Speaker: Becky Eastman

Title: Separable homology of graphs and the Whitehead complex

Abstract: Given a finite-type surface Σ and a collection 𝒞 of curves on Σ, a natural question to ask is, do lifts of curves in 𝒞 generate the homology of every finite cover of Σ? Versions of this question have been studied by numerous authors (Farb-Hensel '16, Koberda-Satharoubane '16, Malestein-Putman '19, Klukowski '23, Boggi-Putman-Salter '23). Our version of this question is: let Γ be a finite regular cover of the rose. We identify the fundamental group of the rose with a finite-rank free group. Is the fundamental group of Γ generated by lifts of elements in a proper free factor of the free group? We define a locally infinite 1-complex Wh(Γ) called the Whitehead complex of Γ which is connected if and only if the answer to this question is yes. Interestingly, if Γ represents a characteristic subgroup or Γ is the rose, Wh(Γ) admits an action of Out(F_n) by isometries. Every component of Wh(Γ) has infinite diameter, and the Whitehead complex of the rose is nonhyperbolic.

Speaker: Carlos Ospina

Title: Real rel flow

Abstract: I will explain what the real rel flow is in the setting of translation surfaces with two cone angles.

Speaker: Elizabeth Crow

Title: Projective Structures on the Circle

Abstract: In this talk we will describe the topological space of projective structures on the circle.

Speaker: Leslie Mavrakis

Title: Branched Manifolds and Thurston's Geometries

Abstract: In joint work with Daryl Cooper and Priyam Patel we prove that for every model geometry, G, there is a compact branched manifold, W(G), with the property that a closed 3-manifold M has a G-structure if and only if there is an immersion of M into W(G). In this talk, I will give some background on branched manifolds and describe W(G) when G is S^3.

Speaker: Michael Zshornack

Title: Arithmetic with Mickey Mouse

Abstract: "Bending" is a way of deforming representations of surface groups to new ones with more interesting properties. For instance, it is one way of constructing quasi-Fuchsian groups from Fuchsian ones. An oft overlooked aspect of this deformation is its underlying arithmetic nature, but this perspective has proven especially useful in studying higher Teichmüller spaces, whose arithmetic properties have interesting relations to higher-rank lattices. In this talk, I'll outline the general idea behind this deformation and how its arithmetic nature has been useful in studying the Hitchin component, one example of a higher Teichmüller space. This is partially based on joint work with Jacques Audibert.

Speaker: Rebecca Rechkin

Title: Normal Stalling's Completion

Abstract: A group G is residually finite if for every nontrivial element g, there exists a finite quotient of G where g is not in the kernel of the quotient. Stallings completion provides a geometric algorithm for finding a finite index subgroup of the rank n free group that does not contain a specific word g by constructing an appropriate cover of the wedge of n circles. However, the algorithm does not usually construct a normal cover of the wedge of circles. In this talk I will give a new algorithm that produces a normal subgroup of the rank n free group that does not contain a specific word g by constructing normal covers of the rose. This work is motivated by a body of research dedicated to quantifying residual finiteness for certain classes of groups.

Speaker: Ryan Dickmann

Title: Mapping class groups of surfaces with noncompact boundary

Abstract: We will talk about sliced Loch Ness monsters, disks with handles, and the classification of surfaces with noncompact boundary due to Brown and Messer. We will also see some general theorems on the mapping class groups of surfaces that were made possible through understanding the Brown and Messer classification.

Speaker: Thomas Hill

Title: Large-Scale Geometry of Pure Mapping Class Groups of Infinite-Type Surfaces

Abstract: The work of Mann and Rafi gives a classification surfaces Σ when Map(Σ) is globally CB, locally CB, and CB generated under the technical assumption of tameness. We restrict our study to the pure mapping class group and give a complete classification without additional assumptions. In stark contrast with the rich class of examples of Mann–Rafi, we prove that PMap(Σ) is globally CB if and only if Σ is the Loch Ness monster surface, and locally CB or CB generated if and only if Σ has finitely many ends and is not a Loch Ness monster surface with (nonzero) punctures.

Speaker: Vivian He

Title: Random walks on groups and superlinear divergent quasi-geodesics

Abstract: When studying random walks on a group, a natural question is how far is the n-th step from the origin. The central limit theorem says that, asymptotically, it follows a normal distribution. We proved this for groups containing superlinar divergent quasi-geodesics. The advantage of this setting compared to previous versions of CLT is that it is invariant under quasi-isometry.

In this talk, I will delve into the superlinear divergence property, as well as its geometric consequences that led to the theory of random walks on groups containing superlinear divergent quasi-geodesics. This is based on joint work with Kunal Chawla, Inhyeok Choi, and Kasra Rafi.

Speaker: Wiktor Mogilski

Title: A discrete four vertex theorem for hyperbolic polygons

Abstract: There are many four vertex type theorems appearing in the literature, coming in both smooth and discrete flavors. The most familiar of these is the classical theorem in differential geometry, which states that the curvature function of a simple smooth closed curve in the plane has at least four extreme values. This theorem admits a natural discretization to Euclidean polygons due to O. Musin. In this talk we adapt the techniques of Musin and provea discrete four vertex theorem for convex hyperbolic polygons.

Speaker: Ziqiang Li
Title: Winding Number for the Spherical Point-in-Polygon Problem
Abstract: We revisit the spherical point-in-polygon problem and discuss how the winding number could be made sense of, for a very practical subclass of spherical polygons: boundary antipode-excluding spherical polygons.