Columbia Undergraduate Math Society

Fall 2014« Spring 2015 Lectures »Summer 2016

Wednesdays, 7:30pm; Room 507, Mathematics
ums [at]
Date Speaker Title Abstract
Feb 4 Qixiao Ma Analytic Theory of
Abelian Varieties
Feb 11 Mitchell Faulk The Modern Formulation
of TQFTs
I'll try to motivate and introduce the modern formulation of a topological quantum field theory (TQFT), inspired by Segal's and Atiyah's original axioms, as a symmetric monoidal functor from a certain bordism category to the category of vector spaces. For motivation, I'll look at classical field theories in physics, and explain how a topological quantum field theory can be viewed as a ``quantization'' of a certain classical model, called the non-linear sigma model. If time permits, I'll discuss examples of TQFTs and their applications to other areas.
Feb 18 Joshua Seaton Extrapolating factorials:
the Gamma function,
classical and p-adic
Graph the points \((n, n!)\) in the plane for positive integers \(n\); there are a continuum of curves that go through these points. So why does the Gamma function -- among the infinitely-many other candidates -- deserve to be regarded as the 'factorial function'? The Bohr-Mollerup-Artin Theorem tells us why, which reveals that a function satisfying a couple of mild 'factorial'-like properties necessarily has to be the Gamma function. We first go through this brief, neat result. Then, in the second part of this talk, we look at the p-adic side of things (we will introduce p-adic numbers gently here, I promise.) In the ring of p-adic integers, the usual integers are dense. So -- in high contrast to the setting over the real numbers -- an integer-valued function on the integers (if p-adically continuous) will lift uniquely to a function on the p-adic integers! After some minor tweaking to the function that maps \(n\) to \(n!\), we will take a brief look at its (unique) p-adic extension, the p-adic Gamma function.
Feb 25 Linus Hamann Contemplation: A Journey through Elliptic Curves In this talk, we will begin with a classical overview of the definition of an elliptic curve and associated properties. Afterwords, we will introduce some of the tools and terminology used in the algebraic geometry of smooth curves, including the Picard group, differentials, and Riemann-Roch. The remainder of the talk will be dedicated to using this new terminology to illuminate various features of the classical view of elliptic curves and thereby give insight into their importance.
March 4 Irit Huq-Kuruvilla
March 11 Karsten Gimre
March 18 Spring break
March 25 Wille Dong
April 1 Sander Mack-Crane
April 8 Max Kieff
April 15 Matei Ionita
Nilay Kumar
April 22 Cosmin Pohoata
April 29 Xiangwen Zhang
designed by Nilay Kumar