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Record of 2021 April - 2022 March

Date ： 29 May

Place ： Online by Zoom

15:00-16:00

Speaker ： Toru Kajigaya （Tokyo University of Science）

Title : Equivariant realizations of Hermitian symmetric space of noncompact type

Abstract ：

**Let M=G/K be a Hermitian symmetric space of noncompact type. We provide
a way of constructing K-equivariant embeddings from M to its tangent space
at the origin by using the polarity of the K-action. As an application,
we reconstruct the**

**16:15-17:15
Speaker ： Shunsuke Saito （Tokyo University of Science）
Title : On polytopeal sufficient conditions for uniform relative K-polystability
and relative K-instability of polarized toric varieties
Abstract ：**

The contents of the talk are based on a joint
work with Yasufumi Nitta (Tokyo University of Science).

**Date ： 24 July
Place ： Online by Zoom
15:00-16:00
Speaker ： Shota Hamanaka （Chuo University）
Title : Closed Ricci flow with integral or pointwise bound of the scalar
curvature
Abstract ：
** In this talk,

a closed manifold has a unique short time solution for any initial metric. The next immediate question is the so-called ``maximal existence time'' for the Ricci flow with respect to initial metric. Hamilton proved that T < +∞ is the maximal existence time of a closed Ricci flow

(M, g(t)) (t ∊ [0,T), dim M ≧ 2) if and only if its norm of the Riemannian curvature tensor is unbounded as $t \rightarrow T$. Therefore a uniform bound for the norm of the Riemannian curvature tensor on M × [0,T) is enough to extend the Ricci flow smoothly over T. Moreover, another known such sufficient conditions to extend the flow smoothly are, for example,

uniformly bound of the norm of the Ricci curvature tensor on M × [0,T) (by Sesum) or

certain integral bounds of geometric quantities rather than point-wise one (by Wang).

Di-Matteo generalized Wang's results using mixed integral norms (i.e., space and time)

which is parametrized by (α, β) ∊ (1, ∞).

In the talk, we will give new extendable result on the closed four dimensional Ricci flow

under the conditions corresponding to (α, β) = (p, +∞) (p > 2) and (+∞, 1). If there is

time, we also discuss the condition sup_{|R| | x ∊ M × [0,T)} < +∞ (the uniformly bound of

the scalar curvature).

Title : Liouville theorems for harmonic map heat flow

along ancient super Ricci flow via reduced geometry

Abstract ：

This talk is based on the joint work with Keita Kunikawa (Utsunomiya university).
We have obtained a Liouville theorem for heat equation along ancient super
Ricci flow via Perelman’s reduced geometry. Recently, we generalize the
target spaces, and formulate several Liouville theorems for harmonic map
heat flow. In this talk, I will introduce such Liouville theorems. In our
Liouville theorems, we impose a growth condition concerning Perelman’s
reduced distance. For non-positively curved target spaces, our growth condition
is sharp. On the other hand, for positively curved target spaces, it is
not clear that our growth condition is sharp. However, in the static case,
the growth condition can be improved, which is almost sharp in view of
the example constructed by Schoen-Uhlenbeck. I will discuss the sharpness
in more detail in the talk.

Place ： Online by Zoom

15:00-16:00

Speaker ： Tomoki Fujii （Tokyo Uiniversity of Science）

Title : Graphical translating solitons for the mean curvature flow and isoparametric functions

Abstract ：

In this talk, I will discuss the condition that the graph of a function over a Riemannian

manifold M whose level sets give isoparametric foliation is a translating soliton for the

mean curvature flow.This translating soliton is given as the graph of a function which

is given as a composition of

ordinary differential equation.

explain the classification of the shape of such translating solitons in case where the

Riemannian manifold M is the n-dimensional sphere (n≧2).

16:15-17:15

Speaker ： Takuma Tomihisa （Waseda University）

Title :The Rarita-Schwinger operator and some related topics

Abstract:

The Rarita-Schwinger fields play an important role in physics by describing
the gravitino.

The Rarita-Schwinger
operator is also important to define the Rarita-Schwinger fields.

Recently, in mathematics, there are many researches about the Rarita-Schwinger
operator,

the "spin-3/2 version" of the Dirac operator.

(1) spectra of the Rarita-Schwinger operators on some symmetric spaces
(joint work with

Yasushi Homma), **
**

(2) the Rarita-Schwinger fields on nearly Kähler manifolds (joint work with Soma Ohno).

Organizers : Naoyuki Koike, Makiko Tanaka,

Yasufumi Nitta,

Shunsuke Saito，Tsukasa Takeuchi

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