KIAS Geometry and Analysis Seminar

2021

June 1st(T), 3rd(R), 8th(T), 10th(R), 15th(T), 5 pm in Korea (online)

Simon Brendle (Columbia University)

1st Tuesday : Minimal surfaces and the isoperimetric inequality

The isoperimetric inequality has a long history, going back to the legend of Queen Dido. In this lecture, I will discuss the isoperimetric inequality and how it relates to the calculus of variations, the notion of mean curvature, and minimal surface theory.

Zoom ID : 817 2528 0484

3rd Thursday : Singularity models in 3D Ricci flow

The Ricci flow is a natural evolution equation for Riemannian metrics on a given manifold. From a PDE perspective, the Ricci flow is a system of linear parabolic equations, which can be viewed as the heat equation analogue of the Einstein equations in general relativity. The central problem in the field is to understand singularity formation. In other words, what does the geometry look like at points where the curvature is large? In his spectacular 2002 breakthrough, Perelman achieved a qualitative understanding of singularity formation in dimension 3; this is sufficient for topological conclusions. In this lecture, we will discuss recent developments which have led to a complete classification of all the singularity models in dimension 3.

Zoom ID : 897 6591 0165

8th Tuesday : The proof of the isoperimetric inequality for minimal surfaces

In this lecture, we will discuss a Michael-Simon-type inequality which holds for submanifolds in Euclidean space of arbitrary dimension and codimension. Our inequality is sharp if the codimension is at most 2. As a special case, we obtain a sharp isoperimetric inequality for minimal submanifolds in Euclidean space of codimension at most 2, thereby answering a question going back to work of Carleman. The proof is inspired by, but does not actually use, optimal transport.

10th Thursday : Classification of ancient solutions to 3D Ricci flow, part I

15th Tuesday : Classification of ancient solutions to 3D Ricci flow, part II

It was shown by Perelman that any finite-time singularity of the Ricci flow in dimension 3 is modeled on an ancient solution which is kappa-noncollapsed and has bounded and nonnegative curvature. In this lecture, we will discuss the classification of these ancient kappa-solutions in dimension 3. It turns out that the only noncompact examples are the shrinking cylinders (and their quotients), and the rotationally symmetric Bryant soliton. This confirms a conjecture of Perelman.

Zoom ID : 828 0166 7463

March 8th Monday & 9th Tuesday, 2 pm in Korea (online)

Woocheol Choi (Sungkyunkwan University)

1st lecture : An introduction to the convex optimization

In this talk, we review the basic concepts and results for the convex optimization, especially the gradient descent method.

2nd lecture : Communication-Computation balanced distributed convex optimization

Distributed convex optimization has received a lot of interest from many researchers since it is widely used in various applications, containing wireless network sensor and machine learning. Recently, Berahas et al (2018) introduced a variant of the distributed gradient descent called the Near DGD+ which combines nested communications and gradient descent steps. They proved that this scheme finds the optimum point using a constant step size when the target function is strongly convex and smooth function. In the first part, we show that the scheme attains O(1/t) convergence rate for convex and smooth function. In addition we obtain a convergence result of the scheme for quasi-strong convex function. In the second part, we use the idea of Near DGD+ to design a variant of the push-sum gradient method on directed graph. This talk is based on a joint work with Doheon Kim and Seok-bae Yun.

Zoom ID : 886 4774 5485

2020

December 19th Saturday, 9 am in Korea (online)

Sébastien Picard (University of British Columbia)

PDEs on Non-Kahler Calabi-Yau Manifolds

We will discuss the non-Kahler Calabi-Yau geometry introduced by string theorists C. Hull and A. Strominger. We propose to study these spaces via a parabolic PDE which is a nonlinear flow of non-Kahler metrics. This talk will survey works with T. Collins, T. Fei, D.H. Phong, S.-T. Yau, and X.-W. Zhang.

Zoom ID : 820 5588 3006

December 18th Friday, 9 am in Korea (online)

Luca Spolaor (University of California San Diego)

Regularity of the free boundary for the two-phase Bernoulli problem

I will describe a recent result obtained in collaboration with G. De Philippis and B. Velichkov concerning the regularity of the free boundaries in the two phase Bernoulli problems. The novelty of our work is the analysis of the free boundary at branch points, where we show that it is given by the union of two C1 graphs. This completes the work started by Alt, Caffarelli, and Friedman in the 80’s.

Zoom ID : 822 9064 3684

December 11th Friday, 9 am in Korea (online)

Liming Sun (University of British Columbia)

Ancient finite entropy flows by powers of curvature in R^2.

Ancient flows have been intensively studied in the mean curvature flow, a higher dimensional version of the curve-shortening flow. In particular, ancient mean curvature flows are useful to investigate singularities. In this talk, I will be talking about our study of the ancient solution of alpha-curve-shortening flow in R^2. Daskalopoulos, Hamilton, and Sesum classify ancient solutions for alpha=1 case, however, for alpha<1, very few are known and especially for small ones. Along this direction, we first construct a family of non-homothetic ancient flows whose entropy is finite. We determine the Morse indices and kernels of the linearized operator of self-shrinkers to the flows. Conversely, we are able to classify all the ancient solutions with finite entropy. It turns out all ancient solutions have the same asymptotic as the ones we have constructed. This work is joint with Keysongsu Choi.

Zoom ID : 890 3177 5805

December 5th Saturday, 8 am in Korea (online)

Otis Chodosh (Stanford University)

Soap bubbles and topological obstructions to positive scalar curvature

The Geroch conjecture (proven by Schoen-Yau and Gromov-Lawson) says that there does not exist a metric of positive scalar curvature (PSC) on the 3-torus. I will explain how to generalize this in certain directions using soap bubbles (minimizers of a prescribed mean curvature functional). In particular I will explain why a closed 5-dimensional aspherical manifold does not admit a PSC metric. This is joint work with Chao Li

Zoom ID : 882 5608 4613

November 20th Friday, 5 pm in Korea (online)

Felix Schulze (University of Warwick)

Mean curvature flow with generic initial data

We show that the mean curvature flow of generic closed surfaces in R^3 avoids asymptotically conical and non-spherical compact singularities. We also show that the mean curvature flow of generic closed low-entropy hypersurfaces in R^4 is smooth until it disappears in a round point. The main technical ingredient is a long-time existence and uniqueness result for ancient mean curvature flows that lie on one side of asymptotically conical or compact shrinking solitons. This is joint work with Otis Chodosh, Kyeongsu Choi and Christos Mantoulidis.

Zoom ID : 871 5601 4956

November 20th Friday, 8 am in Korea (online)

Christos Mantoulidis (Brown University)

Ancient mean curvature flows, gradient flows, and Morse index

This talk will be about joint work with Kyeongsu Choi regarding closed ancient solutions to gradient flows of elliptic functionals in Riemannian manifolds. Our methods classify ancient solutions coming out of a critical point, with very mild decay assumptions, establishing that they are parametrized by unstable eigenfunctions of the critical point. Applied to mean curvature flow, these methods imply an arbitrary dimension and codimension classification of ancient mean curvature flows of closed submanifolds of S^n with low area, with stronger results when n=2 and when n=3.

Zoom ID : 821 8958 6559