Michael Zhao Memorial Student Colloquium (Spring 2023)

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 ams2637@columbia.edu or rh3101@columbia.edu.


Inbar Klang
(Homotopical) fixed point theory
Given a map \(f\) from a topological space \(X\) to itself, one can ask whether it has fixed points, and whether it can be modified to remove the fixed points. I will discuss invariants in algebraic topology that answer these questions, the Lefschetz number and the Reidemeister trace. Time permitting, I will briefly discuss variations of this problem (equivariant, or in families), in which homotopy theory becomes very useful.
Andrew Hanlon
Homological mirror symmetry for projective space
The homological mirror symmetry conjecture aims to relate natural categories in algebraic and symplectic geometry. Namely, it predicts a derived equivalence between the category of coherent sheaves on certain spaces and the Fukaya category on a mirror space. We will discuss aspects of this conjecture and its proof for projective space while pointing out more general principles.
Milind Hegde
A universality class for highly correlated random systems
In classical probability theory, the Gaussian or normal distribution occupies a fundamental role. This is because it arises in a dizzying number of situations: the most well-known example statement (and there are many more) is the central limit theorem, which says that the normal distribution is the limit of a suitably rescaled sum of independent and statistically identical random variables, no matter the details of the distribution. As such, there is a Gaussian universality class for systems with a high degree of independence in the probabilistic sense. In this talk I will discuss another universality class that has been the subject of a great deal of study over the past few decades: it is known as the KPZ universality class, and seems to cover random systems which are highly correlated. I will showcase two or three very different looking models which are all part of the same class and, time permitting, try to highlight some of the connections to other parts of probability theory and mathematics: chaos, functional analysis, algebraic combinatorics, and more.
Mike Miller
Instantons: From Physics to Exotic Topology
Yang-Mills isntantons are the global minima of the Yang-Mills energy functional, whose dynamics govern the nuclear forces in particle physics. For the past 40 years, topologists have seen that these also have a tendency to govern exotic phenomena in 4-dimensional topology. I will explain the geometric definition of the defining equations, the physical origin of their study, and the story that leads to the result of Donaldson and Freedman that there exist many topological 4-manifolds with no smooth structure. I will take for granted that people have some familiarity with manifolds and de Rham cohomology. Some other terms will be mentioned (Riemannian manifold, vector bundle, Hodge star), but I do not expect familiarity with these.
Chen-Chih Lai
Free boundary problem for a gas bubble in a liquid
This talk concerns the dynamics and asymptotic behaviors of a gas bubble deforming in a liquid. Generally, there are three damping mechanisms for energy dissipation of bubble dynamics: viscous damping, radiation damping, and thermal damping. I will give a brief overview on the models capture the viscous and the radiation damping mechanisms. Special emphasis in this talk will be given to thermal relaxation dynamics of a spherically symmetric gas bubble in an incompressible liquid. We discuss the approximate model proposed by A. Prosperetti in 1991. This model is a coupled system of a quasilinear PDE with a free boundary and an ODE for the bubble radius. The system admits a one-parameter manifold of spherical equilibria, parametrized by the bubble mass. I will present recent uniqueness and stability results (joint work with Michael I. Weinstein) of the manifold of spherical equilibria. Finally, if time permits, we discuss works in progress and future directions.
Spring Break
Daniele Alessandrini
Group actions on homogeneous spaces
Homogeneous spaces carry an interesting geometry, and the natural groups of transformations preserving this geometry can move any point to every other point. Examples are (real or complex) projective spaces, Grassmannians, flag manifolds, hyperbolic spaces and many others. In higher Teichmüller theory we can construct interesting actions of surface groups on these homogeneous spaces. There are many questions about understanding the dynamical, geometric and topological properties of such actions.
Elena Giorgi
The Stability of Charged Black Holes
Black hole solutions in General Relativity are parameterized by their mass, spin, and charge. In this talk I will present the main properties of the black hole solutions and I will motivate why the charge of black holes add interesting dynamics to solutions of the Einstein equation thanks to the interaction between gravitational and electromagnetic radiation.
Konstantin Aleshkin
Equivariant elliptic cohomology (Math 307 @ 7:30)
Usual cohomology and K-theory of a topological space (manifold/variety) fit in the framework of generalized cohomology theories. Elliptic cohomology is another example of such and is a natural step behind the K-theory in many contexts. In the talk I will focus on one of its flavors - equivariant elliptic cohomology. In particular, I plan to discuss why it not an abelian group but a sheaf and compute a few basic examples.
Siddhi Krishna
Genus problems (in low-dimensional topology)
Many lines of inquiry within low-dimensional topology, algebraic geometry, and geometric analysis have the following form: "Suppose I have a space and some distinguished subspace of it. When is that subspace 'minimal' with respect to my favorite notion of 'minimal'?". In this talk, I'll make this vague question more precise, state examples of these types of questions in the fields I mentioned above, and survey some concrete questions and answers in the context of 3- and 4-manifold topology. I will not assume any background in topology. All are welcome!

Past Seminars

Fall 2022

Spring 2022

Fall 2021

Spring 2021

Fall 2020

Spring 2020

Fall 2019

Spring 2019

Fall 2018

Spring 2018