日本数学会は，1946年6月12日にそれまでの日本数学物理学会から 日本物理学会とともに分離独立し，創設されました． 2016年度はその70周年にあたり，これを記念して下記のような企画を行います．
- 森 重文
- 代数多様体の有理曲線について ---極小モデルと端射線---
- I have been studying algebraic varieties through rational curves on them. I was first interested in a special problem called the Hartshorne Conjecture, and when I solved it I encountered a notion called an extremal ray as a biproduct, through which I got attracted to the biregular classification and the minimal model program, and furthermore to a general theory of higher dimensional birational classification. Reviewing them, I will also touch the study of 3-dimensional extremal contractions which I have been interested in.
- 石井 仁司
- It is about 35 years since the notion of viscosity solution for partial differential equations was introduced by M. G. Crandall and P.-L. Lions. The theory of viscosity solutions has developed with a close connection with its applications to asymptotic problems on partial differential equations. In my talk, I will discuss the development of study in viscosity solutions and asymptotics problems featuring on the vanishing discount problem for fully nonlinear second-order elliptic equations including Hamilton-Jacobi-Bellman equations.
- 江口 徹
- String theory and moonshine phenomenon
- Several years ago we have discovered a curious phenomenon in string theory, i.e. the appearance of exotic new discrete symmetry in its spectrum. Specifically we have found that the group M24 acts on the elliptic genus of string theory compactified on the K3 surface. This phenomenon appears somewhat similar to the famous monstrous moonshine in the sense that the symmetry under some esoteric discrete group suddenly appears in the series expansion of modular functions or (mock) Jacobi forms. We were able to compute analogues of McKay-Thompson series and the group action of M24 has been proved mathematically in all orders of expansion of K3 elliptic genus in q. Unfortunately, however, we still do not have simple understanding or physical explanation of this moonshine phenomenon. Recently several more examples of new moonshine phenomena have been discovered and are receiving much attentions. There are still many puzzles and mysteries which are waiting for our resolutions.
- 加藤 和也
- I describe several big progresses in number theory which happened in these twenty years. One subject is to introduce the great progresses in Langlands correspondences which led to the solution of Sato-Tate conjecture. Another subject is the progresses in (generalized) Iwasawa theory. I also hope to introduce some works in the area which I myself had studied (ramification theory, etc.).
- 熊谷 隆
- In this talk, we summarize results concerning anomalous behaviour of random walks and diffusions on disordered media. Examples of disordered media include fractals and various models of random graphs, such as percolation clusters, random conductance models, Erdős-Rényi random graphs and uniform spanning trees. Geometric properties of such disordered media have been studied extensively and their scaling limits have been obtained. Our focus here is to analyse properties of dynamics on such media. Due to the inhomogeneity of the underlying spaces, we observe anomalous behaviour of the heat kernels and obtain anomalous diffusions as scaling limits of the random walks. We will give a chronological overview of the related research, and describe how the techniques have developed from those introduced for exactly self-similar fractals to the more robust arguments required for random graphs.
- 小林 俊行
- Birth of new branching problems
- The local to global study of geometries was a major trend of 20th century, with remarkable developments achieved particularly in Riemannian geometry. In contrast, surprising little was known 30 years ago about global properties of locally homogeneous spaces with indefinite-metric, which led us to the theory of discontinuous groups beyond the Riemannian setting. Concerning linear actions (representation theory), one of fundamental problems is to understand how things are built from smallest objects (irreducible decomposition). Branching problems are typical case, but were supposed to be out of control for reductive Lie groups. Breakthrough ideas for branching problems in representation theory emerged partially from the study of discontinuous groups beyond Riemannian setting, and conversely, they have opened new research such as global analysis of indefinite-Riemannian locally symmetric spaces (e.g. anti-de Sitter manifolds). Based on the developments over the last two decades, we present a program on branching problems, from the general theory to concrete construction of symmetry breaking operators.
- 深谷 賢治
- Homological Mirror symmetry is prposed by M. Kontsevitch in 1996 as a way to understand Mirror symmetry beyond the coincidence of various numbers. It decribe Mirror symmetry as an equivalence of two categories: one is derived category of coherent sheaves on complex manifold, the other is so called Fukaya category, whose object is a Lagrangian submanifold of a symplectic manifolds (together with certain data) and the morphisms between two objects are Floer homology. At the begining homological Mirror symmetry was a difficult subject to study and thought widely believed to be correct, examples where it was confirmed was restricted to elliptic curve and partially a higher dimensional complex torus. Recent developement changes the situation and now there are many cases where homological Mirror symmetry is proved, and various approaches exists for its proof. I would ilke to survey some of them.
- 砂田 利一
- 今年（2016年）はリーマンの没後150年という記念すべき年である。 ユークリッドからリーマンに至る歴史の中で、 幾何学自身が大きな変化を遂げてきたことは言うまでもないことであるが、 一方で数学と数学者を取り巻く社会的環境の変化にも自然と目が向かう。 環境の変化はこれからも絶え間なく続くことになり、 基礎研究の代表格である数学もその姿を大きく変えていくことだろう。 過去から現在、そして未来へと、この変化を語ることが目的なのであるが、 本講演では、数学への「個人的思い入れ」を中心にした話をしようと思う。