Each week, the Michael Zhao Memorial Student Colloquium holds 45minute 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.
This Fall the Seminar is organized inperson. It meets on Tuesdays at 6:00p.m. EDT in Mathematics Room 507.
Organizers: Georgy Gaitsgori and Patrick Lei.
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  dgAlgebras, twisted homology, and homotopy commutativity
Given an algebraic object, there is a long history of studying deformations of this object. When A is a dgalgebra (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 dgalgebra 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 $\cup_1: A^p \otimes A^q \to A^{p+q1}$ which encodes the failure of A to be a commutative algebra. In particular, when $H^*(A)$ is also torsionfree, 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 grouplike objects associated to the variety. I will discuss a Hall algebra associated to CalabiYau 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 "torsionfree 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 padic 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 lowdimensional topology for building 3manifolds. 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 4manifolds: From Casson to Conway and back
Fourdimensional topology is famous for its "exotic" phenomena, differences between continuous and differential topology. In this talk, I will discuss the close connections between 4manifold 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 nonexistence of a smoothly embedded disk in 4space bounded by the "Conway knot" (proven by Piccirillo in 2018). I will close by explaining where my work fits into this broader story. 