The graduate student seminar meets from 4 to 5:00 pm Thursdays in Goldsmith somewhere. Want to give a talk? Email Matt Cordes at mcordes@etc... for details.

# January 23

## Diophantine approximation: from Pigeon-hole principle to Dynamical systems.

In this (pizza) talk, I will (as usual) introduce the field of Diophantine approximation. I will discuss about the three main methods of the field: continued fractions, pigeon-hole principle, and dynamical system on the space of lattices.

# January 30

## Right Angled Artin Groups and their Subgroups.

Right angled artin groups (RAAGs) are a simple generalization of free and free abelian groups. Unlike those groups, they exhibit a rich collection of subgroups. We'll look at some of the surprising subgroups found inside them, and discuss the metric implications of some of these embeddings.

# February 27

## Ricci Flow and the Poincare Conjecture

In this talk, I will mainly talk about the proof of finite time extinction of Ricci flow. And show you how to use the finite time extinction to prove the poincare conjecture.

# March 13

## Automaticity of the Braid Group

An automatic structure gives us finite state machines that can identify certain words and relationships between them in our group. In particular, an automatic structure gives a solution to the word problem in quadratic time. In this talk we will focus on the braid group and sketch out a construction of an automatic structure for it. This relies heavily on the intimate relationship between the braid group and the symmetric group.

# March 20

## An Introduction to Modular Forms

Modular Forms are one of the fundamental objects in Number Theory. In this talk, I will introduce basics of the classical Theory of Modular Forms along with some examples and try to emphasize their importance in Number Theory. If I have enough time at the end, I will talk briefly about some more interesting (and advanced) facets of theory of Modular Forms ( e.g. Congruences of Modular Forms, Families of Modular Forms, Hecke Algebras and Galois Representations, etc.)

# March 27

## Spanning Trees of the Hypercube

I will present some of the results of Olivier’s paper: On the Spanning Trees of the Hypercube and Other Products of Graphs. I aim to make the talk accessible to all, so I will begin with an introduction to generating functions. We’ll talk about what they are, why they are so useful, and how they can be used to count spanning trees.

# April 3

## The partition function and its ilk mod p

A conjecture from the 1960s about the parity of the partition function: one expects it to be even just about half the time. Easy to state, but tricky to prove! I will discuss some recent hope-progress by Serre, Nicolas, and Bellaiche using modular forms mod 2, as well as some related conjectures mod p > 2.

The main goal will be to share some of the more accessible and appealing aspects of the theory. Partition bijections may be dwelled upon or not depending on interest.

# April 10

## Braids, Graphs, and ROBOTS!

Mathematicians and engineers alike are interested in studying graph braid groups, which arise in the study of robotics and motion planning. I'll start by defining braid groups and configuration spaces, and then define the analogous graph braids groups and the topological spaces associated with them. Heavily motivated and explained through example, we'll talk about the different types of configuration spaces associated to a graph braid group, how to compute exactly what these spaces are, and (time permitting), we'll discuss how those spaces are themselves related.

This introductory talk is based on work by Aaron Abrams & Rob Ghrist.

# April 24

Previous Semesters