in a single quantum grading, is free abelian, and its rank reflects whether the link is fibered. We extend these results to (p,q)-cables of positive knots whenever $q \geq p$. We then turn to ongoing work investigating how odd-Khovanov homology and Khovanov-Rozansky homology can be used to construct obstructions to these positivity notions.
Posted September 1, 2025
Last modified October 9, 2025
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Matthew Haulmark, UT Rio Grande Valley
Cubes from Divisions
Actions on CAT(0) cube complexes have played an important role in advances in low-dimensional topology. Most notably, they are central to Wise's Quasiconvex Hierarchy Theorem and Agol's proof of the Virtual Haken Conjecture. In group theory, one way of obtaining an action on a cube complex is via the Sageev construction. Given a group G and a collection of codimension-1 subgroups of G, Sageev's construction gives an isometric action on a CAT(0) cube complex. In recent work with Jason Manning, we give an alternate route to the Sageev construction, which is potentially applicable to new situations. Much of this talk will be spent on background. We will introduce the notion of a wall space, as well as the cube complex dual to a wallspace. We will then construct an action on a CAT(0) cube complex given a group action on a sufficiently nice topological space and a system of divisions of that space.
Posted September 1, 2025
Last modified September 23, 2025
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Chen Zhang, Stony Brook University
TBA
TBA
Posted August 21, 2025
Last modified October 9, 2025
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Matthew Zaremsky, University at Albany (SUNY)
On the Sigma-invariants of pure symmetric automorphism groups
An automorphism of the free group F_n is "pure symmetric" if it sends each generator to a conjugate of itself. The group of all pure symmetric automorphisms of F_n, sometimes called the "McCool group" of F_n, is an interesting and important group with connections to braid groups, motion planning, and mathematical physics. The "Sigma-invariants" of a group are a family of geometric invariants due to Bieri, Neumann, Strebel, and Renz, which are notoriously difficult to compute in general, but reveal a wealth of information about the group and its fibering properties. In recent joint work with Mikhail Ershov, we compute large parts of the Sigma-invariants of the McCool groups, and in particular prove that they are always either empty or dense in the relevant character sphere. One key tool to highlight is an underutilized criterion due to Meinert, which seems likely to have additional future applications.
Posted September 1, 2025
Last modified September 23, 2025
Geometry and Topology Seminar Seminar website
3:30 pm TBA
Jayden Wang, University of Michigan
TBA
TBA
Posted September 10, 2025
Last modified September 23, 2025
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Corey Bregman, Tufts University
TBA
TBA