The seminar usually takes place in Lecture Hall 5F, Cosmology Building at 4-5PM every Thursday, unless noted otherwise.
Organizers: Yng-Ing Lee 李瑩英 (NTU), Ulrich Menne 孟悟理 (NTNU), Chung-Jun Tsai 蔡忠潤 (NTU), Mao-Pei Tsui 崔茂培 (NTU)
Jesse Madnick (NCTS): Associative Submanifolds of the Berger Space
In complex geometry, given a special Lagrangian cone in a Calabi-Yau cone (e.g., R^6), its link is a minimal Legendrian in a Sasaki-Einstein manifold (e.g., S^5). Similarly, in the theory of exceptional holonomy, given a Cayley cone in a Spin(7)-holonomy cone (e.g., R^8), its link is a (minimal) “associative 3-fold” in a “nearly-parallel 7-manifold” (e.g., S^7). In this talk (joint work with Gavin Ball), we consider the Berger space B^7 = SO(5)/SO(3) — a positively-curved compact 7-manifold with a homogeneous nearly-parallel structure — and construct the first complete associative 3-folds in B. Our examples are circle bundles over genus g surfaces for every g ≥ 0, thus providing infinitely many topological types of compact associative 3-folds in B.
Andrea Galasso (NCTS): Equivariant fixed point formulae and Toeplitz operators under Hamiltonian torus actions
Suppose given a holomorphic and Hamiltonian action of a compact torus on a polarized Hodge manifold. Assume that the action lifts to the quantizing line bundle, so that there is an induced unitary representation on the associated Hardy space which decomposes into isotypes. In this talk we consider the description of asymptotics along rays in weight space of traces of equivariant Toeplitz operators composed with quantomorphisms for torus actions. The main ingredient in the proof is the microlocal analysis of the equivariant Szegö kernels. As a particular case we obtain a simple approach for asymptotics of the Lefschetz fixed point formula and traces of Toeplitz operators in the setting of ladder representations.
Ryosuke Takahashi 楊劼之 (NCKU): On the index theorem for Z/2-graphs
In this talk, I will introduce the concept of Z/2-graphs which were classified zero loci of Z/2-harmonic spinors defined on 3-dimensional, closed, oriented manifolds. We will discuss the structure of the moduli space of Z/2-graphs. We will also discuss the difference between Z/2-graphs and Z/2-embeddings, how the index theorem changes and the main difficulty to establish the index theorem for Z/2-graphs. This is an ongoing project with Andriy Haydys.
Jie Zhou (Tsinghua, Beijing): Topology of Surfaces with Finite Willmore Energy
In this presentation, we talk about a critical case of the Allard regularity theorem. Combining with Reifenberg's topological disk theorem, we get a critical Allard-Reifenberg type $C^{\alpha}$-regularity theorem. As a main result, we get the topological finiteness for a class of properly immersed surfaces in $R^n$ with finite Willmore energy. Especially, we prove the removability of isolated singularity of multiplicity one surfaces with finite Willmore energy and a uniqueness theorem of the catenoid under no a priori topological finiteness assumption.
Siao-Hao Guo (NTU): Analysis of Velázquez's solution to the mean curvature flow
Velázquez constructed a solution to the mean curvature flow with a type II singularity, at which the tangent flow is the Simons' cone. In this talk we will introduce some features of the solution such as its singularity model and investigate whether the mean curvature stays bounded or not.
Jui-En Chang (NTU): Generic singularities of the network flow
In this talk, I'll present the problem I'm working on and some partial results. In the network flow, singularities may form. They can be described as self-similar shrinking solutions called regular shrinkers. An important problem is that if we perturb the initial network, will the new network flow to the same singularity? All networks with 2 or more enclosed regions can be perturbed away. Therefore, the problem reduces to the network with less than 2 enclosed regions. There are finitely many of them and they are completely classified. Here, I use the entropy argument as in Colding and Minicozzi's work to show that the 4-ray star, the 5-ray star, the fish, and the rocket can be perturbed away.
Thierry De Pauw (East China Normal University, Shanghai, and University Paris Diderot): Flat and rectifiable G chains in metric space
I will report on a work in common with R. Hardt from Rice University. We generalise the tools of homological integration developed by H. Federer and W.H. Fleming in their study of normal and integral currents. The generalisation both concerns the ambient metric space and the group of coefficients and encompasses previous work by W.H. Fleming, L. Ambrosio and B. Kirchheim, B. White, and U. Lang. We will insist on the modern point of view that emphasises the regularity of 0-dimensional slices with respect to its variable in the range space. We will discuss the classical theorems, mainly the rectifiability of flat G chains of finite mass with coefficients in a White group.
Wei-Bo Su (Academia Sinica): Construction of Lagrangian translating solitons
Lagrangian translating solitons are Lagrangian submanifolds that evolve by translations under mean curvature flow, and they are natural candidates for the local models of finite-time singularities. In this talk, I will describe a gluing construction of Lagrangian translating solitons, which allows us to construct new examples with complicated geometries. The basic idea is to first build approximate solutions by gluing in Lawlor necks to the intersection points of Grim Reapers and then solve the nonlinear PDE near the approximate solutions by a fixed point argument. The main difficulty is the invertibility of the linearized operator, which we get round by introducing suitable weighted Sobolev spaces on the approximate solutions.
Chuan-Shen Peter Hu (NTNU): A bridge between mathematical morphology and topological data analysis
Mathematical morphology is a classic field in set theory and digital image processing. Topological data analysis (TDA), on the other hand, is a rising field in both algebraic topology and machine learning. One-parameter persistent homology, a main tool in TDA, relies on having a one-parameter filtration which is a sequence of sets satisfying nested subset relations. It has been widely studied and applied. Recently, the theory of multi-parameter persistent homology has been proposed; however, real world applications or concrete examples of multi-parameter filtrations are very limited. In this talk, I will introduce a framework for constructing multi-parameter filtrations of binary images by using the classic opening/closing operations in mathematical morphology. The framework not only offers geometric insights, but also provides concrete examples to the TDA field. As an application, a denoising algorithm in digital images is developed based on the geometric and topological information extracted from the framework. This is a joint work with Yu-Min Chung at UNC Greensboro, and Sarah Day at College of William and Mary.
Albert Wood (NTU): Singularities of Equivariant Lagrangian Mean Curvature Flow
After Ricci flow, the mean curvature flow of submanifolds in a Riemannian manifold is the most natural and famous geometric flow, describing the gradient descent for the volume functional. It is also of interest to symplectic geometers, since in an ambient Kahler-Einstein manifold, the flow preserves the class of Lagrangian submanifolds. It is therefore of interest to understand the behaviour of the flow, in particular the nature of singularities.
In this talk I hope to show the difference between the behaviour of singularities in the general case and the Lagrangian case, and give an exposition of my results regarding equivariant submanifolds.
