Each week, the Michael Zhao Memorial Student Colloquium holds 45-minute talks by Columbia mathematics faculty about their own research. The talks are intended for current PhD students in mathematics at Columbia. If you are an undergraduate student or external graduate student and would like to come, please email firstname.lastname@example.org or email@example.com.
This Fall the Seminar is organized online. It usually meets on Wednesdays at 4:00p.m./Thursdays at 9:30a.m. EDT virtually in Zoom, followed by informal gathering through a platform Gather Town.
Organizers: Chuwen Wang, Hindy Drillick, Georgy Gaitsgori and Patrick Lei.
|Date and time||Speaker||Title and abstract|
|Wednesday, September 23, 4:00p.m. EDT||Simon Brendle||Heat diffusion and geometry|
|Wednesday, September 30, 4:00p.m. EDT||Ivan Corwin||Random permutations, partitions and PDEs
We start with a seemingly innocuous question - what do large random permutations look like? Focusing on the structure of their "increasing subsequences" we encounter some remarkable mathematics related to symmetric functions (e.g. Schur and Macdonald), random matrices, and stochastic PDEs. No prior knowledge of any of this will be assumed.
|Thursday, October 8, 9:30a.m. EDT||Nicholas Salter||Framed mapping class groups, or the topology of families of translation surfaces
Riemann surfaces are among the most ubiquitous of mathematical objects, and many problems can be formulated as understanding a family of Riemann surfaces. The study of such families draws on many areas of mathematics - complex analysis and algebraic geometry, low-dimensional topology, dynamics, geometric group theory, and more. In many situations, the family under study is equipped with the extra data of a preferred section of some line bundle, e.g. a holomorphic 1-form. I will discuss some new tools for understanding the behavior of these families.
|Thursday, October 15, 9:30a.m. EDT||Francesco Lin||Spin structures on surfaces and the 28 bitangents to a plane quartic
It is a classical result in algebraic geometry that a general planar quartic admits exactly 28 bitangent lines. In this talk, I will discuss a purely topological approach to this geometric result due to Atiyah. Along the way, I will introduce spin groups and the basic ideas of index theory.
|Thursday, October 29, 9:30a.m. EDT||Dusa McDuff||Some questions in symplectic geometry
This will be an elementary talk that explains what a symplectic structure is, describes some of the basic questions and results, and then mentions some problems of current interest.
|Thursday, November 5, 9:30a.m. EDT||Chao Li||The Beilinson-Bloch conjecture
The Beilinson-Bloch conjecture generalizes the celebrated Birch and Swinnerton-Dyer conjecture from elliptic curves to higher dimensional varieties. We will give an introduction to this conjecture and mention some recent progress.
|Thursday, November 12, 9:30a.m. EDT||Mikhail Khovanov||Introduction to categorification
We'll discuss categorification and its relation to link homology and representation theory.
|Thursday, November 19, 9:30a.m. EDT||Akash Sengupta||Manin's Conjecture: birational geometry and arithmetic
I'll give an introduction to Manin's conjecture about rational points and discuss the birational geometric invariants and the geometric thin set involved in the conjecture.
|Thursday, December 3, 9:30a.m. EDT||Mohammed Abouzaid||Is there a hyperbolic Lagrangian in the quintic 3-fold?
I will explain the state of our understanding about Lagrangian embeddings in projective varieties, focusing on the case of hypersurfaces. A particularly intriguing puzzle occurs for the degree 5 hypersurface in CP^4, which physicists predict should admit a Lagrangian manifold which admits a hyperbolic metric, but which no one has seen yet.
|Thursday, December 10, 4:00p.m. EDT||Giulia Sacca||Fano varieties and hyper-Kahler manifolds
Within the classification of smooth projective algebraic varieties, Fano manifolds and hyper-Kahler manifolds stand very far a part. For example, Fano manifolds have no non trivial global holomorphic forms while the geometry of hyper-Kahler manifolds is completely controlled by a holomorphic symplectic form. Yet, starting with certain Fano manifolds, several geometric constructions have allowed the construction of some hyper-Kahlers. In this talk I will give an overview of the relation between these two apparently very different classes of algebraic varieties.
|Thursday, December 17, 12:00p.m. EDT||Mu-Tao Wang||How does a black hole rotate?
Einstein's theory of general relativity stipulates that space, time, and gravity are unified by his eponymous equation. One would expect the theory to give a precise description of elementary properties of a black hole, such as its mass and angular momentum. However, essential difficulties from the gauge invariance and nonlinear nature of the theory have prevented such an understanding. In this talk, I will discuss these features and some recent applications of the mathematical theory of PDE and isometric embedding to evaluate angular momentum in general relativity.