Student Mathematics Colloquium

Fall 2018

Each week, the Student Mathematics 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 me!

The colloquium meets at 6:00pm, Wednesdays in Mathematics Room 507.
The talks will be followed by dinner.
Date Speaker Title
September 19 Will Sawin  Number theory over function fields
September 26 Kyler Siegel

Symplectic geometry originally emerged in the 19th century as the natural geometric setting for Hamiltonian mechanics. However, it was unclear whether this geometric perspective had any serious implications for classical physics until 1985, when Gromov introduced J-holomorphic curves and used them to prove a number of deep results about the nature of symplectic geometry.

Since then, J-holomorphic curves have been used to define a large number of elaborate structures and invariants, many also related to modern theoretical physics and appearing in Homological Mirror Symmetry. In this talk I will give a highly biased introduction to the field, explaining some of the key objects, tools, and open problems.

October 3 Konstantin Matetski

The Kardar-Parisi-Zhang (KPZ) universality class contains models which describe one-dimensional random growing interfaces and which have the unusual dynamical scaling exponent z = 3/2. On the physical level such models describe real-world processes, like evolutions of fire fronts, bacterial colony boundaries, growing liquid crystals and coffee stains.

Depending on characteristics of models, the two universal objects arise in their scaling limits: the KPZ equation and the KPZ fixed point. A solution theory of the first object was a part of the breakthrough of M. Hairer in 2010, while a complete characterisation of the second object was obtained only in the last year by Matetski, Quastel and Remenik. This talk is an introduction to the KPZ universality.

October 9 (Tuesday 7pm) Andrei Okounkov  Geometric representation theory and enumerative geometry
October 17 Mu-Tao Wang

I would like to discuss the role mathematics plays in the historical development and the frontier research of general relativity.

October 24 Alexandra Florea  Moments and zeros of L-functions over function fields
October 31 Dusa McDuff

An elementary introduction to symplectic geometry inspired by the question: what can you say about the shape of the image of a round ball in 4 dimensions under a transformation of the space that preserves the symplectic structure?

November 7 Daniela De Silva  Free boundary problems
November 14 Postponed (no talk)  Postponed (no talk)
November 21 Thanksgiving (no talk)  Thanksgiving (no talk)
November 28 Chiu-Chu Melissa Liu  Mirror symmetry and enumerative geometry
December 10 (Monday) Michael Harris

An arithmetic group is a discrete subgroup of a semisimple (or reductive) Lie group such that the quotient space has finite invariant volume. The starting point for the applications of the Langlands program to number theory is the observation that the cohomology of an arithmetic group, viewed as an abstract group, can be computed in terms of automorphic forms. Most of the talk will be devoted to explaining this relation and how it serves as a bridge between Galois representations and analysis on Lie groups.

It has long been known that a general arithmetic group has cohomology repeated in several degrees. Venkatesh has proposed a conjectural framework that extracts additional number-theoretic information from this structure. I will explain how this works in the simplest examples.

December 12 Chao Li  Moonshine and the BSD conjecture
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Organizer: Robin Zhang