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.

- When: Tuesday 6:00 - 7:00 PM ET
- Where: Mathematics Building, Room 520
- Organizers: Kevin Chang, Vivian Yu, Rafah Hajjar Muñoz, Alex Scheffelin

- 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.
- 10/26 (Wednesday, Room 507)
- Dorian Goldfeld
**The Template Method in Number Theory**- A “template” is a pattern used as guide for producing other similar things. We will show how Borel Eisenstein Series in the theory of automorphic forms can be used as a “template" to deduce properties of other automorphic forms. This lecture (based on joint work with S. Miller and M. Woodbury) will include a brief introduction to automorphic forms and Eisenstein series.
- 11/01
- Johan Asplund
**Legendrian knot theory and the Chekanov-Eliashberg dg-algebra**- Legendrian knots are knots in 3-dimensional space obeying certain geometric restrictions, and are important objects in contact geometry and symplectic geometry. The aim of this talk is to give a short introduction to Legendrian knot theory and define the Chekanov-Eliashberg dg-algebra.
- 11/15
- Michele Fornea
**Plectic Stark-Heegner points: remarkable solutions to elliptic curves' equations**- In this talk I will explain how p-adic analytic methods can be used to conjecturally produce remarkable algebraic solutions to elliptic curves' equations.
- 11/22
- Allen Yuan
**A brief tour of higher algebra**- Spectra are among the most fundamental objects in algebraic topology and appear naturally in the study of generalized cohomology theories, higher K-groups and cobordism invariants. My goal is to explain the modern perspective that spectra are natural homotopical analogues of abelian groups in a theory of “higher algebra,” where one has topological analogues of algebraic structures like rings, modules, and tensor products. This perspective promotes new interactions with other areas of mathematics. On the one hand, the existence of additional “chromatic primes” in higher algebra (interpolating between characteristic 0 and characteristic p) has shed light on mod p phenomena in geometry, number theory, and representation theory. On the other hand, the extension of algebraic ideas to higher algebra has been fruitful in topology: I will discuss work, joint with Robert Burklund and Tomer Schlank, which proves a higher algebra analogue of Hilbert’s Nullstellensatz. In addition to initiating the study of “chromatic algebraic geometry,” this work resolves a form of Rognes’ “chromatic redshift” conjecture in algebraic K-theory.
- 11/29
- Nathan Chen
**What are… rational maps?**- I will talk about rational maps and use them to give an introduction to measures of irrationality.
- 12/06
- Tudor Padurariu
**On derived categories of coherent sheaves on a variety**- I will mention some of the fundamental constructions and results in the study of derived categories of coherent sheaves on a variety. I will then discuss a conjecture of Bondal-Orlov and its consequences.