談話会 Alfonso Sorrentino「Mathematical Billiards」、寺本央「Toward Molecular Propagation through Degenerated Electron Energy Level Crossings」、秋田利之「カンドルと対称群の中心拡大 」

開催日時
2017年   10月 5日 14時 45分 ~ 2017年   10月 5日 17時 50分
場所
理学部4号館5階若手研究者交流室(4-501)
講演者
Alfonso Sorrentino 氏(ローマトルベルガータ大学)、寺本央 氏(北海道大学電子科学研究所)、秋田利之 氏(北海道大学大学院理学研究院)
 
スケジュール:
14:45-15:35 Alfonso Sorrentino 氏(ローマトルベルガータ大学)
15:35-16:00 teatime(場所:5階コモンスペース)
16:00-16:50 寺本央 氏(北海道大学電子科学研究所)
16:50-17:00 teatime(場所:5階コモンスペース)
17:00-17:50 秋田利之 氏(北海道大学大学院理学研究院)

Alfonso Sorrentino 氏
Title: Mathematical Billiards
Abstract:
A mathematical billiard is a system describing the inertial motion of a point mass inside a domain, with elastic reflections at the boundary. This simple model has been first proposed by G.D. Birkhoff as a mathematical playground where “the formal side, usually so formidable in dynamics, almost completely disappears and only the interesting qualitative questions need to be considered”.

Since then billiards have captured much attention in many different contexts, becoming a very popular subject of investigation. Despite their apparently simple (local) dynamics, their qualitative dynamical properties are extremely non-local. This global influence on the dynamics translates into several intriguing rigidity phenomena, which are at the basis of several unanswered questions and conjectures.

In this talk I shall focus on several of these questions. In particular, I shall describe some recent results related to the possibility of inferring dynamical information on the billiard map from its length spectrum (i.e., the collection of lengths of its periodic orbits), and on the classification of integrable billiards (also known as Birkhoff conjecture).

This talk is based on several works in collaboration with V. Kaloshin and with G. Huang and V. Kaloshin.

寺本央 氏
タイトル:Toward Molecular Propagation through Degenerated Electron Energy Level Crossings
アブストラクト:
When a molecule propagates through electron level crossings, the molecule experiences non-adiabatic transitions, which are one of the most important processes in chemical reaction dynamics. Since a pioneering work of Hagedorn [1], mathematical understanding of molecular propagation through non-degenerated electron energy level crossings has been developed. However, understanding of that through *degenerated* electron energy level crossings is still limited only for one-degree-of-freedom systems. Such degenerated energy level crossings are expected to appear in a molecule in a certain symmetry or a molecule under a control of external fields, understanding of which is invaluable to control chemical reactions of the molecule.

Toward understanding of molecular propagation through degenerated electron energy level crossings, first, we systematically classify degenerated electron energy level crossings of Type I [1] in terms of singularity theory, developed recently [2,3]. To understand non-adiabatic transitions through such degenerated electron energy level crossings, we need to solve a Schrodinger equation in a neighborhood of the crossings. To do that, we consider the principal symbol of the operator in the Schrodinger equation and try to construct a simple normal form of that by following [4,5]. To generalize the results in [4,5] to denegerated cases, we need to solve a bifurcation problem, which is yet to be solved again by singularity theory. In this talk, we discuss our future perspectives of this problem.

[1] G. A. Hagedorn, Molecular Propagation through Electron Energy Level Crossings, Mem. Amer. Math. Soc. 111, 536 (1994).
[2] H. Teramoto, K. Kondo, S. Izumiya, M. Toda and T. Komatsuzaki, J. Math. Phys., 58, 073502 (2017).
[3] S. Izumiya, M. Takahashi, and H. Teramoto, in preparation.
[4] P. J. Braam and J. J. Duistermaat, Indag. Mathem., N. S. 407 (1993).
[5] Y. Colin De Verdiere, Annales de l'Institut Fourier, 1023 (2003).

秋田利之 氏
タイトル:カンドルと対称群の中心拡大
アブストラクト:
カンドルは1980年代に定義された比較的新しい代数系です。
n文字の置換全体の集合は共役を演算としてカンドルになります。
本講演では置換のなすカンドル、対称群の中心拡大、ブレイド群などの関わりを中心にお話ししたいと思います(学部レベルの群論しか仮定しません)。

関連項目

研究集会・セミナー・集中講義の一覧へ