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.

There is no reason why Feynman diagrams couldn't be taught in an advanced calculus class.  I will discuss something actually useful: how to compute asymptotic series for certain exponential integrals.  We will start with one variable, where the asymptotic series will be sums over graphs, and try to get to matrix integrals (the Feynman diagrams for which involve surfaces boundary, i.e. `interacting strings').

We will spend some time building important notions and general theory of mathematical logic before turning our attention to the theory of algebraically closed fields. For a given characteristic, the theory of algebraically closed fields is extremely “nice” being categorical and therefore also model complete. These properties will allow us to prove the first-order Lefschetz principle which allows an algebraic geometer to transfer statements about $\mathbb{C}$ to any other algebraically complete field of characteristic zero, in particular, $\bar{\mathbb{Q}}$. Furthermore, the compactness of first-order logic will allow us to show that any statement true of the algebraic closure of finite fields for all but finitely many characteristics is true of all algebraically closed fields of characteristic zero. An easy corollary of this powerful reduction is the Ax-Grothendieck theorem which states that injective polynomial functions $\mathbb{C}^n \to \mathbb{C}^n$ must be surjective. This gives an example of a purely algebraic result proven elegantly using primarily model-theoretic tools.

We give an introduction to operads and A-infinity spaces. We show how these objects naturally arise when trying to answer the question of what topological spaces are homotopy equivalent to topological groups.

We will uncover the rich structure hidden in two-dimensional conformal field theories from a purely mathematical perspective. In particular, we'll show that the correlators of WZW theory satisfy a remarkable system of differential equations called the KZ equations. These correlators are matrix elements of certain operators between representations of affine Lie algebras.

In the talk I will try to explain some mathematical applications of string theory. I will touch on dualities such as mirror symmetry, large N duality, geometric transition, GW/DT correspondence and symplectic duality.