Wednesday January 30, 2019 1:00 PM  2:00 PM Math Tower P131
 Ying Hong Tham, Stony Brook University
Character varietiesGiven a closed surface $S$ and a Lie group $G$, one defines the character variety to be roughly the space of representations of $π_1(S)$ in $G$. This space has connections to geometry, topology, physics, combinatorics, knot theory, and many other fields. In this talk, we will discuss some of these connections and develop some properties of the character variety.

Wednesday February 06, 2019 1:00 PM  2:00 PM Math Tower P131
 Yi Wang, Stony Brook University
Flat connections on Riemann surfacesI will explain how to identify the moduli space of flat connections, which is a differential geometric object, with corresponding moduli spaces in topological and holomorphic contexts. The key example we will focus on is Jacobian variety of a Riemann surface. In general, when the structure group is unitary, this is a theorem of Narasimhan and Seshadri, generalized by DonaldsonUhlenbeckYau; When the structure group is complex reductive, this is DonaldsonCorlette's theorem on harmonic metrics and HitchinSimpson's theorem on Higgs bundles.

Wednesday February 13, 2019 1:00 PM  2:00 PM Math Tower P131
 Matthew Dannenberg, Stony Brook University
KAM Theory and the Collapse of Integrable DynamicsMany simple models in physics and symplectic geometry exhibit an algebraic property called (Liouville) complete integrability. This algebraic property gives rise to rich topological structure and simple dynamics, yet itself vanishes under small perturbations of the model. KAM Theory provides a mechanism to show that, despite this loss, the topological and dynamical structures persist on an extremely large set. In this talk I'll discuss what makes an integrable system so nice, show how to obtain the large set of persistence, and hint at the onset of chaos beyond KAM Tori.

Wednesday February 20, 2019 1:00 PM  2:00 PM Math Tower P131
 Taras Kolomatski, Stony Brook University
An Infinite Quantum Ramsey TheoremNik Weaver (2015) showed an intriguing noncommutative version of the classical Ramsey's theorem on graphs: Let $\mathcal{V}$ be a subspace of $M_n(\mathbb{C})$ which contains the identity matrix and is stable under the formation of Hermitian conjugates. If $n$ is sufficiently large, then there is a rank $k$ orthogonal projection such that $\dim (P\mathcal{V}P)$ is $1$ or $k^2$. These are the minimal and maximal possibilities for this dimension, and in these cases such a projection is called a quantum $k$anticlique or quantum $k$clique, respectively.
Weaver further showed that both the classical and quantum Ramsey's theorems are special cases of a general Ramsey theorem on \textit{quantum graphs}, which are modelled on such matrix spaces with the additional algebraic structure of being a bimodule of some matrix $*$algebra. Investigation of such objects was initially motivated by quantum information theory, in which quantum graphs provided an analogue of the confusability graph in classical communication over a noisy channel. Weaver's work follows a long list of results successfully generalising classical results to this context, such as the definition of quantum Shannon capacity by Duan, Severini and Winter (2013).
In this talk, I will look at salient examples that demonstrate the difference between the classical and quantum contexts, sketch Weaver's results, and describe the process by which we successfully adapted Weaver's work demonstrate a quantum analogue of the classical infinite Ramsey's theorem in Kennedy, Kolomatski, Spivak (2017). Working in this infinite dimensional setting required functional analysis, and invited plenty of delightful nuance in topological considerations.

Wednesday February 27, 2019 1:00 PM  2:00 PM Math Tower P131
 Diljit Singh, Stony Brook University
An Introduction to Additive Number TheoryWe will give an overview of certain questions in the field, interesting connections to other topics in math, and important theorems. The talk will be flexible and we can focus on topics of the audiences choosing.

Wednesday March 06, 2019 1:00 PM  2:00 PM Math Tower P131
 Jiahao Hu, Stony Brook University
Formal groups, complex manifolds and elliptic curvesA (commutative) formal group law on a ring R is a formal power series f(x,y) with coefficients in R satisfying the axioms that define a group and f(x,y)=f(y,x). A stably almost complex manifold X is a smooth (real) manifold with its tangent bundle TX being stably almost complex, that is after direct summing with a finite rank trivial bundle it becomes a complex vector bundle. An elliptic curve E is a genus 1 Riemann surface. I will introduce a very surprising connection among these three objects, and explain what it means by saying "signature and Ahat genus correspond to degenerate elliptic curves".

Wednesday March 13, 2019 1:00 PM  2:00 PM Math Tower P131
 Shamuel Auyeung, Stony Brook University
Morse Theory and Hamiltonian Floer HomologyOne rendition of Arnold's famous conjecture is the following: The number of 1periodic solutions of a nondegenerate timedependent Hamiltonian system on a compact symplectic manifold $(M,ω)$ is greater than or equal to $\sum_k dim(H_k(M; \mathbb{Z}_2))$.
The goal of this talk is to tell a story of Arnold's conjecture, beginning with Morse theory and ending with an answer to the conjecture by way of an infinite dimensional Morse theory variant: Floer theory. Along the way, I'll define what we need to understand the conjecture itself and also introduce the main players: wellchosen functions and their critical points, vector fields and their trajectories, transversality and compactness.

