Columbia Undergraduate Math Society

Summer 2014« Fall 2014 Lectures »Spring 2015


Wednesdays, 7:30pm; Room 507, Mathematics
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ums [at] math.columbia.edu
Date Speaker Title Abstract
Sept 17 Karsten Gimre Discrete groups and limits of metric spaces Discrete groups are of fundamental importance in many fields, including differential geometry, topology and number theory. We will survey some of the mathematics behind a important theorem due to Mikhael Gromov which studies the subclass of discrete groups given by those of polynomial growth. A key novel idea introduced by Gromov's proof is to view groups as geometric objects, reducing the study of groups of polynomial growth to a problem about metric spaces.
Sept 24 Jo Nelson Contact Structures and Reeb Dynamics Contact geometry is the study of geometric structures on odd dimensional smooth manifolds given by a hyperplane field specified by a one form which satisfies a maximum nondegeneracy condition called complete non-integrability; these hyperplane fields are called contact structures. The associated one form is called a contact form and uniquely determines a vector field called the Reeb vector field on the manifold. If that was all gibberish to you, no worries - I'll introduce the notion of manifolds and differential forms, which are just slight generalizations from what you've seen in multivariable calculus and analysis. This talk will have lots of cool pictures and animations illustrating these fascinating concepts in differential geometry and many concrete examples will be given.
Oct 1 Rob Castellano Introduction to
kind-of-Gromov-Witten invariants
Gromov-Witten invariants play a crucial role in string theory and a philosophy known as mirror symmetry which describes connections between symplectic geometric and algebro-geometric phenomena. Rather than describe this, I will explain how the classical problems of counting curves of a particular geometric type (this goes back hundreds of years) give rise to the definition of Gromov-Witten invariants. Although I will not formally define Gromov-Witten invariants, I will describe the problems that arise in defining them and why they are difficult to compute. We will compute a Gromov-Witten invariant when these problems are easily resolved.
Oct 8 Pak-Hin Lee Why is the Ramanujan constant
almost an integer?
It is no coincidence that the Ramanujan constant \(e^{\pi\sqrt{163}}\) is almost an integer, and a good explanation is provided by the theory of complex multiplication, which, roughly speaking, studies elliptic curves with extra symmetries. I will give a gentle introduction to this theory and explain some of the fascinating links between elliptic curves, algebraic number theory and modular functions, before deriving the near integrality of \(e^{\pi\sqrt{163}}\) as a by-product.
Oct 15 Zhengyu Zong Quantum cohomology and Gromov-Witten
theory of projective spaces
The quantum cohomology ring of a smooth projective variety is a deformation of the usual cohomology ring. This new structure can be defined by genus 0 Gromov-Witten theory. In this talk, I will talk about the quantum cohomology ring of projective spaces and do some explicit computations.
Oct 22 Anand Deopurkar TBA TBA
Oct 29 Mike Wong TBA TBA
Nov 5 Connor Mooney TBA TBA
Nov 12 Sebastien Picard TBA TBA
Nov 19 Natasha Potashnik TBA TBA
Nov 26 Thanksgiving break
Dec 3 Xiangwen Zhang TBA TBA