Differential Geometry Seminar (2024)

As a project of OCAMI, we shall promote the seminar on differential geometry in the wide sense of including the areas related to geometric analysis, topology, algebraic geometry, mathematical physics, integrable systems, information sciences etc.

Differential Geometry Seminar (list by year)

Differential Geometry Seminar (2024) Talks

Date July 19 (Fri.) 2024, 16:45-18:15(Japan time)
Speaker Wafaa Batat (Ecole Nationale Polytechnique d'Oran Maurice Audin)
Title

Homogeneous Structures on Three- and Four-dimensional Lie groups

Place Dept. of Mathematics, Faculty of Science Bldg., F405
Abstract

In this talk, we will introduce the notion of homogeneous pseudo-Riemannian structures and demonstrate how to establish homogeneity and natural reductiveness of 3- and 4-dimensional Lie groups through a tensor satisfying certain geometric partial differential equations involving the metric and the curvature of a given manifold. These equations are known as Ambrose-Singer equations. We will begin by examining homogeneous structures on three-dimensional unimodular and non-unimodular Lie groups, proving the existence of homogeneous Lorentzian structures that differ from the canonical ones without being naturally reductive, a phenomenon with no Riemannian counterpart. Using these homogeneous structures, we will show how to classify naturally reductive 3-dimensional Lorentzian manifolds. Our focus will then shift to the geometric properties of four-dimensional nilpotent Lie groups endowed with a family of non-flat left-invariant Lorentzian metrics. We will conduct a comprehensive classification of homogeneous structures for each metric and meticulously examine the distinctive properties characterizing each structure. Additionally, we will provide a specific example demonstrating the presence of a naturally reductive, non-flat, left-invariant Lorentzian metric on the 2-nilpotent Lie group, where the center exhibits degeneracy. Furthermore, we will establish the existence of a non-canonical homogeneous structure. As an application, we will demonstrate the existence of naturally reductive left-invariant Lorentzian metrics on the four-dimensional 3-nilpotent Lie group.

Date July 5 (Fri.) 2024, 16:45-18:15(Japan time)
Speaker Christopher Mahadeo (University of Illinois at Chicago)
Title

Topological recursion and twisted Higgs bundles

Place Dept. of Mathematics, Faculty of Science Bldg., F405
Abstract

Prior works relating meromorphic Higgs bundles to topological recursion have considered non-singular models that allow the recursion to be carried out on a smooth Riemann surface. I will discuss some recent work where we define a "twisted topological recursion" on the spectral curve of a twisted Higgs bundle, and show that the g=0 components of the recursion compute the Taylor expansion of the period matrix of the spectral curve, mirroring a result of for ordinary Higgs bundles and topological recursion. I will also discuss some current work relating topological recursion to a new viewpoint of quantization of Higgs bundles.

Date May 24 (Fri.) 2024, 16:45-18:15(Japan time)
Speaker Yasuhiko Asao (Fukuoka University)
Title

Homology of metric spaces

Place Faculty of Science Bldg., E101

Differential Geometry Seminar (2024) Organizers

Name Tel E-mail
Hiroshi Tamaru 06-6605-2615 tamaru [at] omu.ac.jp
Yoshinori Hashimoto yhashimoto [at] omu.ac.jp
Hiroaki Ishida hiroaki.ishida [at] omu.ac.jp
Shin Kato 06-6605-2616 shinkato [at] omu.ac.jp
Takayuki Koike tkoike [at] omu.ac.jp
Ushio Tanaka utanaka [at] omu.ac.jp
Kaname Hashimoto h-kaname [at] sci.osaka-cu.ac.jp