Spring 2021
Spring 2021
We need graduate student speakers for the following Thursdays of Spring 2021:
- Feb. 11 -- Speaker: Ian Montague
Title: Pin(2)-Equivariant Seiberg-Witten Floer Homology and a Disproof of the Triangulation Conjecture
Abstract: In this talk I will provide motivation for, and an outline of Ciprian Manolescu's proof from 2013 that there exist non-triangulable n-dimensional manifolds for every n>=5. Here's a nice Quanta article about the subject that may be of interest:
https://www.quantamagazine.org/triangulation-conjecture-disproved-20150113/
- March 04 -- Speaker: Alex Semendinger
Title: CAT(0) Groups With Ill-Defined Connected Boundaries
Abstract: It has long been known that CAT(0) groups do not have well-defined boundaries. However, every previously-known example of this fact involved a CAT(0) space whose boundary is not path-connected. In 2019, Michael Ben-Zvi and Robert Kropholler showed that there are right-angled Artin groups that act on CAT(0) spaces with non-homeomorphic boundaries which are connected (and in fact can be chosen to be n-connected). I will present this result along with the relevant background on CAT(0) groups and their boundaries.
- March 11 -- Speaker: Simon Huynh
Title: Configuration Spaces on Robotics
Abstract: Robert Ghrist described configuration spaces as a useful model for studying autonomous agents in an environment. Besides providing solutions to the problem of motion planning, they also have connections to the study of Braid Groups and applications in other scientific areas. In this talk, we will specifically look at configuration spaces on graphs as a way to study autonomous robots whose movements are restricted on a system of rail-tracks. We will show that these spaces have interesting topological structures and describe methods for understanding their topological properties by presenting two important results discovered by Aaron Abrams and Robert Ghrist.
- March 18 -- Speaker: Zihao Liu
Title: A New Proof of Gromov’s Theorem on Groups of Polynomial Growth
Abstract: The growth rate of a group is a well-defined notion from asymptotic analysis. To say that a finitely generated group has polynomial growth means the number of elements of length (relative to a generating set) at most n is bounded above by a polynomial function p(n). The order of growth is then the least degree of any such polynomial function p. It is not so difficult to prove that a finitely generated nilpotent group has a polynomial growth function. In 1981, Gromov proves the converse of that statement. Let G be a finitely generated group of polynomial growth, then G is virtually nilpotent. In this talk, I will introduce some backgrounds regarding word metrics and growth function as well as the sketches of a new proof of Gromov’s theorem on groups of polynomial growth.
- March 25 -- Speaker: Changxin Ding
Title: The rotor-routing torsor and the Bernardi torsor
Abstract: A torsor for a group H is a set S together with a simply transitive action of H on S. Fix a graph G. Let S be the set of the spanning trees of G and H be the Picard group of G. The rotor-routing torsor and the Bernardi torsor are two different torsor structures on S for H. In 2017, Matthew Baker and Yao Wang proved that these two torsors coincide when G is planar. We prove the conjecture raised by them that the two torsors disagree when $G$ is non-planar. I will define everything necessary and introduce these results.
- April 01 -- Speaker: Charles Stine
Title: Knot Theory and the Poincare Conjecture
Abstract: I will give a general sketch of the history of the Poincare conjecture, and outline the key concepts which were used to prove it in each dimension. The last open case of the conjecture is the smooth case in dimension four: where the conjecture is widely believed to be false. In the last twenty years there has been a growing sense that knot theory might provide a way to demonstrate a counter example. Several papers, published in the last year, indicate that the mathematical community may soon be able to realize these hopes. I will survey this recent work, and explain how it fits into the broader context of what is known about smooth 4-manifolds.
- April 15 -- Speaker: Jill Stifano
Title: Extending Group Actions on Metric Spaces in the Case of Finitely-Generated Groups
Abstract: Given a group G and a subgroup H of G, let us consider an action of H on a metric space. We ask the question: when it is possible to extend this to an action of G on a possibly different metric space? I will give the known results and some reasoning for the case of finitely generated groups. This presentation is based on the paper “Extending Group Actions on Metric Spaces” by Carolyn Abbott, David Hume, and Denis Osin.
- April 29 -- Speaker: Tarakaram (Ram) Gollamudi
Title: Adjoint L-function (Joël's p-adic L- function)
Abstract: I will give a brief outline of the construction of Joël's adjoint L- function. I will introduce overconvergent modular forms and modular symbols and then I will outline the construction of adjoint L-function.
If time permits we can work out a specific example. I am not assuming any prerequisites beyond the tensor product of vector spaces. However, some familiarity with modular forms (https://en.wikipedia.org/wiki/Modular_form ) would be helpful but not required.