# Michael Zhao Memorial Student Colloquium

## Fall 2021

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 ggaitsgori@math.columbia.edu or plei@math.columbia.edu.

## Seminar Plan

This Fall the Seminar is organized in-person. It meets on Tuesdays at 6:00p.m. EDT in Mathematics Room 507.

Organizers: Georgy Gaitsgori and Patrick Lei.

## Past Seminars

Date and time Speaker Title and abstract
Tuesday, September 21, 6:00p.m. EDT Simon Brendle Minimal surfaces and the isoperimetric inequality

Tuesday, September 28, 6:00p.m. EDT Michael Miller dg-Algebras, twisted homology, and homotopy commutativity

Given an algebraic object, there is a long history of studying deformations of this object. When A is a dg-algebra (think: "cochains on a space X") and M is a module over A (think: "cochains on a space mapping to X"), then recently guided by questions in Floer homology I have been interested in studying deformations of the differential on M. These deformations are governed by certain sequences of elements of A called twisting sequences, and these twisting sequences have an intrinsic notion of homotopy, so that the homology of the deformation is determined by knowing the twisting sequence up to homotopy. I will discuss how to study these twisting sequences and how to control what happens to them as you pass from one dg-algebra to another. This turns out to be impossible when A has no further structure, but accessible when A is also equipped with the structure of a homotopy commuting algebra: it has an additional product $\cup_1: A^p \otimes A^q \to A^{p+q-1}$ which encodes the failure of A to be a commutative algebra. In particular, when $H^*(A)$ is also torsion-free, we can completely solve the question: can I determine algorithmically when two twisting sequences are homotopic?
Tuesday, October 19, 6:00p.m. EDT Tudor Padurariu Hall algebras and enumerative geometry

Hall algebras are a general device that constructs an algebra out of a category, where the multiplication between two objects encodes information about possible extensions of those objects. When the category is representations of a quiver (for example, a Dynkin diagram), Hall algebras recover quantum groups associated to the quivers. Hall algebras of smooth varieties have been studied and one expects that in dimensions at most 2 they are related to quantum group-like objects associated to the variety. I will discuss a Hall algebra associated to Calabi-Yau 3 folds and mention the expected connections between its structure and counting curves on the 3 fold.
Tuesday, November 9, 6:00p.m. EDT Allen Yuan Algebraic models for spaces

One fundamental goal of algebraic topology is to assign algebraic invariants to topological spaces. As one develops more and more sophisticated homotopy invariants for spaces, it becomes natural to ask: how much of a space is captured by these invariants? Can one somehow completely capture a space, up to homotopy equivalence? In my talk, I will survey some answers to this question, starting with Sullivan's work on rational homotopy theory, which describes the "torsion-free part" of a space via a certain commutative differential graded algebra of cochains. Time permitting, we will then explore how more sophisticated homotopy theoretic ideas go into Mandell's work on p-adic homotopy theory and beyond.
Tuesday, November 16, 6:00p.m. EDT Elena Giorgi The stability of black holes

Black holes are fundamental objects in our understanding of the universe. The mathematics behind them has surprising geometric properties, and their dynamics is governed by hyperbolic PDEs. We will see how one we can answer to the basic question of whether these solutions to the Einstein equation are stable under small perturbations, which is a typical requirement to be physically meaningful, and how the dispersion of gravitational waves plays a key role in the stability problem.
Tuesday, November 23, 6:00p.m. EDT Marco Castronovo The Laurent phenomenon in topology

In the early 2000s, Fomin and Zelevinsky introduced a method to generate families of Laurent polynomials whose coefficients have curious properties. I will explain in what sense one can hope to use this to classify Lagrangians in a compact symplectic manifold.
Tuesday, November 30, 6:00p.m. EDT Siddhi Krishna Dehn Surgery: Why and How

In this talk, I'll introduce Dehn surgery, a prominent technique within low-dimensional topology for building 3-manifolds. Dehn surgery can be studied using a variety of tools, including hyperbolic geometry, representation theory, and Floer homology. I'll provide an overview of major themes, questions, and results, as well as types of tools developed along the way. No background in topology is expected; all are welcome!
Tuesday, December 14, 6:00p.m. EDT Kyle Hayden Exotic 4-manifolds: From Casson to Conway and back

Four-dimensional topology is famous for its "exotic" phenomena, differences between continuous and differential topology. In this talk, I will discuss the close connections between 4-manifold topology and knot theory, focusing on two results proven nearly 40 years apart: the existence of exotic smooth structures on R^4 (proven by Donaldson and Freedman in the early 1980's) and the non-existence of a smoothly embedded disk in 4-space bounded by the "Conway knot" (proven by Piccirillo in 2018). I will close by explaining where my work fits into this broader story.