MATH-IMS Joint Colloquium Series
(Pure Mathematics)

• Organizers: Yi-Jen Lee and Martin Li
• Hosts: Department of Mathematics & Institute of Mathematical Sciences, CUHK
• Contact: Please email to or martinli@math.cuhk.edu.hk for any inquiries. The recording of the talks are available upon request.

• Remark: Due to current COVID-19 situation, we will be hosting the colloquium talks online via ZOOM until safe travels are permitted. You can access the talk via the ZOOM link below.

Upcoming talk

Date/Time/Venue Talks
Oct 28, 2021 (Thur)
10:00AM
Speaker: Thomas Lam (University of Michigan)
Title: From Grassmannians to Catalan numbers

Abstract: The binomial coefficients have a well-studied q-analogue known as Gaussian polynomials. These polynomials appear as Poincare polynomials (or point counts) of the Grassmannian of k-planes in $\mathbf{C}^n$ (or $\mathbf{F}_q^n$).
Another family of important combinatorial numbers is the Catalan numbers, and they have two well-studied q-analogues from the 1900s, due to Carlitz and Riordan, and to MacMahon respectively. I will explain how these q-analogues appear as the Poincare polynomial and point count, respectively, of an open (non-compact) subvariety of the Grassmannian known as the top positroid variety. The story involves connections to knot homology and to the geometry of flag varieties.
The talk is based on joint work with Pavel Galashin.

2021-22 Fall schedule

Date/Time/Venue Talks
Sep 30, 2021 (Thur)
10:00AM
Speaker: Richard Bamler (University of California, Berkeley)
Title: Recent developments in Ricci flow

Abstract:
Ricci flows are a powerful geometric-analytical tool, as they have been used to prove important results in low-dimensional topology.
In the first part of this talk I will focus on Ricci flows in dimension 3. I will briefly review Perelman’s construction of Ricci flow with surgery, which led to the resolution of the Poincaré and Geometrization Conjectures. Then I will discuss recent work of Lott, Kleiner and myself on an improved version of this flow, called “singular Ricci flow”. This work allowed us to resolve the Generalized Smale Conjecture, concerning diffeomorphism groups, and a conjecture concerning the contractibility of the space of positive scalar curvature metrics on 3-manifolds.
In the second part of the talk, I will focus on Ricci flows in higher dimensions. I will present a new compactness theory, which can be used to study the singularity formation of the flow, as well as its long-time asymptotics. I will discuss these and some further consequences. I will also convey some intuition of the new terminology that had to be introduced in connection with this compactness theory.
Oct 7, 2021 (Thur)
10:00AM
Speaker: Jonathan Hanselman (Princeton University)
Title: The Cosmetic Surgery Conjecture and Heegaard Floer homology

Abstract: The cosmetic surgery conjecture is a basic open question concerning Dehn surgery on knots, a fundamental operation in low-dimensional topology. Generalizing the knot complement problem settled by Gordon and Luecke, it asserts that two different surgeries on the same knot never produce the same 3-manifold. I will give an overview of the conjecture and discuss some recent progress. I will also discuss the machinery behind these results, which is of independent interest. The work I will describe makes use of Heegaard Floer homology, a powerful suite of invariants for both 3-manifolds and knots. While these invariants have been around for nearly two decades, new results were facilitated by a recent reinterpretation of these invariants due to Rasmussen, Watson, and myself. In particular, there is an equivalence between the algebraic objects Heegaard Floer theory traditionally associates to knots and certain geometric objects---collections of immersed curves in the punctured torus. This leads to a beautiful interplay between algebraic and geometric techniques which, among many other applications, points to strong obstructions to cosmetic surgeries.
Oct 21, 2021 (Thur)
10:00AM
Speaker: Dan Cristofaro-Gardiner (University of Maryland)
Title: Periodic Floer homology and surface dynamics

Abstract: Periodic Floer homology (PFH) is an algebraic invariant associated to area-preserving surface diffeomorphisms. Lee and Taubes have shown that PFH is isomorphic to a version of Seiberg-Witten Floer homology, and so PFH links topology and dynamics in a novel and fruitful way. We recently used this bridge to settle several longstanding problems in surface dynamics. I will explain a bit about the ideas for this in the case of two of these problems: our resolution of the Simplicity Conjecture, which states that the group of compactly supported area-preserving homeomorphisms of the two-disc is not simple; and our resolution of a version of the smooth closing lemma, which implies that a generic smooth area-preserving diffeomorphism of a closed surface has a dense set of periodic points. A kind of Weyl law recovering the classical Calabi invariant from the asymptotics of PFH plays a key role in both proofs. These are joint works with Seyfaddini and Humiliere in the first case, and Prasad and Zhang in the second.
Oct 28, 2021 (Thur)
10:00AM
Speaker: Thomas Lam (University of Michigan)
Title: From Grassmannians to Catalan numbers

Abstract: The binomial coefficients have a well-studied q-analogue known as Gaussian polynomials. These polynomials appear as Poincare polynomials (or point counts) of the Grassmannian of k-planes in $\mathbf{C}^n$ (or $\mathbf{F}_q^n$).
Another family of important combinatorial numbers is the Catalan numbers, and they have two well-studied q-analogues from the 1900s, due to Carlitz and Riordan, and to MacMahon respectively. I will explain how these q-analogues appear as the Poincare polynomial and point count, respectively, of an open (non-compact) subvariety of the Grassmannian known as the top positroid variety. The story involves connections to knot homology and to the geometry of flag varieties.
The talk is based on joint work with Pavel Galashin.
Nov 4, 2021 (Thur)
10:00AM
Speaker: Chao Li (Courant Institute, New York University)
Title: Stable minimal hypersurfaces in $\mathbf{R}^4$

Abstract: In this talk, I will discuss the Bernstein problem for minimal surfaces, and the recent solution to the stable Bernstein problem for minimal hypersurfaces in $\mathbf{R}^4$. Precisely, we show that a complete, two-sided, stable minimal hypersurface in $\mathbf{R}^4$ is flat. Corollaries include curvature estimates for stable minimal hypersurfaces in 4-dimensional Riemannian manifolds, and a structural theorem for minimal hypersurfaces with bounded Morse index in $\mathbf{R}^4$. This is based on joint work with Otis Chodosh.
Nov 11, 2021 (Thur)
10:00AM
Speaker: Robin Neumayer (Carnegie Mellon University)
Title: TBA

Abstract: TBA
Nov 18, 2021 (Thur)
10:00AM
Speaker: Ovidiu Savin (Columbia University)
Title: TBA

Abstract: TBA
Nov 25, 2021 (Thur)
4:00PM
Speaker: Vincent Humilière (Institut de Mathématiques de Jussieu)
Title: TBA

Abstract: TBA
Dec 2, 2021 (Thur)
10:00AM
Speaker: Lan-hsuan Huang (University of Connecticut)
Title: TBA

Abstract: TBA
Dec 9, 2021 (Thur)
4:00PM