2017年度秋季総合分科会(於:山形大学)

SUMMARY: We survey the recent development of the representation theory of affine W-algebras and the Higgs branch conjecture in the four dimensional \(N=2\) superconformal field theories.

SUMMARY: We consider the well-posedness and the ill-posedness in the Sobolev space of the Caucy problem for the third order nonlinear Schrödinger equation (3NLS) on the one dimensional torus. Especially, we fucus on what role the smoothing type estimates of the cubic nonlinearity play in the well-posedness issue. First, I talk about the time local well-posedness in the negative Sobolev space and the nonuniqueness of solutions without auxiliary spaces for (3NLS). Second, I talk about the ill-posedness in the Sobolev space for (3NLS) with Raman scattering term. In the latter case, I also present the result of the Cauchy-Kowalevsky type on the local unique solvability in the analytic function space. These topics show that the nonlinear interaction often yields the smoothing effect.

SUMMARY: E. Schrödinger proposed the following problem in his 1932’s paper. Suppose that there exist N particles in a subset A of 3-dimensional Euclidean space and each particle moves independently, with a given transition probability, to a different subset B of 3-dimensional Euclidean space. He tried to find the maximal probability of such events, provided the number of particles in each point in A and B are fixed. Though he did not succeed in finding the maximal probability, he obtained Euler’s equation, for the variational problem above, which is called Schrödinger’s functional equation. After S. Bernstein’s talk in ICM 1932, E. Schrödinger’s problem has been developed as the study of Bernstein process (or reciprocal process) and that of Doob’s h-path process. It is also known that the problem is closely related to E. Nelson’s stochastic mechanics. In this talk, we focus on our research about Schrödinger’s problem and its generalization as stochastic optimal transportation problem, its application to Nelson’s stochastic mechanics, Monge’s problem as the zero-noise limit of stochastic optimal transportation problem and relation to mean field PDEs.

SUMMARY: In the first half of 20th century, general frameworks for the study of functions of several complex variables have been developed by Behnke, Thullen, Cartan, Oka, Grauert and others. They clarified the conditions for a domain to be holomorphically complete and showed sufficiently many existences of global holomorphic functions on such domains. The results were crystallized to the theory of Stein manifolds, and were sheaf theoretically formulated (Theorem A and B) by Cartan and Serre.

On the other hand, new interactions in the last few decades between mathematics and physics (e.g. string theory, gauge theory, quantum field theory, etc.) gave new impetus to mathematicians to understand global analytic functions. For instance, mirror symmetry between complex geometry and symplectic geometry was first observed by physicists, and was later formulated by Konstsevich in language of categories. Inspired by Douglas’ stability condition in physics, Tom Bridgeland introduced the space of stability conditions for such a triangulated category.

These spaces (of stability conditions) are complex manifolds of interest in mathematics and in physics. In examples, they are Stein manifolds, but do not seem to be classical symmetric domains. New problems appear: develop and understand global analytic functions on such spaces. The problem seems to be mirror symmetric to the construction of certain higher dimensional ”automorphic forms” on the period domains which I shall describe in the lecture. Our understanding of global holomorphic functions for such domains is still quite limited, compared with the rich theory of the one variable case, where there exist strong means like Fourier analysis and Eisenstein series.

In the lecture, I will describe some aspects of these problems from the perspective of the period map theory, with which I have been engaged.

SUMMARY: Around the early 1990’s, Kenji Fukaya introduced the notion of an \(A_{\infty }\) category, roughly speaking, whose objects are Lagrangian submanifolds in a symplectic manifold and the morphism spaces are the Floer chain complexes equipped with \(A_{\infty }\) structures defined by moduli spaces of holomorphic maps form a \(2\)-dimensional disc to the symplectic manifold with Lagrangian boundary conditions. Now it is called Fukaya category. M. Kontsevich used this category to formulate his homological mirror symmetry conjecture. In this talk I will try to give a brief introduction and discuss some aspects of the Fukaya category with emphasis on mirror symmetry. Based on my joint works with K. Fukaya, Y.-G. Oh, K. Ono, and also with M. Abouzaid.

SUMMARY: The classical Dixmier–Douady theory describes the structure of continuous trace \(C^*\)-algebras in terms of the third cohomology of its spectrum. In 1989, Rosenberg formulated twisted K-theory in full generality as the K-theory of a continuous trace \(C^*\)-algebra with its spectrum homeomorphic to a prescribed space and with a prescribed third cohomology class. Since then twisted K-theory has been extensively studied, partly because its relationship with string theory was reveled in the late ’90s.

On the other hand,in the Elliott program of the classification of amenable \(C^*\)-algebras, the importance of a certain class of \(C^*\)-algebra with very simple structure had been recognized among the specialists long before the formal definition as strongly self-absorbing \(C^*\)-algebras was introduced in 2007. Recently,a surprising and unexpected application of them was found by Dadalart–Pennig,who showed that the Dixmier–Douady theory can be generalized to every strongly self-absorbing \(C^*\)-algebra in that the classical Dixmier–Douady theory is for the trivial \(C^*\)-algebra, the complex numbers. Moreover, a generalized cohomology theory arises from every strongly self-absorbing \(C^*\)-algebra, whose characteristic classes have higher terms beyond the third cohomology. In this talk, I will give an account of this theory for non-specialists.

SUMMARY: It is well known that in some nonlinear diffusive systems, we can observe various interesting phenomena such as spatial pattern formation, traveling waves and complex spatio-temporal dynamics. Recently, it was revealed that some simple nonlinear diffusive systems can exhibit irregular behavior of solutions by a sort of the butterfly effect. In this talk, I will present a few examples and explain the mechanism of stabilization and destabilization.

SUMMARY: As a \(p\)-adic version of local Langlands correspondence, \(p\)-adic local Langlands correspondence for \(\mathrm {GL}_2(\mathbb {Q}_p)\), which is a correspondence between two dimensional irreducible \(p\)-adic representations of \(\mathrm {Gal}(\overline {\mathbb {Q}}_p/\mathbb {Q}_p)\) and irreducible \(p\)-adic Banach representations of \(\mathrm {GL}_2(\mathbb {Q}_p)\), was recently established by Breuil, Berger–Breuil, Colmez, Kisin, Paskunas. As this correspondence also encodes information about \(p\)-adic variations of the both sides, it is expected to have many applications to some important problems in number theory concerning relationships between Galois side and automorphic (or analytic) side. For example, Kisin and Emerton independently applied it to prove Fontaine–Mazur conjecture on the modularity of two dimensional geometric odd \(p\)-adic representation of \(\mathrm {Gal}(\overline {\mathbb {Q}}/\mathbb {Q})\).

In our talk, I’d like to explain these topics and the recent developments. In particular, I’d like to explain another application of \(p\)-adic local Langlands correspondence, precisely, its application to the rank two case of global and local epsilon conjectures on the functional equation of Kato’s Euler systems associated to Hecke eigencuspforms.