Postdoc Seminar Series

Organizers: Xi Sisi Shen, Siddhi Krishna, Nathan Chen

Goal: Bring together postdocs in the Columbia math department for broad-audience math talks!

These will be held roughly once a month during the academic year 2023-2024. See below for the talk details:

Title: Finding arithmetic structures in certain complex manifolds

Abstract: An arithmetically interesting algebraic variety is often produced by showing that a complex algebraic manifold can in fact be defined over a number field. The most famous examples in number theory are Shimura varieties, and they can define what it means for a transcendental object to be arithmetic. For example, one can define what it means for a modular form, a priori some highly symmetric holomorphic function, to be defined over the rational numbers. The situation is even more interesting if there is no evident underlying algebraic variety - for some non-algebraic complex manifolds, we still see hints of arithmetic structures in the cohomology! We will talk about the journey of finding arithmetic structures in certain transcendental objects (e.g. flag domains, Penrose transforms, twistor spaces).

Title: Categorical dynamics on stable module categories

Abstract: Given a mathematical object X and an endomorphism f of X, entropy assigns to this pair a number h(f) measuring the dynamical complexity of f. Initially defined for measure spaces and topological spaces, it has also been generalized to measure the complexity of endomorphisms of categories of sheaves arising in algebraic geometry and homotopy theory. I will discuss a result showing that categorical entropy measures homotopically meaningful information and work in progress to bring a dynamical perspective to homotopy theory.

Title: Legendrian knots and complex Grassmannians

Abstract: In recent years there has been remarkable cross-pollination between the fields of Legendrian knot theory and cluster algebras. One particular instance of this is a certain relation between a class of Legendrian knots and positroid varieties; they are subvarieties of complex Grassmannians whose coordinate rings are cluster algebras. In  this talk we will explore this relation further, explaining the result that inclusion of positroid varieties corresponds to Lagrangian cobordism of the Legendrian knots. This reports on joint work with Y. Bae, O. Capovilla-Searle, M. Castronovo, C. Leverson and A. Wu.

The talks below are from the past academic year (2022-2023).

Title: An invitation to higher algebra

Abstract: Ever find yourself struggling to generalize a theorem from characteristic 0 to characteristic p?  In this talk, I'll try to convince you that higher algebra is a powerful tool for doing so (with examples from arithmetic/symplectic/manifold geometry and representation theory).  Higher algebra is a generalization of algebra coming from algebraic topology which has "intermediate" characteristics interpolating between 0 and p.  Most of the talk will be a gentle introduction to how this picture arises, but I'll end with a brief discussion of joint work with Burklund and Schlank on a "higher Hilbert's Nullstellensatz."

Title: Gauged Gromov-Witten theory and affine Grassmannians

Abstract: The study of the geometry and topology of Riemann surfaces has captured the interest of geometers for decades. They lie in the intersection of many areas of mathematics: we can aim to understand representations of their fundamental groups, we can attempt to classify differential equations on them, or perhaps study their algebraic vector bundles. It is also tempting to try to classify all compact Riemann surfaces themselves. Moduli theory provides a way to understand many of these classification problems, by turning our focus to the geometry of a parameter space. These parameter spaces, called moduli spaces, provide us in turn with a lot of examples of complex varieties with rich geometric features. In this talk I will discuss some recent techniques developed to construct such moduli spaces and study their geometry. With time permitting, I will also try to explain what it means to count vector bundles on compact Riemann surfaces, and why such counts are given by combinations of certain special values of transcendental functions.

Title: Cohomologies of moduli spaces of sheaves and applications

Abstract: Given an algebraic variety, one can construct new varieties, called moduli spaces, which parametrize objects (such as points, curves, or vector bundles) on the initial variety. Even if one starts with a small dimensional variety with fairly simple geometry, the moduli spaces obtained are interesting examples of intricate higher dimensional varieties which carry rich structures with connections to many other areas of mathematics.

In this talk, I will discuss explicit computations of various cohomologies (some classical, some algebraic, and some inspired by physics) of moduli spaces of points for the affine two and three dimensional spaces. These computations have applications in representation theory and inspire the construction of new spaces in noncommutative algebraic geometry.

Title: A brief guided tour of the Kardar-Parisi-Zhang universality class

Abstract: Two classical objects in probability theory are the random walk and its scaling limit, Brownian motion. These objects are closely related to the well-known Gaussian distribution, and are associated to what is often called the Gaussian universality class, reflecting the wide variety of situations in which they arise. A common feature of members of the Gaussian universality class is a scaling exponent of 1/2: for instance, regarding a random walk as a path of length n, the typical size of random fluctuations at the midpoint is of order n^{1/2}. In the past few decades, a new universality class has been heavily investigated in probability theory, known as the Kardar-Parisi-Zhang (KPZ) universality class. This class has a different characteristic scaling exponent (1/3) and new universal scaling limits replacing the classical ones like the Gaussian distribution and Brownian motion. In this talk I will introduce some simple models which are members of the KPZ universality class, explain where the scaling exponents come from, and perhaps discuss some results. I will assume essentially no background in probability theory.

Title: Singularities and the minimal model program

Abstract: The minimal model program proposes a way of classifying algebraic varieties up to birational equivalence, by breaking down a variety into simpler varieties. It turns out that even if we are interested in smooth varieties, we inevitably encounter singularities while running the minimal model program. I'll discuss the singularities appearing in birational geometry and give an overview of the minimal model program.