Differential Geometry Seminar (2023)

Curvature flows for hypersurfaces and their geometric applications

Isoperimetric inequality is one of the oldest problems in mathematics, which relates with convex geometry, differential geometry and geometric PDEs, etc. Recently, the isoperimetric type inequalities in hyperbolic space have been widely investigated by using the hypersurface curvature flows, including the inverse curvature flows, quermassintegral preserving curvature flows, contracting curvature flows, and locally constrained curvature flows. In this talk, I will survey the recent progress in this direction, which is based on my joint works with Ben Andrews (ANU), Yong Wei (USTC), Changwei Xiong (SCU), Yingxiang Hu (Beihang U.).

A conical approximation of constant scalar curvature Kähler metrics of Poincaré type and algebro-geometric stability

Extended Hitchin equation for cyclic Higgs bundles associated with a quasi-subharmonic function, and its Dirichlet problem

Generating operators for symmetry breaking

We introduce the notion of "generating operator" for a family of differential operators between two manifolds and develop this construction in the framework of covariant differential operators acting on sections of holomorphic vector bundles over homogeneous spaces. An explicit formula for the generating operator associated with the Rankin--Cohen brackets will be discussed.

A Hilbert-Mumford criterion for nilsolitons

Iterative minimization algorithm on mixture family

Iterative minimization algorithms appear in various areas including machine learning, neural network, and information theory. The em algorithm is one of the famous iterative minimization algorithms in the former area, and Arimoto-Blahut algorithm is a typical iterative algorithm in the latter area.
However, these two topics had been separately studied for a long time.
In this paper, we generalize an algorithm that was recently proposed in the context of Arimoto-Blahut algorithm. Then, we show various convergence theorems, one of which covers the case when each iterative step is done approximately. Also, we apply this algorithm to the target problem in em algorithm, and propose its improvement. In addition, we apply it to other various problems in information theory.
This paper is available in https://arxiv.org/abs/2302.06905

A gerby deformation of complex tori and the homological mirror symmetry

J-equations on fiber spaces

Some limit theorems for the total scalar curvature

Cohomogeneity one actions on symmetric spaces

An isometric action on a Riemannian manifold is of cohomogeneity one if the minimum codimension of its orbits is one or, equivalently, it has hypersurfaces as its regular orbits. While the classification of homogeneous hypersurfaces in real hyperbolic spaces follows from Cartan's classical work on isoparametric hypersurfaces, obtaining similar results for other symmetric spaces of noncompact type has proven to be difficult: For example, it was not until recently that the classification of homogeneous hypersurfaces on the quaternionic hyperbolic spaces was obtained. In noncompact symmetric spaces of higher rank, several structural results by J. Berndt and H. Tamaru allowed to obtain similar classifications, but only for certain irreducible spaces of rank two.

In this talk I will report on the state of the classification of cohomogeneity one actions in symmetric spaces focusing on the case of symmetric spaces of noncompact type. Recently, in a joint work with J.Carlos Díaz-Ramos and Miguel Domínguez-Váquez we have proposed a generalized procedure for the classification of such actions on a given symmetric space of noncompact type. This allowed us to produce the first classification result of homogeneous hypersurfaces in a space of rank > 2, namely on the spaces SL(n,R)/SO(n).