Calendar
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 October 29, 2025
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Chen Zhang, Simons Center for Geometry and Physics
Plane Floer homology and the odd Khovanov homology of 2-knots
In this talk, I will discuss joint work with Spyropoulous and Vidyarthi in which we prove a conjecture of Migdail and Wehrli regarding the maps which odd Khovanov homology associates to knotted spheres. Our main tool is the spectral sequence from reduced OKH to Plane Floer homology.
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 November 10, 2025
Geometry and Topology Seminar Seminar website
3:30 pm Locket 233
Jayden Wang, University of Michigan
From Euclid’s first postulate to Lorentzian polynomials
Imagine a world where our basic intuition about points, lines, and planes no longer applies—a world where three points in a three-dimensional linear space need not lie in any plane. This is the tropical world. I will tell a story about tropical linear spaces, where many familiar incidence properties of classical linear geometry fail in surprising ways. I will also discuss how both the fulfillment and the violation of these properties resonate across other areas of mathematics, including algebraic curves, Lorentzian polynomials, and matroid combinatorics.
Posted November 5, 2025
Last modified November 17, 2025
Geometry and Topology Seminar Seminar website
1:30 pm Online
Advika Rajapakse, UCLA
Four Steenrod squares on Khovanov homology
The odd Khovanov spectrum is a space-level link invariant that, after taking (reduced) cohomology, recovers odd Khovanov cohomology. We use the second Steenrod square to disprove several conjectures regarding the odd Khovanov spectrum. We furthermore prove that there exists more than one even Khovanov spectrum, answering another question from Sarkar-Scaduto-Stoffregen.
Posted September 10, 2025
Last modified December 2, 2025
Geometry and Topology Seminar Seminar website
1:30 pm Lockett 233
Corey Bregman, Tufts University
Diffeomorphism groups of reducible 3-manifolds
Let M be a smooth, compact, connected orientable 3-manifold. A classical result of Kneser and Milnor states that M admits a connected sum decomposition into prime factors, unique up to reordering. We introduce a topological poset of embedded 2-spheres in M and use it to study the classifying space BDiff(M) for the diffeomorphism group of M. We prove that if M is closed then BDiff(M) has finite type, and if M has non-empty boundary then BDiff(M rel ∂M) is homotopy equivalent to a finite CW complex. The proof will take us on crash course through classical 3-manifold topology and geometrization. This is joint work with Rachael Boyd and Jan Steinebrunner.
Posted January 28, 2026
Geometry and Topology Seminar Seminar website
1:30 pm 233 Lockett Hall
Konrad Wrobel, University of Texas at Austin
Measure equivalence classification of Baumslag-Solitar groups
We complete the classification of Baumslag-Solitar groups up to measure equivalence by showing all Baumslag-Solitar groups with nonunimodular Bass-Serre tree are measure equivalent (i.e., BS(r,s) with r between 1 and s). The proof makes critical use of combinatorial descriptive set theory tools in the measure class preserving setting and passes through the world of measure equivalence of nonunimodular locally compact groups. In particular, as an intermediate step we obtain measure equivalence couplings between all groups of the form Aut(T_{r,s}) for r between 1 and s where T_{r,s} is the directed tree with r incoming edges and s outgoing edges at each vertex. This is joint work with Damien Gaboriau, Antoine Poulin, Anush Tserunyan, and Robin Tucker-Drob.
Posted January 28, 2026
Last modified February 3, 2026
Geometry and Topology Seminar Seminar website
1:30 pm 233 Lockett Hall
Joshua Sabloff, Haverford College
On the Non-Orientable Genera of a Knot: Connections and Comparisons
We define a new quantity, the Euler-normalized non-orientable genus, to connect a variety of ideas in the theory of non-orientable surfaces bounded by knots. We use this quantity to explore the geography of non-orientable surfaces bounded by a fixed knot in 3 and 4 dimensions. In particular, we will use the Euler-normalized non-orientable genus to reframe non-orientable slice-torus bounds on the (ordinary) non-orientable 4-genus and to bound below the Turaev genus as a measure of distance to an alternating knot. This is joint work with Julia Knihs, Jeanette Patel, and Thea Rugg.
Posted January 28, 2026
Last modified February 17, 2026
Geometry and Topology Seminar Seminar website
1:30 pm 233 Lockett Hall
Nilangshu Bhattacharyya, Louisiana State University
Steenrod Square on Khovanov Homology
Khovanov homology assigns a knot or a link to a bigraded homology theory that categorifies the Jones polynomial. It has concrete applications, for instance Rasmussen’s $s$-invariant, extracted from Lee’s deformation, which gives a lower bound on the smooth slice genus. At the same time, while the theory is very combinatorial and closely tied to the representation theory of $U_q(\mathfrak{sl}_2)$, it can be hard to see the underlying geometric picture directly from the homology groups. The stable homotopy refinement, introduced by Lipshitz and Sarkar, upgrades Khovanov homology to a space-level invariant: a spectrum whose cohomology recovers Khovanov homology while supporting additional structure that is invisible at the level of homology. This refinement induces stable cohomology operations, such as Steenrod squares, on Khovanov homology. In this talk, I will explain how to compute $Sq^1$ and $Sq^2$ on Khovanov homology.
Posted January 28, 2026
Last modified February 17, 2026
Geometry and Topology Seminar Seminar website
1:30 pm 233 Lockett Hall
Nir Gadish, University of Pennsylvania
Letter braiding invariants of words in groups
How can we tell if a group element can be written as k-fold nested commutator? One way is to find a collection of computable function that vanish only on nested commutators. This talk will introduce letter-braiding invariants - these are elementarily defined functions on words, inspired by the homotopy theory of loop-spaces and carrying deep geometric content. They give a universal finite-type invariant for arbitrary groups, extending the influential Magnus expansion of free groups that already had countless applications in low dimensional topology. As a consequence we get new geometric formulas for braid and link invariants, and a way to linearize automorphisms of general groups that specializes to the Johnson homomorphism of mapping class groups.
Posted January 19, 2026
Last modified March 3, 2026
Geometry and Topology Seminar Seminar website
1:30 pm Virtual
Ettore Marmo, Università degli Studi di Milano-Bicocca
Toric arrangements and Bloch-Kato pro-p groups
For every field K we can define a distinguished extension K^{sep} called its separable closure. The maximal pro-p quotient G_K(p) of the Galois group $G_K = Gal(K^{sep}/K)$ is called the maximal pro-p Galois group of K, many arithmetical properties of the field are encoded in the structure of this group. It is interesting to ask which pro-p groups can be realized as the maximal pro-p Galois group of some field. It is known that any such group must also satisfy a cohomological property called the Bloch-Kato property. In this talk we will discuss some families of pro-p groups arising as from toric arrangements and some techniques to study the Bloch-Kato property in this context. This talk is based on joint work with Th. Weigel and E. Delucchi.
Posted March 17, 2026
Geometry and Topology Seminar Seminar website
1:30 pm Virtual
Jonathan Fruchter, University of Bonn
Virtual homological torsion in low dimensions
A long-standing conjecture of Bergeron and Venkatesh predicts that in closed hyperbolic 3-manifolds, the amount of torsion in the first homology of finite-sheeted normal covers should grow exponentially with the degree of the cover as the covers become larger, at a rate reflecting the volume of the manifold. Yet no finitely presented residually finite group is known to exhibit such behaviour, and meaningful lower bounds on torsion growth are rare. In this talk I will explain how a particular two-dimensional lens offers a clearer view of some of the underlying mechanisms that create homological torsion in finite covers, and how they might relate to its growth. If time allows, I will also discuss how these ideas connect to the question of profinite rigidity: how much information about a group is encoded in its finite quotients.
Posted March 27, 2026
Last modified April 6, 2026
Geometry and Topology Seminar Seminar website
1:30 pm Lockett 233
Chris Manon, University of Kentucky
Toric tropical vector bundles
A toric vector bundle is a vector bundle over a toric variety which is equipped with a lift of the action action of the associated torus. As a source of examples, toric vector bundles and their projectivizations provide a rich class of spaces that still manage to admit a combinatorial characterization. Toric vector bundles were first classified by Kaneyama, and later by Klyachko using the data of decorated subspace arrangements. Klyachko's classification is the foundation of many interesting results on toric vector bundles and has recently led to a connection between toric vector bundles, matroids, and tropical geometry. After explaining some of this background, I'll introduce the notion of a tropical toric vector bundle over a toric variety. These objects are discrete analogues of vector bundles which still have notions of positivity, a sheaf of sections, an Euler characteristic, and Chern classes. The combinatorics of these invariants can reveal properties of their classical analogues as well as point the way to new theorems for tropical vector bundles over a more general base. Time permitting, I will discuss some new results on higher Betti numbers of a tropical vector bundle.