Columbia Undergraduate Math Society

Summer 2014« Fall 2014 Lectures »Spring 2015

Wednesdays, 7:30pm; Room 507, Mathematics
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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 How to count using (co)homology How many times in a day can you switch the two hands of a clock and get a valid time? How many lines in three-space intersect four given lines? How many conics in the plane are tangent to five given conics? Come and you will know the answers, along with a powerful modern tool to answer many more such questions.
Oct 29 Mike Wong The manifold questions about manifolds Manifolds are very important objects in various areas of mathematics. In this talk, we will define many different types of manifolds, attempt the impossible task of imagining a few higher-dimensional ones, (pretend to) understand the classifications of some of them, be shocked by some crazy facts, and introduce some unsolved problems.
Nov 5 Connor Mooney Unique continuation If all the derivatives of a holomorphic function vanish at a point, then the function vanishes everywhere. This remarkable rigidity property is known as unique continuation. We will discuss two proofs, one by cheating and one using more robust PDE methods. We will also construct an interesting counterexample to illustrate how far the PDE ideas can be pushed.
Nov 12 Sebastien Picard The Lelong-Poincare formula The Lelong-Poincare formula is a fundamental identity in multivariable complex analysis and complex geometry. The first part of this talk should be easily accessible and will be about a simple case of this equation occurring in the complex plane. The second half of this talk will involve more advanced topics such as the complex Monge-Ampere equation, the Chern class of holomorphic line bundles, and a conjecture worth one million dollars.
Nov 19 Natasha Potashnik The number of lines
on a cubic surface
Imagine a smooth cubic surface (the manifold cut out by a degree three polynomial). Does it always contain a line? Can it contain more than one? In this talk we'll explain why every smooth cubic surface contains exactly 27 lines. We will need to set up the right framework to ask this question, and along the way we'll encounter fun ideas from algebraic geometry.
Nov 26 Thanksgiving break
Dec 3 Po-ning Chen Morse theory In this talk, we will discuss the main idea and application of the Morse theory. The basic insights, according to Marston Morse, is that a typical differentiable function on a manifold will reflect the topology of the manifold. In particular, the critical points of the differentiable function will play key role in understanding the topology of the manifold.
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