Fall 2023

Abstract: We discuss the known result:

Theorem. Every ANR is homotopy equivalent to a simplicial complex with the metric topology.

This theorem, with “simplicial complex with the metric topology” replaced by “CW complex”, follows from:

The Whitehead-Milnor Theorem. Every space that is dominated by a CW complex is homotopy equivalent to one.

An ingenious proof of the Whitehead-Milnor Theorem via the “Eilenberg-Mazur swindle” was inspired by an argument in a 1965 paper by M. Mather. This proof is presented in the Appendix of Hatcher’s Algebraic Topology (pages 528-529). We explain how to modify this proof to replace “CW complex” by “simplicial complex with the metric topology”.  

Abstract: We show that groups acting properly and cocompactly by type-preserving automorphisms on buildings satisfy a weak Tits alternative: they are either virtually abelian or contain a non-abelian free subgroup. This is joint work with Damian Osajda and Piotr Przytycki.

Abstract: One of the many guises of the braid group is as the fundamental group of the space of monic squarefree polynomials. From this point of view, there is a natural “equicritical stratification" according to the multiplicities of the critical points. These equicritical strata form a natural and rich class of spaces at the intersection of algebraic geometry, topology, and geometric group theory, and can be studied from many different points of view; their fundamental groups (“stratified braid groups”) look to be interesting cousins of the classical braid groups. I will describe some of my work on this topic thus far, which includes a partial description of the relationship between stratified and classical braid groups, and some progress towards showing that the equicritical strata are K(\pi,1) spaces.

Abstract: We present a uniform description of SU(3)-structures in dimension 6 as well as G2-structures in dimension 7 in terms of a characterising spinor and the spinorial field equations it satisfies. We explain the relation to Berger’s holonomy theorem as well as the classification of transitive sphere actions. Joint work with Thomas Friedrich +, Simo Chiossi, and Jos Hoell.

Abstract: We consider the set of closed geodesics on a hyperbolic surface. Given any non-negative integer k, we are interested in the set of primitive essential closed geodesics with at least k self-intersections. Among these, we investigate those of minimal length. In this talk, we will discuss their self-intersection numbers.

Abstract: The goal of this talk is to give an elementary exposition of some results due to Bestvina–Mess on the local connectivity of the boundary of a one-ended hyperbolic group.  Along the way we will show that one-ended hyperbolic groups are semistable at infinity and their boundaries are linearly connected.  Geoghegan first observed that local connectivity implies semistability using some deep results of shape theory.  Bonk–Kleiner originally proved linear connectedness of the boundary using subtle analytic methods.  We show how all three results follow directly from elementary methods inspired by the proofs of Bestvina–Mess.

Abstract: In the late 20th century, the notions of automaticity and biautomaticity were developed to better understand the geometric structure of a group. Epstein and Thurston developed an automatic structure on Seifert manifold groups and Neumann and Reeves expanded upon it by developing a biautomatic structure on the same.

In this talk, we strengthen the theorem of Neumann–Reeves by proving that a Seifert manifold equipped with a given framing can be given a biautomatic structure compatible with the framing. We will see this as two different results.

We will first see how we can generate a 1-quasi cocycle which will form the notion of "height" on this Seifert Manifold group, such that it is bounded on the elements generated by curves on a framing. The proof of this is inspired by recent work done by Hagen, Rusell, Spriano and Sisto. We will then see how given this 1-quasi cocycle, we can generate a compatible Neumann-Reeves biautomatic structure.

Abstract: Recently, cellular sheaves have become a useful tool in applied topology for piecing together data parameterized by a discrete space. In this talk, I will introduce cellular sheaves and survey some applications in opinion dynamics, engineering and machine learning. 

Abstract: In his famous paper ``How to draw a graph" in 1962, Tutte proposed a simple method to produce a straight-line embedding of a planar graph in the plane, known as Tutte's spring theorem. This construction provides not only one embedding of a planar graph, but infinite many distinct embeddings of the given graph. This observation leads to a surprisingly simple proof of a classical theorem proved by Bloch, Connelly, and Henderson in 1984 stating that the space of geodesic triangulations of a convex polygon is contractible. In this talk, we will introduce spaces of geodesic triangulations of surfaces, review Tutte's spring theorem, and present this short proof. We will briefly report the recent progress in identifying the homotopy types of spaces of geodesic triangulations of more complicated surfaces. This is joint work with Tianqi Wu and Xiaoping Zhu.

Abstract: This pair of talks aims to first overview Z-compactifications, Z-structures and the ideal class of groups where such structures may exist (Type F groups). We then start to describe an extension of the theory to a new type of groups with a closely related property, Type F_n groups.

Abstract: William Thurston proposed regarding the map induced from two circle packings in the plane with the same tangency pattern as a discrete conformal map. A discrete analogue of the Riemann mapping is deduced from Koebe-Andreev-Thurston theorem. One question is how to extend this theory to Riemann surfaces and relate classical conformal structures to discrete conformal structures. Since circles are preserved under complex projective transformations, it is natural to consider circle packings on surfaces with complex projective structures. Kojima, Mizushima and Tan conjectured that for a given combinatorics the deformation space of circle packings is homeomorphic to the Teichmueller space. In this talk, we report its progress and explain its connection to discrete harmonic functions. Particularly, we extend the conjecture for the Weil-Petersson class in the universal Teichmueller space. 

Abstract: We will introduce an analogue of big mapping class groups as defined by Algom-Kfir and Bestvina that hopes to answer the question: What is "Big Out(F_n)?" These will arise as groups of proper homotopy classes of proper homotopy equivalences of locally finite graphs. Similar to the surface setting, these groups are not finitely nor compactly generated. As such, one must take more care when attempting to use the standard tools of geometric group theory. We will discuss new results that classify when these groups have a well-defined quasi-isometry type. This is joint work with Hannah Hoganson and Sanghoon Kwak.

Abstract: The twist number of a simple closed geodesic on a finite area hyperbolic once-punctured torus is given by the length from the projection of the cusp on the left of the geodesic to the cusp projection on the right. The relative twist number is the ratio of the twist number to the hyperbolic length. Simple closed geodesics on a punctured torus are naturally parameterized by a rational number (aka the slope), and, with symmeries of the modular torus in mind, it is natural to restrict to rationals in [0,1]. We will show that the graph of the relative twist number function is dense in the square [0,1]^2. Consequently, in contrast to the hyperbolic length, the twist number function does not extend continuously to the space of measured laminations.

Abstract: In this talk, I will discuss the notions of the coarse homotopy between coarse maps and the coarse homotopy groups.

I will also describe a coarse version of the mapping cylinder construction and show that it has the coarse homotopy extension property.

This result leads to a coarse version of Whitehead's theorem. If a coarse map between coarse CW complexes induces an isomorphism on the coarse homotopy groups, then the map is a coarse homotopy equivalence.