Michael Zhao Memorial Student Colloquium (Spring 2022)

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 kyc2130@columbia.edu or qy2266@columbia.edu.


Will Sawin (Zoom talk)
Finite quotients of 3-manifold groups
Let G and H be two finite groups. Does there exist a 3-manifold whose fundamental group admits G as a quotient but not H? We prove a theorem that determines the answers to questions of this type. The proof relies on a probabilistic argument - we estimate the probability that a random 3-manifold (according to a distribution defined by Dunfield and Thurston) has a fundamental group with certain quotients but not others. Our methods grew out of work in number theory on the class groups of random number fields. This talk will discuss the work in topology and its connections to number theory. This is joint work with Melanie Wood.
01/26 (Wednesday)
Konstantin Aleshkin (Zoom talk)
Higgs-Coulomb correspondence in GLSM
Gauged Linear Sigma Models (GLSM) provide interesting models of enumerative geometry. They generalize curve counting for hypersurfaces in toric varieties and Landau-Ginzburg models. Higgs-Coulomb correspondence for GLSM states that generating functions of GLSM invariants ("Higgs branch") can be written as certain explicit Mellin-Barnes type integrals ("Coulomb branch"). In the talk I will explain the idea behind this correspondence and try to show how it allows us to produce remarkable results in enumerative geometry.
Robert Friedman
Singularities and their deformations
We discuss isolated singularities in algebraic and complex analytic geometry, and describe how their deformations are related to topology and geometry.
Inbar Klang
Introduction to factorization homology
This talk will introduce factorization homology from a configuration spaces perspective. We'll also talk about nonabelian Poincare duality, which relates factorization homology to mapping spaces.
Mu-Tao Wang
How do black holes rotate?
The event GW150914 in the first observation of gravitational waves by LIGO and Virgo corresponds to a binary black hole merger: two black holes rotate about each other and eventually settle down to a single rotating black hole. The initial black hole masses are 36 units and 29 units and the final black hole mass is 62 units, with 3 units of mass radiated away in gravitational waves. One naturally wonders how much "angular momentum" radiated away. This turned out to be a more subtle and challenging question due to the presence of "supertranslation ambiguity," which was discovered in the 1960’s. In this talk, I will explain how a recent development in the theory of mathematical relativity identities a new definition of angular momentum and completely resolves such ambiguity.
Melissa Liu
Mirror symmetry for projective complete intersections and quasimap wall-crossing
This is an expository talk on a mirror theorem for complete intersections in projective spaces, first proved by Givental and Lian-Liu-Yau in 1996-7, and the more recent proof by Ionut Ciocan-Fontanine and Bumsig Kim via quasimap wall-crossing.
Eric Urban
Euler systems of rank two for adjoint modular Galois representations
I will explain how the study of congruences between modular forms of various weights and levels provides a way to construct a rank two p-adic Euler system for adjoint modular Galois representations and its application towards the Bloch-Kato conjecture for its twists by Dirichlet character of level prime to p in the non ordinary Fontaine-Lafaille case.
Giulia Saccà
Moduli spaces on K3 categories are irreducible symplectic varieties
From classical work of Mukai, to more recent work of Kuznetsov and others, it has been established that the stable locus of certain moduli spaces on K3 surfaces and related objects have a holomorphic symplectic structure. I will give an overview of these topics and then talk about recent results dealing with global properties of these moduli spaces.
Mikhail Khovanov
Introduction to categorification and link homology
Categorification lifts quantum invariants of links, such as Jones polynomial and Reshetikhin-Turaev invariants, to bigraded homology theories of links. We'll give a broad survey of these structures.
Aleksander Doan
Invariants of Calabi-Yau threefolds
One of the central problems of geometry is to classify manifolds which admit special geometric structures. Calabi-Yau manifolds provide a particularly interesting class of examples. Since their discovery forty years ago they have stimulated extraordinary research activity, leading to new, beautiful mathematics and surprising connections with physics. A distinctive feature of Calabi-Yau manifolds is that they lie at the intersection of three branches of geometry: algebraic, differential, and symplectic. As a result, there is an abundance of examples of these manifolds as well as tools to understand them. The talk will focus on invariants of Calabi-Yau manifolds of complex dimension three, which are defined by counting holomorphic curves. Towards the end of the talk, I will outline a proposal for defining a new invariant, which counts holomorphic curves together with solutions to partial differential equations originating from gauge theory; the technical challenges of this proposal will lead us to explore uncharted territories in geometry and analysis.
Daniela De Silva (Room 312)
On Free Boundary Problems
In this talk, we will present an overview of techniques and results concerning the regularity theory for Free Boundary Problems (FBP), that is problems in which one must solve a PDE and along the way find out the region in which the PDE holds. FBP naturally arise in a variety of applications and research in this area has been very fruitful and active for several decades. Using the Bernoulli one-phase problem as a basic elliptic model, we will highlight main contributions and open questions often originating from a striking resemblance with the regularity theory for minimal surfaces. We will further consider parabolic problems, including the classical Stefan problem. If time permits it, we will describe so-called thin free boundary problems, in which the free boundary occurs on a lower dimensional subspace, and that arise in connection with non-local phenomena.
Amadou Bah (Room 312)
The Ramification theory of local fields: from perfect to imperfect residue fields
The ramification theory of discrete valuation fields plays a crucial role in various subfields of Arithmetic Geometry. In this expository talk, I will recall enough of the classical picture (perfect residue field) to introduce the key idea in A. Abbes and T. Saito's generalisation of the theory to arbitrary residue fields. I will then explain how such a generalisation enters into Takeshi Saito's extension to arbitrary dimensions of the classical Grothendieck-Ogg-Shafarevich formula for the Euler characteristic of an l-adic sheaf on a curve over a perfect field.

Past Seminars

Fall 2021

Spring 2021

Fall 2020

Spring 2020

Fall 2019

Spring 2019

Fall 2018

Spring 2018