The seminar will run on Fridays from 3:30 to 4:30 PM in MC 107.

Coffee and Snacks are usually provided.

A list of suggested topics for expository talks of MSc Students can be accessed here.

## Organizational Meeting (Friday Jan 17, 2020)

3:30-4:30 PM, MC107

## Brandon Doherty. A model structure for cubical categories (Friday Jan 24, 2020)

Simplicial categories, categories enriched over simplicial sets, are one of the most well-known models for the theory of ($\infty$,1)-categories. Their homotopy theory is studied by means of a model structure on the category of simplicial categories, due to Bergner. Recently there has been some interest in categories enriched over cubical sets as an alternative model for ($\infty$,1)-categories. In this talk, I will discuss a model structure on the category of cubical categories which is analogous to the Bergner model structure for simplicial categories. If time permits, I will also discuss an adjunction between the categories of simplicial sets and cubical categories, due to Kapulkin and Voevodsky, and sketch the proof that it is a Quillen adjunction when the category of simplicial sets is equipped with the Joyal model structure.

## Prakash Singh, Hofer's geometry on the group of Hamilton diffeomorphism of the 2-sphere (Friday Jan 31, 2020)

Given an isotopy (or flow of a vector field) on a Symplectic manifold $(M,w)$, the trajectories of this isotopy may form a complicated system of curves, but one can study the isotopy as a curve in some topological group $(Ham (M,w))$. In fact, we can put a metric (Hofer's metric) on $Ham(M,w)$ and study the geometric properties of this metric itself. We will look at some basic questions that one can ask about this metric and discuss the diameter of this topological group under this metric. In particular, we will try to prove that $Ham(S^2)$ has infinite diameter with respect to this metric.

## Mohabat Tarkeshian, Jacobians of graphs and starfish (Friday Feb 7, 2020)

Graphs can be viewed as discrete analogues of various mathematical structures, such as algebraic curves and Riemann surfaces. A finite graph has an associated finite abelian group called the Jacobian group, denoted $Jac(G)$ (also called the critical, sandpile, or Picard group). We will discuss these groups and a special type of $k$-regular graph called a starfish graph. A starfish is similar to a hyperelliptic curve since each can be viewed as a (ramified) double-cover of a simply connected object. It is a graph $S$ exhibited as a double-cover $S \to T$ of a tree $T$ with ramification locus equal to the leaves of $T$. We will discuss how varying the starfish can give rise to natural questions such as how often $S$ is Ramanujan and how $S \to T$ can be viewed as a harmonic morphism of graphs. We will also briefly examine induced maps on Jacobians and their connection to harmonic morphisms.

## Luis Scoccola, Monoidal categories, string diagrams, and Markov categories (Friday Feb 14, 2020)

Monoidal categories are categories with a notion of product that is weaker than the categorical product. This relaxation allows for many interesting examples coming from topology, logic, physics, and probability theory. I will introduce monoidal categories and string diagrams, a graphical calculus to work with diagrams in monoidal categories, and I will go over several examples, paying particular attention to the ones coming from probability theory. This will motivate the definition of Markov category, an abstraction that lets us study probabilistic concepts such as marginalization, conditionals, Markov processes, almost sure equality, and deterministic maps diagrammatically.

## Andrew Herring, Arithmetic Dynamics: Uniform Boundedness, and two Galois Representations (Friday Feb 28, 2020)

Arithmetic dynamics is a relatively new area of research in which we ask number theoretic questions about dynamical systems. An easy source of problems is by analogy with arithmetic geometry: starting from some problem in arithmetic geometry (solved or not), we can consider its dynamical analogue (for example by replacing the words "torsion points on an abelian variety" with the words "(pre)periodic points of a rational map.") In this talk we'll look at the (dynamical) uniform boundedness conjecture of Morton and Silverman (an analogy of Merel's Theorem on the uniform bound on torsion of an elliptic curve), and two Galois representations-the arboreal Galois representation, and the dynatomic Galois representation, each of which might be regarded as an analogy of the l-adic representation of an abelian variety. This talk is meant to be a gentle introduction to some of the basic problems in arithmetic dynamics; the only pre-requisites are an open heart and an open mind.

## Jeremy Gamble, Spin structures and Dirac operators (Friday Mar 6, 2020)

Spin structures and Dirac operators play an important role in many areas of mathematics, such as mathematical physics, index theory and differential geometry. The Dirac operator is like a "square root" of the Laplacian, and can always be defined on a manifold which has a spin structure. In the talk, I will introduce Clifford algebras, Spin and some elements from differential geometry in order to define spin structures and Dirac operators. I will motivate these objects with some examples from physics, and present some theorems such as an obstruction for the existence of a spin structure, some basic properties of Dirac operators. Some familiarity with differential geometry will be assumed.

## Alejandro Santacruz, The compensated compactness method applied to a system of balance laws (Friday Mar 13, 2020) (Cancelled or posponed to next season)

Systems of balance laws are composed by partial differential equations in the form $$u(x,t)_t - F(u)_x = g,$$ where $u(x,t)$ is a vector valued function usually interpreted as densities of quantities balanced in a physical system. When the function $F$ is non-linear these systems usually have no classical solution, a standard method used to prove the existence of weak solutions is to study an approximated perturbed system, solve it, and then form a sequence of solutions converging to a solution for the original problem. When the flux function $F$ is non-linear, proving the weak continuity of the flux is a major source of complications, this is where the compensated compactness method can be applied for some of this systems. In this talk I will give an example of a system associated with the ARG (Aw-Rascle-Greenberg) traffic model whose solution can be proven using this method and I will briefly explain each major step in its application. No previous knowledge in PDE's will be assumed.

## Nathan Joseph Pagliaroli, Applying Random Matrix Theory to Noncommutative Geometry (Friday Mar 20, 2020) (Cancelled or posponed to next season)

A probability measure can be constructed on the space of Dirac operators that form a spectral triple with a specified Hilbert space and algebra. For certain cases the Dirac operators can be expressed in terms of random matrices. This allows us to deduce the statistical behaviour of the spectrum of Dirac operators by analyzing the behaviour of the eigenvalues of random matrices. Using well known techniques from Random Matrix Theory (RMT) we were able to explicitly find the distribution of certain Dirac operators’ spectrums. This talk will focus much more on RMT than NCG. I will first give a very brief introduction to RMT then talk about the ideas and techniques used to find the aforementioned results. The only background needed will be linear algebra and basic probability theory.