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.

The PDF version of the abstracts 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.