Title

On integral p-adic cohomology theory

Date

2021.10.27 (Wed)　16:45-17:45 　　

Place

Zoom

Speaker

Atsushi Shiho (University of Tokyo)

Abstract

　When an algebraic variety over a perfect field k of positive characteristic p is proper and smooth, its crystalline cohomology is a finitely generated W-module (W is the Witt ring of k) and it gives a good p-adic cohomology theory. However, when the variety is non-proper or non-smooth, it is not necessarily a finitely generated W-module. On the other hand, rigid cohomology is a good p-adic cohomology theory which is a finite-dimensional K-vector space (K is the fraction field of W) for any algebraic variety over k, but it does not define a finitely generated W-module strucure (integral structure).
In this talk, we explain several approaches to define an integral p-adic cohomology theory for possibly non-proper and non-smooth algebraic varieties which is compatible with log crystalline cohomology and rigid cohomology. This is joint work with Veronika Ertl and Johannes Sprang.

Title

Paracontrolled calculus and regularity structures

Date

2021.10.20 (Wed)　16:45-17:45 　　

Place

Zoom

Speaker

Masato Hoshino (Osaka University)

Abstract

　In the past several years, the study of singular stochastic PDEs has significantly progressed by two groundbreaking theories: the theory of regularity structures by Hairer and the paracontrolled calculus by Gubinelli, Imkeller, and Perkowski. In this talk, we prove the equivalence statement between some notions of these theories under slightly restrictive but harmless assumptions, which are naturally satisfied by the regularity structures introduced by Bruned, Hairer, and Zambotti for the study of a large class of singular stochastic PDEs.
This talk is based on a joint work with Ismaël Bailleul (Université Rennes 1).

Title

Analysis of Laplacians on fractals

Date

2021.10.13 (Wed)　16:45-17:45 　　

Place

Zoom

Speaker

Naotaka Kajino (RIMS, Kyoto University)

Abstract

　The research field of analysis of Laplacians on fractals is aimed at establishing rigorous mathematical theory of description and analysis of fundamental physical phenomena like heat conduction and wave propagation. Since the usual notion of partial derivatives do not make sense on fractals, it is a highly non-trivial problem how one should define a canonical ``Laplacian'' on a given fractal (and what criteria one should apply to decide whether it is ``canonical''). Answering this question requires careful thoughts on the geometric features of individual fractals and has been successfully achieved only for limited classes of fractals, although the speaker has made some progress in the last several years.
　The first half of this talk will discuss the classical case of Euclidean self-similar fractals studied extensively since late 1980s, explain the construction of Laplacians and present fundamental results such as asymptotic behavior of the heat kernel and the eigenvalues. Then the latter half will present the speaker's recent results, obtained by his studies since 2015, that ``a geometrically canonical Laplacian can be constructed and satisfies Weyl's eigenvalue asymptotics'' on (some examples of) circle packing fractals invariant under the action of certain Kleinian groups (discrete groups of Moebius transformations on the Riemann sphere). Here the principal order term of the eigenvalue asymptotics is given in terms of the Hausdorff dimension and measure of the fractal, which is the central reason why this Laplacian is considered as ``geometrically canonical''.

Title

Towards Function theory on Teichmüller space

Date

2021.7.14 (Wed)　16:45-17:45 　　

Place

Zoom

Speaker

Hideki Miyachi (Kanazawa University)

Abstract

　Teichmüller space is the moduli space of marked Riemann surfaces, and is realized as a bounded domain in the complex Euclidean space under the natural complex structure. It is natural to develop the function theory on Teichmüller space. In this talk, we will give a recent progress on the study of Function theory on Teichmüller space.

Title

Crystalline mean curvature flow with a volume constraint

Date

2021.7.7 (Wed)　16:45-17:45 　　

Place

Zoom

Speaker

Norbert Pozar (Kanazawa University)

Abstract

　In this talk I will discuss new results on the crystalline mean curvature flow with nonlocal forcing given by a volume constraint. We establish existence of solutions for initial data with a certain reflection property given by the symmetries of the Wulff shape, which we show is preserved in the evolution. This talk is based on joint work with Inwon Kim and Dohyun Kwon.

Title

On the extension problem of quasimorphisms on groups

Date

2021.6.30 (Wed)　16:45-17:45 　　

Place

Zoom

Speaker

Takahiro Matsushita (University of the Ryukyus)

Title

Long time behavior of solution to the nonlinear Schrödinger equation with delta potential

Date

2021.6.23 (Wed)　16:45-17:45 　　

Place

Zoom

Speaker

Junichi Segata (Kyushu University)

Abstract

　We summarize recent progress on long time behavior of solution to the one dimensional nonlinear Schrödinger equation with a delta potential. We first consider the case where potential is repulsive and prove the (modified) scattering for small global solutions. Next we mention the case where potential is attractive and establish the asymptotic stability of the family of solitary waves.

Title

Homological mirror symmetry and the gamma integral structures for invertible polynomials

Date

2021.6.9 (Wed)　16:45-17:45 　　

Place

Zoom

Speaker

Atsushi Takahashi (Osaka University)

Title

A default contagion model for pricing defaultable bonds from an information based perspective

Date

2021.6.2 (Wed)　16:45-17:45 　　

Place

Zoom

Speaker

Hidetoshi Nakagawa (Hitotsubashi University)

Abstract

(Joint research with Prof. Dr. Hideyuki Takada)
　In this study, we introduce an extended model of the information based model of credit risk proposed by Brody, Hughston and Macrina (2010) to a multi-name case to investigate how default contagion risk influences the price fluctuation of defaultable discount bonds. Under the model with a couple of obligors, we derive a stochastic differential equation for one defaultable zero-recovery discount bond price process to reflect default contagion risk of a counterpart debt obligor. As a consequence, we find that the excess rate of the return in the trend term of the bond consists of not only the issuer's hazard rate but also the counterpart obligor's hazard rate adjusted with the "pseudo-default loss" rate. We also find that the bond price can jump at the default time of the counterpart by the amount dependent on the correlation between the issuer and the counterpart. Moreover, we numerically examine the impact of default contagion risk on some bond price components within the model.
Reference :
　Hidetoshi Nakagawa and Hideyuki Takada, "A default contagion model for pricing defaultable bonds from an information based perspective," FS-2020-E-001,HUB FS Working paper series (submitted)
URL http://www.fs.hub.hit-u.ac.jp/inc/files/staff-research/workingpaper/FS-2020-E-001.pdf

Title

Hamilton-Jacobi equations on metric spaces

Date

2021.4.21 (Wed)　16:45-17:45 　　

Place

Rm420, Research Institute for Mathematical Sciences, Kyoto University

Speaker

Atsushi Nakayasu (Kyoto University)