2024年１月９日～２４日の期間、KTGU Special Weeksとして4名の海外研究者をお招きし、「スーパーグローバルコース数学特別講演会」が下記の要領で開催されます。
【講演者/所属/日程】
Prof. Francesco Lin（Columbia University）Jan 9（Talk 1）・11（Talk 2），2024
Prof. Kazuo Yamazaki（University of Nebraska-Lincoln）Jan 15，2024
Prof. Alexander Bertoloni Meli（University of Bonn）Jan 17，2024
Prof. Dongho Chae（Chung-Ang University）Jan 24，2024
【要申込】参加希望者は下記ＵＲＬの Googleフォームにて申込を行って下さい。
URL：https://forms.gle/q8PyQqJgBUip8yMs6
締切日：2024年１月４日（木）17時厳守
【会場】後日 個別にお知らせいたします。
（状況により参加をお受けできない場合がございます）
＜ Prof. Francesco Lin ＞
Date and Time : January 9（Talk 1）・11 （Talk 2） 2024
（Talk 1）
Date and Time：January 9，2024 10:00～11:00
Title：Homology cobordism and the geometry of hyperbolic three-manifolds
Abstract：
The three-dimensional homology cobordism group is a fundamental object in low-dimensional topology. A major challenge in the study of its structure is to understand the interaction between hyperbolic geometry and homology cobordism. In this talk, after introducing the main protagonists and explain the role they play in topology, I will discuss how monopole Floer homology (an invariant of three-manifolds obtained by suitably counting solutions to the Seiberg-Witten equations) can be used to study some basic properties of certain subgroups of the homology cobordism group generated by hyperbolic homology spheres satisfying natural geometric constraints of Riemannian and spectral nature.
（Talk 2）
Date and Time：January 11，2024 10:00～11:00
Title： The Dirac equation on spectrally large three-manifolds
Abstract：
To a Riemannian three-manifold equipped with a torsion spin^c structure one can naturally associate a b_1-dimensional family of twisted Dirac operators. Even though the topological properties of such family are explicitly described by the Atiyah-Singer index theorem in terms of the triple cup product, its geometric features are much less understood. In this talk, I will discuss how monopole Floer homology can be used to provide information on the latter for manifolds with a large spectral gap on coexact 1-forms, with a focus on concrete examples.
＜ Prof. Kazuo Yamazaki ＞
Date and Time ：January 15，2024 15:00～16:00
Title：Convex integration on stochastic partial differential equations
Abstract：
Convex integration has its roots in the work of Nash in geometry and evolved very recently to a powerful new tool that can prove non-uniqueness for many equations. We know from undergraduate PDE course that one-dimensional Burgers’ equation can be proven to exhibit finite-time shock via characteristics; however, such an approach was limited and could not shed much light on other complicated models, a primary example being the Navier-Stokes equations due to being vector-valued and the presence of its pressure term, etc. Via convex integration we now know non-uniqueness at a relatively low regularity level for many systems of equations; examples include Euler equations, Navier-Stokes equations, MHD system, Boussinesq system, active scalars such as the surface quasi-geostrophic equations and porous media equations, etc. This phenomenon has very recently spilled over to non-uniqueness in the compressible case and stochastic case, the latter being when the equation is forced by random noise. The purpose of this talk is to give an overview of the convex integration technique and review recent developments and open problems, primarily focused on the stochastic case.
＜ Prof. Alexander Bertoloni Meli ＞
Date and Time：January 17，2024 15:00～16:00
Title：The Langlands Correspondence and Local Shimura Varieties
Abstract:：
I will discuss my work on the cohomology of local Shimura varieties using global methods and its relation to the local Langlands correspondence for p-adic fields. In particular, I will explain how explicit descriptions of cohomology imply the compatibility of the
Langlands correspondences of Fargues–Scholze and Arthur in certain cases.
＜ Prof. Dongho Chae ＞
Date and Time：January 24 ，2024 15:00～16:00
Title：On the Liouville type problem in the stationary Navier-Stokes equations
Abstract：
In this lecture we discuss the Liouville type problem in the stationary Navier-Stokes equations.
The problem is to prove or construct a counter example of solutions for the 3 dimensional stationary Navier-Stokes equations
in the whole of R^3 that vanishes at infinity, and satisfying the condition of finite Dirichlet integral.
We present some of the classical results as well as more recent partial progresses, and discuss difficulties of various trials.
