Following J.P. Serre’s paper of the same title as this talk, I will give an algebraic proof of the Ax-Grothendieck Theorem - which appears to be a theorem of complex analysis - using finite fields. I will discuss what makes this argument work, and if time permits, I will prove another result that appears in Serre’s paper.

The study of QFTs has inspired many modern mathematical constructions and results. QFTs which are unchanged by diffeomorphism are called topological; we will play around with the structure of such QFTs in (1+1) dimensions and prove a baby version of the celebrated Verlinde formula. If time permits, we’ll define gauge theories and their quantizations, and apply the baby Verlinde formula to them to get some interesting group/representation theoretic identities.

From generic nice functions, we could get information on the topological invariant of the manifold. Andreas Floer generalized this idea to functionals on loop space of a closed symplectic manifold to prove Arnold's conjecture under some further conditions. This construction has far richer structure than just the homology of the manifold and actually encodes all of its symplectic info. I will show some connection with the talks before on symplectic manifold and TQFT if time allows.

(Canceled Due to Snow)
Consider a 2nd order ODE: z(1-z)f'' + (1-2z)f' - (1/4) f = 0 known as hypergeometric differential equation. In the first part of my talk, I will briefly discuss it's solution found by Euler and studied systematically by Gauss known as Gauss hypergeometric series. Then, in the second part of my talk, I will discuss some seemingly completely unrelated formula (Igusa's formula) about counting points of Elliptic curves over characteristic $p$ (of Legendre form). For the rest of the talk, I will try to tell the audience why and how these two things are related.

We will describe Khovanov's groundbreaking "categorification" of the Jones polynomial of a knot. Khovanov homology is a topological invariant of knots and links with an easy combinatorial definition. It has been used to answer some longstanding questions in three-dimensional topology such as the Milnor Conjecture. The objective of this talk is to define Khovanov homology, compute it for some examples, and state some of its properties.

In this talk, I will motivate and explain various aspects and techniques of Alexandrov Geometry as well as applications to geometric questions.

The goal of this talk is to outline the proof that the moduli space of Riemann surfaces of genus $g$ is $3g-3$ complex dimensional. Along the way, we hope to illustrate how the combinatorics of simple closed curves on a surface give a great deal of information about the geometry of the moduli space of Riemann surfaces of genus $g$. Our proof of the main result heavily relies on understanding the hyperbolic geometry of pairs of pants, that is, closed disks with two boundary components. We will first study the hyperbolic geometry of a pair of pants.  After this, we will be able to introduce the Teichm\"{u}ller space of a surface and its associated Fenchel--Nielsen coordinates, which will enable us to prove the main result.

While it is easy to dissect a square into any even number of triangles of equal area, it turns out to be impossible to do the same with an odd number of triangles. The only proof of this result to date, due to Monsky (1970), requires input from both combinatorial topology (!) and the 2-adic numbers (!!), which is surprising given that the problem is geometric in nature. In this talk, we will introduce all the necessary concepts and explain Monsky's proof in detail.