Spring 2024

Abstract: Combinatorial Hodge theory, originally developed by Eckmann in 1944, has seen sprawling applications in applied topology, topological signal processing, spectral graph theory, discrete exterior calculus, cellular sheaf theory and other areas in the last decade. Despite this, it is still an expert subject not widely known to practitioners in signal processing. The purpose of this seminar is to show how one can construct exposition of the subject, highlighting how - perhaps surprisingly - modest prerequisites in linear algebra suffice.

Abstract: In continuation of the previous seminar on combinatorial Hodge Laplacians, we will discuss cellular sheaves and how the Hodge Laplacian generalized to the cellular Sheaf Laplacian. On the way we will cover the basic idea of cellular Sheaf Cohomology.

Abstract: Is every closed geodesic on the complex plane minus two points cut out by algebraic equations? I'll start with some motivation and explanation for this question, which I asked on Mathoverflow (link here). I'll describe a (very cool!) partial answer given by Ian Agol, and I'll try to indicate how much more there is to do.

Abstract: In this talk, we will introduce a geometric flavored boundary construction for proper geodesic spaces and finitely generated groups. We define the quasi-redirecting boundary and prove that it is QI-invariant and frequently compact. We will also show that this boundary is strongly connected to the sublinearly Morse boundaries and thus is often connected to the Poisson boundaries. An important application of this construction is the asymptotically tree-graded spaces. If time permits, We will discuss several interesting and accessible questions in this new direction. This is based on work in progress joint with Kasra Rafi.

Abstract: I will discuss strong hyperbolicity—a metric notion intermediate between CAT(-1) and Gromov hyperbolicity. Part of the motivation comes from the analytic theory of hyperbolic groups. As an application, I will sketch a short proof of the fact that hyperbolic groups admit proper actions on Lp-spaces, for sufficiently large p.

Abstract: A Kleinian action of a group is a proper, isometric action on the real hyperbolic space H³.  Every Kleinian group acts by homeomorphisms on the 2–sphere at infinity of H³.  The dynamical properties of such an action are well understood.  However, it is unknown whether any group action on S² with these dynamical properties must always be a Kleinian action.  For instance, the well-known Cannon Conjecture asks: If G is a word hyperbolic group with boundary homeomorphic to S², must G be a Kleinian group?  This question was initially proposed as a potential route towards resolving Thurston's Geometrization Conjecture (solved by Perelman using unrelated methods), but the Cannon Conjecture remains a complete mystery to this day.

Many relatively hyperbolic groups have a boundary that embeds in S², and sometimes one can get the action on the boundary to extend to an action on S².  In this case, one hopes to recognize whether the action on S² is actually a Kleinian action.  Indeed, many groups with planar boundary do arise as Kleinian groups.  I will discuss this phenomenon (which does not resolve the Cannon Conjecture) and some ideas that go into its proof.  This is joint work with Genevieve Walsh.

Abstract: We say an action of a group G on a space X recognizes all stable subgroups if every stable subgroup of G is quasi-isometrically embedded in the action on X. The problem of constructing or identifying such spaces has been extensively studied for many groups, including mapping class groups and right angled Artin groups- these are well known examples of acylindrically hyperbolic groups. In these cases, the recognizing spaces are the largest acylindrical actions for the group. One can therefore ask the question if a largest acylindrical action of an acylindrically hyperbolic group (if it exists) is a recognizing space for stable subgroups in general. We answer this question in the negative by producing an example of a relatively hyperbolic group whose largest acylindrical action fails to recognize all stable subgroups. This is joint work with Marissa Chesser, Alice Kerr, Johanna Mangahas and Marie Trin.

Abstract: In his thesis, Milnor described a certain nilpotent quotient of the fundamental group of the complement of a finite link---nowadays called the Milnor group of the link---which determines whether or not the link can be homotoped to the unlink through a homotopy in which distinct components are not allowed to intersect. The goal of this talk is to present new phenomena for infinite links. I will describe a family of infinite, periodic links whose Milnor groups are different from those of the infinite unlink, even though both have the same nilpotent quotients.

Each of these periodic links covers a knot K in a handlebody H and the Milnor group of the infinite link provides an obstruction to homotoping the knot to an unknot through a homotopy with contractible double point loops. It also provides a place to investigate the question whether the knot K bounds a pi_1-null disk in H x I (that is, an immersed disk all of whose double point loops are contractible) which---for some specific knots---is a variant of the four-dimensional surgery conjecture. Joint work with Peter Teichner. 

Abstract: In this talk, we will recall what it means to be a Z(n)-compactification.  Then focusing on n=0, we will show the end-point compactification is a Z(0)-compactification and any Type F_1 group admits a Z(0)-structure. 

Abstract: This talk will be a follow up to my prior talk focusing on Z(0)-compactifications. Besides a quick recap, the focus of this talk will be to prove the statement that all type F_1 groups (ie. finitely generated groups) admit a Z(0)-structure. 

Abstract: Cluster algebras are commutative rings with a set of recursively-defined generators. Many cluster algebras with desirable properties arise from a surface with marked points (S,M) in the sense that they can be viewed as the coordinate ring of the corresponding decorated Teichmüller space of (S,M). Oguz-Yildirim and Pilaud-Reading-Schroll have recently exhibited a method to use the set of order ideals of a poset to give a direct formula for any generator of a cluster algebra of surface type. We use this construction to give ​"skein relations", which will be multiplication formulas for elements of the cluster algebra which arise from resolving intersections of the corresponding curves on the surface. By working with surfaces which could have internal marked points, called "punctures", we generalize previously known skein relations from Musiker-Schiffler-Williams and Canakci-Schiffler, who largely only work in unpunctured surfaces. Our results are an important step forward to understanding bases of cluster algebras coming from punctured surfaces. This talk is based on joint work with Wonwoo Kang and Elizabeth Kelley, and no familiarity with cluster algebras will be assumed. 

Abstract: CAT(0) cube complexes have become a major player in geometric group theory, thanks to a combination of two factors: they are rigid enough to give useful information about groups that act on them; but they are flexible enough to allow many groups to do so. The main tool for building actions on CAT(0) cube complexes is Sageev's construction. In this talk, I'll discuss Sageev's construction, and describe a generalisation of it that allows one to construct group actions other types of metric space of interest. Based on joint work with Abdul Zalloum and Davide Spriano.

Abstract: Quotients of free products are natural combinations of groups that have been exploited to study embedding problems. These groups have seen a resurgence of attention from a more geometric point of view following celebrated work of Haglund--Wise and Agol.  I will discuss a geometric model for studying quotients of free products initiated by Martin and Steenbock. We will use this model to adapt ideas from Gromov's density model to quotients of free products, their actions on CAT(0) cube complexes, and combination theorems for residual finiteness.  Results discussed will be based on join work with Einstein, and Krishna MS, and Montee, and Steenbock.

Abstract: Fix a hyperbolic surface $X$, and for $L > 0$, let $\mathcal{S}_L(X)$ denote the set of simple closed geodesics of length at most $L$ on $X$. Fixing a norm on $H_1(X, \mathbb{R})$, we may ask what is the statistics of the norm of homology class of $\alpha$, denoted by $[\alpha]$, when $\alpha$ is chosen randomly uniformly from $\mathcal{S}_L(X)$, as $L \rightarrow \infty$? For example, does one expect the norm of $[\alpha]$ to be of order $L$ or smaller?

We answer this question by proving a CLT-type result for the norm of homology of a randomly chosen curve in $\mathcal{S}_L(X)$. We discuss the main steps to reduce the desired CLT to a CLT for the Kontsevich-Zorich cocycle obtained by Forni-Saqban.

Abstract: I will introduce the notion of halfspaces in group splittings and discuss the problem of when these halfspaces are one-ended. I will also discuss connections to JSJ splittings of groups, and to determining whether groups are simply connected at infinity. This is joint work with Michael Mihalik.

Abstract: We study infinite type surfaces and introduce an associated space of marked hyperbolic structures. Analogous to the role Teichmüller space plays in the theory for finite type surfaces, we use the space to study deformations between hyperbolic structures on infinite type surfaces and their big mapping class groups. Following the perspective of Bers' proof of the Nielsen-Thurston classification, we will see that big mapping classes fall into a trichotomy stated in terms of their action on marked hyperbolic structures. All work discussed is joint with Ara Basmajian.

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