Fall 2018

Speaker: Ian Montague
Title: An Introduction to Contact 3-Manifolds and Legendrian Knots
Abstract: A contact structure on a smooth odd-dimensional manifold M is a codimension-one sub-bundle of its tangent bundle TM satisfying a 'complete non-integrability' condition. Motivated by classical mechanics, contact structures arise naturally as constant-energy hypersurfaces of even-dimensional phase spaces of mechanical systems. By work of Giroux, it turns out in the case of 3-manifolds that contact structures have an intimate relationship with the smooth topology of these manifolds. In this talk I will attempt to give an overview of some of the interesting aspects of 3-dimensional contact topology, and if time permits I will mention some of the ways in which Floer-theoretic invariants have been used to shed light on this area of mathematics.

Speaker: Job Rock
Title: Geodesic Laminations
Abstract: We'll define geodesics, laminations, and geodesic laminations. Our focus will be on the disc model of the hyperbolic plane, but there will be other examples. Then we'll alude to the connections between geodesic laminations, mapping class groups, and triangulations of an n-gon.

Speaker: Kewen Wang
Title: Dirichlet prime number theorem
Abstract: Prime numbers have a lot of interesting properties, and one of them is about the cardinality of prime numbers of a special type. It is easy to prove there are infinite prime numbers of the type 4z+1, but it is not straightforward to see it is true in general. This conjecture, states that there are infinitely many primes that are congruent to a modulo d if a and d are coprime, proved by Dirichlet. In this talk, we will give an analytic approach, which is that of Dirichlet himself, to prove the theorem, and we will use some basic properties of Zeta function and L-function.

Speaker: Josh Eike
Title: Regular Languages for Geodesics
Abstract: Given a finite generating set for a group G, we can turn G into a Cayley graph by attaching group elements by edges whenever they differ by a generator. This turns G into a geodesic metric space. An important question in group theory is how can we minimally represent a group element as a product of generators? Put another way, what is the language of geodesics in a group? For hyperbolic groups, the language of geodesics is simple in the sense that it can be produced by a machine with finitely many states. I'll talk about geodesic languages for general finitely-generated groups. In particular, I'll explain work done in collaboration with Abdul Zalloum of SUNY Buffalo studying contracting geodesics.
Paper(s): https://arxiv.org/abs/1809.02692

Speaker: Abhishek Gupta
Title: Galois Representations
Abstract: Galois representations are important objects of study in algebraic number theory. One can attach a Galois representation to various number theoretic objects like modular forms, algebraic varieties etc. These representations encode various properties of the original object under consideration. In this talk, we will give examples of Galois representations and their use in studying number theoretic objects.

Speaker: Zach Larsen
Title: The Spectral Theorems
Abstract: Many theorems in functional analysis generalize results about linear operators on finite-dimensional spaces. I'll discuss two such theorems, as well as an application to representations of locally compact groups. I will only assume material covered in the first year graduate courses.

Speaker: Aritro Pathak
Title: Overview of the Green Tao theorem
Abstract: The celebrated Green Tao theorem states that the primes contain arithmetic progressions of arbitrary finite length. The first proof was given by Terence Tao and Ben Green in 2004. There were further simplifications to the proof, particularly with the formulation of what is called the Relative Szemeredi's theorem by Conlon, Zhao and Fox . We would try to give an overview of the main ingredients of the resulting proof, following an exposition by Conlon, Zhao and Fox; https://arxiv.org/abs/1403.2957.
Paper(s): https://arxiv.org/abs/1403.2957

Speaker: Rose Morris-Wright
Title: Garside Sturcture and Solutions to the Word Problem
Abstract: In 1969, Frank Garside introduced a lattice corresponding to a braid group, and used it to solve the word and conjugacy problems in braid groups. Since then, the Garside structure has become an important tool in the study of braid groups as well as many other groups. In this talk, I will explain what a Garside structure is through the primary example of the braid group. I will also discuss how to use this structure to prove further properties of the braid group and how to generalize these results to other groups.

Speaker: Abhishek Gupta
Title: Group schemes and p-divisible groups
Abstract: Groups schemes are objects in algebraic geometry frequently used in number theory. In this talk we will define and give examples of group schemes and related objects called as p-divisible groups and look at their applications in number theory. Familiarity with basic algebraic geometry will be assumed.

Speaker: Ying Zhou
Title: Two Alternative Definitions of m-Maximal Green Sequences
Abstract: Igusa proved that there are alternative definitions of maximal green sequences using HN filtrations and forward Hom-orthogonal sequences. We have extended this result to m-maximal green sequences and have given equivalent definitions of m-maximal green sequences using HN filtrations and forward Hom^{≤ 0}-orthogonal sequences.

Speaker: Matt Garcia
Title: Functors for Differential Geometry
Abstract: An endofunctor on the category of smooth manifolds which preserves products is represented by a finite dimensional local Artin algebra. This is a result of Michor and Kriegl. We'll unpack the relevant definitions and talk about some of the ingredients of the proof. Time permitting, we will get a glimpse of how this results relates to constructing a cartesian closed category where smooth manifolds embed. Concretely, this means allowing Frechet manifolds to account for infinite dimensional function spaces.