Columbia Undergraduate Math Society

Summer 2018 << Fall 2018 Lectures >> Spring 2019

Wednesdays, 7:30pm; Room 507, Mathematics
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Date Speaker Title Abstract
September 12
Shizhang Li
Hypergeometric Series and
Igusa's Formula
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.
September 19
Semen Rezchikov
Feynman Diagram Techniques
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').
September 26
Ben Church
What Can Mathematical
Logic Do For Me?
A Model-Theoretic Approach
to the Lefschetz principle
and Ax-Grothendieck
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.
October 3
Alex Pieloch
Operads and A-infinity spaces.
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.
October 10
Henry Liu
Vertex Operators and
Knizhnik-Zamolodchikov Equations
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.
October 17
Iakov Kononov
Mathematics of Strings
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.
October 24
Stanislav Atanasov
October 31
Micah Gay
November 7
Maithreya Sitaraman
November 14
Renata Picciotto
November 21
No meeting
Thanksgiving Break
November 28
December 5
Alisa Knizel
December 13
No meeting
Reading Week
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