Michael Zhao Memorial Student Colloquium (Fall 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.

Schedule

09/06
Gyujin Oh
Cohomological degree-shifting actions on locally symmetric spaces
The degree-shifting action on the cohomology of locally symmetric spaces, which has its origins in the representation theory of real reductive groups, enjoys a surprising connection with arithmetic, as expected by the so-called motivic action conjectures of A. Venkatesh. Although these conjectures are expected to hold in great generality, there is a disparity between the algebraic and non-algebraic locally symmetric spaces. We will discuss the nature of the degree-shifting action in both cases using Hodge theory.
09/13
Florian Johne
Positive curvature and topology
Understanding the interplay of curvature and topology is a longstanding question in geometry. We review some classical results and techniques, before we focus on the minimal surface technique: Using the stability inequality for minimal surfaces, one can rule out metrics of positive Ricci curvature on manifolds with topology $M x \Sphere^1$, and metrics of positive scalar curvature on the torus $\Torus^n$. Finally, if time permits, we explain a recent result (joint with S.~Brendle and S.~Hirsch) on the non-existence of metrics of positive intermediate curvature on more general products of tori and spheres.
09/20
Simon Brendle
Singularity models in geometric flows
Geometric flows have played a central role in differential geometry over the past 40 years. The most important example is the Ricci flow for Riemannian metrics. The main problem is to understand singularity formation. Singularities can often be modeled on ancient solutions. These are solutions that have backhistory extending infinitely far into the past. I will discuss recent developments leading to a classification of these singularity models in various settings.
09/27
Andres Fernandez Herrero
Moduli of vector bundles on curves and related moduli problems
In the first part of this talk we will discuss the construction of the moduli space of vector bundles on a fixed smooth projective curve. The construction using Geometric Invariant Theory (GIT) goes back to the work of Mumford. We will try to explain what it means to construct a moduli space, how to parametrize moduli problems using schemes/stacks, explain the role of semistability, and give a superficial overview of the construction of moduli of vector bundles (and more generally sheaves) omitting technical details.
By the end of the talk, I will try to describe some joint work with Daniel Halpern-Leistner, where we apply some stack-theoretic techniques to construct moduli spaces in a broad range of moduli problems related to vector bundles (including Higgs bundles, Bradlow pairs etc. as special cases).
10/11
Francesco Lin
Closed geodesics of hyperbolic manifolds
The study of lengths of closed geodesics of hyperbolic manifolds is a very rich topic with connections to several fields including topology, number theory and spectral geometry. In this talk, after discussing the basics concepts and results in hyperbolic geometry, I will discuss some of these interactions.
10/18
Andrew Hanlon
TBD

Past Seminars

Spring 2022

Fall 2021

Spring 2021

Fall 2020

Spring 2020

Fall 2019

Spring 2019

Fall 2018

Spring 2018