Organization meeting. 4PM at Astro-Math 430

Yi-Sheng Wang 王以晟 (NCTS): Some real life applications of topology

Once an abstract branch of mathematics, topology now finds its applications in many scientific fields. Algebraic topology provides tools resilient to noise and effective in analyzing complicated data sets; persistent homology is one of such emerging tools from algebraic topology; it has been proved successful in analyzing data sets which traditional methods cannot cope with well.

In the first part of the talk, I will give a quick review on persistent homology and basic invariants of filtered space drawn from persistent homology. Some recent applications of persistent homology in sensor network and neuroscience are discussed in the second part.

Ser-Wei Fu 傅斯緯 (NCTS): Introducing the curve complex

A major theme of geometric group theory is the topological and geometric properties of a space that a specific group acts on. The study of the mapping class group associated to a surface of finite-type led to various descriptions and classifications using the set of simple closed curves. In this talk I will introduce the curve complex and discuss various interesting properties. The curve complex of the torus is a special case that can be given an explicit treatment using continued fractions. The higher complexity cases are related to Teichmüller theory and one way to see it is through the shadows of Teichmüller geodesics. The curve complex is hyperbolic and studying the Gromov boundary of the curve complex will give us an interesting result about cylinders and flat surfaces. The conclusion of this talk is joint work with Chris Leininger.

Ulrich Menne 孟悟理 (NTNU): Regularity of and geometric analysis on varifolds with mean curvature

Geometric variational problems involving mean curvature (e.g., area-stationary surfaces, mean curvature flow, and Willmore surfaces) give rise to the study of measure-theoretic surfaces (varifolds). We survey results on their regularity and on the infrastructure for geometric analysis thereon. The first item includes the existence of an approximate second fundamental form, whereas the second covers weakly differentiable (and Sobolev) functions, geodesic distance, and curvatures in the spirit of convex geometry.

Martin Man-chun Li 李文俊 (CUHK): Mean curvature flow with free boundary

Mean curvature flow (MCF) is the negative gradient flow for the area functional in Euclidean spaces, or more generally in Riemannian manifolds. Over the past few decades, there have been substantial progress towards our knowledge on the analytic and geometric properties of MCF. For compact surfaces without boundary, we have a fairly good understanding of the convergence and singularity formation under the flow. In this talk, we will discuss some recent results on MCF of surfaces with boundary. In the presence of boundary, suitable boundary conditions have to be imposed to ensure the evolution equations are well-posed. Two such boundary conditions are the Dirichlet (fixed or prescribed) and Neumann (free or prescribed contact angle) boundary conditions. We will mention some new phenomena in contrast with the classical MCF without boundary. We give a convergence result for mean curvature flow of convex hypersurfaces with free boundary. This is joint work with Sven Hirsch. (These works are partially supported by RGC grants from the Hong Kong Government.)

Pak Tung Ho 何柏通 (Sogang University): Chern-Yamabe Problem

I will first talk about the Yamabe problem and the Yamabe flow on Riemannian manifolds. Then I will explain what the Chern-Yamabe problem is, and talk about the Chern-Yamabe flow which is a geometric flow approach to solve the Chern-Yamabe problem.

Yoshihiro Sugimoto (NCTS): Hamiltonian diffeomorphism groups on symplectic manifolds

A Hamiltonian diffeomorphism on a symplectic manifold is a diffeomorphism which is generated by a time-dependent Hamiltonian vector field. A Hamiltonian diffeomorphism is called autonomous if it is generated by time-independent Hamiltonian vector field. In this talk, I will explain that the complement of the autonomous elements is dense in the Hamiltonian diffeomorphism group.

Ting-Jung Kuo 郭庭榕 (NTNU): A survey of recent developments in Lame equation related to monodromy theory

In this talk, I will talk about the generalized Lame equation from monodromy aspect. As applications, a criterion of existence of solution of a related nonlinear pde will be given in this talk. These are joint works with Z. Chen and C-S Lin.

Xiaowei Wang 王曉瑋 (Rutgers University): Moduli space of Fano Kahler-Einstein varieties

In this talk, I will survey our construction of a proper moduli space for smoothable Fano Kahler-Einstein varieties based on the recent breakthrough on Kahler-Einstein problem. This is based on a joint work with Chi Li and Chenyang Xu.

Albert Chern 陳人豪 (TU Berlin) : Shape from Metric

We study the computational isometric immersion problem: given a 2D Riemannian manifold, find an immersion into 3D that realizes the intrinsic lengths. Classical approaches involve variational problems resembling stiff membrane elasticity. The challenge remains as these methods can yield surfaces that are pinched and tangled. To address this challenge, we develop a discrete theory for surface immersions into 3D. In particular, the theory correctly represents the topology of the space of immersions, i.e., the regular homotopy classes. Our approach relies on spinors to represent 3D orientations and to encode, in the spin connection, the regular homotopy class. With this theory incorporated, we resolve the challenge of pinched surfaces and ensures immersions. We demonstrate our algorithm with several applications from mathematical visualization and art directed isometric shape deformation, which mimics the behavior of thin materials with high membrane stiffness.

Yu-Shen Lin 林昱伸 (Boston University): The SYZ Fibration and Partial collapsing of the Log Calabi-Yau Surfaces

Abstract: The Strominger-Yau-Zaslow conjecture expect that the Calabi-Yau manifolds admits special Lagrangian fibrations and the fibration collapse near the large complex structure limit. In this talk, I will explain both phenomena in the case of log Calabi-Yau surfaces. Given a del Pezzo surface Y with a smooth anti-canonical divisor D. I will briefly explain the existence of the special Lagrangian fibration on the complement X=Y\D using Lagrangian mean curvature flow. Then I will focus on the behavior of the Ricci-flat metric as D degenerating to a nodal rational curve. By studying the Ricci-flat metric on the corresponding rational elliptic surfaces, we will see the collapsing of special fibration near infinity.

This is based on the joint work with T. Collins and A. Jacob.