MATH-IMS Joint Colloquium Series (Pure Mathematics)
2021-22 Term 1


Date/Time/Venue Talks
Sep 30, 2021 (Thur)
10:00AM
Zoom link
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
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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
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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
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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
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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
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Speaker: Robin Neumayer (Carnegie Mellon University)
Title: Quantitative Stability in the Calculus of Variations

Abstract: Among all subsets of Euclidean space with a fixed volume, balls have the smallest perimeter. Furthermore, any set with nearly minimal perimeter is geometrically close, in a quantitative sense, to a ball. This latter statement reflects the quantitative stability of balls with respect to the perimeter functional. We will discuss recent advances in quantitative stability and applications in various contexts. The talk includes joint work with several collaborators and will be accessible to a broad research audience.
Nov 18, 2021 (Thur)
10:00AM
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Speaker: Ovidiu Savin (Columbia University)
Title: The multiple membrane problem

Abstract: For a positive integer N, the N-membranes problem describes the equilibrium position of N ordered elastic membranes subject to forcing and boundary conditions. If the heights of the membranes are described by real functions $u_1, u_2,...,u_N$, then the problem can be understood as a system of N-1 coupled obstacle problems with interacting free boundaries which can cross each other. When N=2 there is only one free boundary and the problem is equivalent to the classical obstacle problem. In my talk I will review the free boundary regularity in obstacle-type problems and discuss some recent work in collaboration with Hui Yu on the multiple membrane problem.
Nov 25, 2021 (Thur)
4:00PM
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Speaker: Vincent Humilière (Institut de Mathématiques de Jussieu)
Title: The group of area-preserving homeomorphisms of the 2-sphere: first secrets revealed

Abstract: I will report on recent results on the algebraic structure of this group which has remained very mysterious for a long time, in contrary to other similar but well-understood groups. I will discuss commutators, normal subgroups, homomorphisms, quasi-morphisms. In particular, I will present some ideas from joint work with Dan Cristofaro-Gardiner, Cheuk-Yu Mak, Sobhan Seyfaddini and Ivan Smith which involves studying the symplectic topology of symmetric products of spheres.
Dec 2, 2021 (Thur)
10:00AM
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Speaker: Lan-hsuan Huang (University of Connecticut)
Title: A geometric boundary value problem in general relativity

Abstract: Constructing Riemannian metrics of zero scalar curvature with prescribed boundary geometry is of fundamental importance in general relativity and differential geometry. This talk will focus on a special class of metrics of zero scalar curvature, called static vacuum. A static vacuum metric produces a Ricci flat manifold of one dimension higher and is related to scalar curvature deformation and gluing. There were very limited examples of static vacuum metrics without symmetry. For example, the celebrated Uniqueness Theorem of Static Black Holes says that any asymptotically flat, static vacuum metric with minimal surface boundary must be rotationally symmetric. In contrast, motivating by his quasi-local mass program, R. Bartnik conjectured that one should always find an asymptotically flat, static vacuum metric with quite arbitrarily prescribed boundary geometry. I will discuss recent progress toward this conjecture. It is based on joint work with Zhongshan An.
Dec 9, 2021 (Thur)
4:00PM
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Speaker: Ye Tian (Morningside Center of Mathematics, Chinese Academy of Sciences)
Title: Congruent Number Problem and L-Functions

Abstract: Recall a positive integer is called a congruent number if it is the area of a right triangle with rational side lengths. For example, 5, 6, 7 are congruent number (Fibonacci) and 1, 2, 3 are not (Fermat). The congruent number problem (CNP for short) is to determine whether a given positive integer is a congruent number or not. The L-function of elliptic curve is an important tool to study CNP. In this talk, we introduce some recent progress.



© Martin Li, Department of Mathematics, The Chinese University of Hong Kong