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.

In this talk, we introduce modular forms and the theory of Hecke operators used to study them. We then shift the focus from the modular forms to the (symmetric) spaces they naturally inhabit, and focus on the symmetries of these spaces. These turn out to be directly related to certain algebraic invariants of the spaces they act on, called cohomologies. Lastly, we discuss recent conjectures concerning the cohomologies with rational coefficients and explain how these fit with other conjectures in number theory.

Broadly speaking, categorification can be thought of as the process of replacing set-theoretic terms to the category-theoretic analogues. In this talk, we will begin by motivating the general idea behind categorification with the example of how the Euler characteristic gets lifted to homology. Then, we will move on to the categorification of familiar operations on everyone’s favourite ring, Z. Finally, we will discuss Khovanov homology, which arises from the categorification of the Jones polynomial.

In this talk, I will give an interesting geometrical interpretation of the standard representation. If i have time, I might interpret the box representation of $S_5$ as well. Proofs will be given.

Anyone who has taken part in an election has probably noticed that the procedure chosen to determine the winner may affect the result. We will investigate some commonly used procedures and show how geometry can help us understand and analyze their results and their shortcomings. We will discover some disturbing truths about just how much the choice of a voting system can affect the outcome. We will wonder if it is possible to find a "perfect" system, that is a system satisfying some number of fairly natural conditions. That will lead us to discuss the famous Arrow's impossibility theorem, which highlights the difficulty of aggregating individual preferences.

In this talk, I will start by giving three short proofs that there are infinitely many primes: one classical, one analytic, and one topological. I will then recast the topological proof in a more conceptual light, and explain how to relate it to the analytic one.

I plan to talk about random matrices: Wigner's semicircle law and topological expansions.