2018年度秋季総合分科会(於:岡山大学)

SUMMARY: We develop a new theory to solve infinite-dimensional stochastic differential equations describing infinite particle systems. At the same time, we introduce various methods: algebraic, analytic, and geometric method for constructing stochastic dynamics of infinite particle systems. Our new method enables us to apply the classical stochastic analysis to various models of infinite particle systems. We use this to construct infinite-dimensional stochastic dynamics related to random matrices. These are infinite particle systems with logarithmic interacting potential. As an application, we prove dynamical universality of random matrices.

SUMMARY: Quantum field theory (QFT) is a branch of theoretical physics where the quantum property of fields is studied, where a field stand for any physical entity extended through time and space. In this talk, I would like to illustrate three aspects of connections between QFT and mathematics though a number of examples:
- the first is to formulate QFT in a mathematical language, i.e. to find mathematics of QFT.
- the second is to obtain mathematical conjectures using the ideas of QFT, i.e. to derive mathematics by QFT,
- and the third is to apply existing mathematics in the analysis of QFT, i.e. to use mathematics for QFT.

SUMMARY: A discrete subgroup of \(\textrm {PSL}(2, \mathbb {C})\) is called a Kleinian group. An element of \(\textrm {PSL}(2, \mathbb {C})\) is a conformal automorphism of the Riemann sphere and also it is a hyperbolic isometry on the hyperbolic 3-space. Hence, the theory of Kleinian makes significant contributions to complex analysis as well as to the hyperbolic geometry. In this talk, we discuss complex analytic properties related to Kleinian groups. In particular, we consider geometric function theoretic properties of regions of discontinuity of Kleinian groups and complex structures of deformation spaces of Kleinian groups.

SUMMARY: The Čech–de Rham cohomology together with its integration theory has been effectively used in various problems related to localization of characteristic classes. Likewise we may develop the Čech–Dolbeault cohomology theory and on the way we naturally come up with the relative Dolbeault cohomology. This cohomology turns out to be canonically isomorphic with the local (relative) cohomology of A. Grothendieck and M. Sato with coefficients in the sheaf of holomorphic forms so that it provides a handy way of representing the latter.

In this talk we present the theory of relative Dolbeault cohomology and give, as applications, simple explicit expressions of Sato hyperfunctions, some fundamental operations on them and related local duality theorems. Particularly noteworthy is that the integration of hyperfunctions in our framework, which is a descendant of the integration theory on the Čech–de Rham cohomology, is simply given as the usual integration of \(C^{\infty }\) differential forms. Also the Thom class in relative de Rham cohomology plays an essential role in the scene of interaction between topology and analysis. This approach also yields a new insight into the theory of hyperfunctions and further leads to a number of results that cannot be achieved by the conventional way.

The talk includes a joint work with N. Honda and T. Izawa.

SUMMARY: In this lecture, I will focus on recent progress on metrics that maximize the first eigenvalue of the Laplacian (under area normalization) on a closed surface. I first discuss Hersch–Yang–Yau’s inequality (1970, 1980), which was the starting point of the above problem. This is an inequality indicating that the first eigenvalue (precisely, the product of it with the area) is bounded from above by a constant depending only on the genus of the surface. Then I overview the recent progress on the existence problem for maximizing metrics together with the relation with minimal surfaces in the sphere. Finally I will discuss Jacobson–Levitin–Nadirashvili–Nigam–Polterovich’s conjecture, which explicitly predicts maximizing metrics in the case of genus two, and the affirmative resolution of it (joint work with Toshihiro Shoda).

SUMMARY: In this talk, we survey some of the results in the field of arithmetic dynamics, focusing on problems which arise as analogs of theorems and conjectures for abelian varieties. More specifically, a central object of study in arithmetic dynamics is the orbit \[ \{P, \phi (P), \phi (\phi (P)), \phi (\phi (\phi (P))), \ldots \}, \] where \(\phi :X\dashrightarrow X\) is a rational map of an algebraic variety \(X\) to itself defined over a number field \(k\) and \(P\) is a \(k\)-rational point of \(X\). Because of the formal analogy between an orbit \[ P\mapsto \phi (P) \mapsto \phi (\phi (P)) \mapsto \phi (\phi (\phi (P)))\mapsto \cdots \] (where each arrow is an application of \(\phi \)) and the subgroup generated by a point in an abelian variety \[ O\mapsto P \mapsto 2P \mapsto 3P \mapsto \cdots \] (where each arrow is an addition by \(P\)), one could hope that theorems and conjectures for abelian varieties over number fields hold when we restrict to a single orbit, even if the entire variety \(X\) does not have a group structure. We will explore this “hope” for questions concerning torsion points, Galois images, integral points, and intersections with subvarieties. We will find that sometimes the dynamical analog holds, sometimes the dynamical analog holds conditionally on deep conjectures, and sometimes the most natural dynamical analog fails completely and the revised analog remains open.

SUMMARY: We consider the inhomogeneous Dirichlet-boundary value problem for the cubic nonlinear Schrödinger equations on the half line. We present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions to the problem by using the classical energy method and factorization techniques.

SUMMARY: There has been much demand for the statistical analysis of dependent observations in many fields, for example, economics, engineering and the natural sciences. A model that describes the probability structure of a series of dependent observations is called a stochastic process. The stochastic processes mentioned here are very widespread, e.g., non-Gaussian linear processes, long-memory processes, nonlinear processes and continuous time processes etc. For them we will develop not only the usual estimation and testing theory but also many other statistical methods and techniques, such as discriminant analysis, cluster analysis, nonparametric methods, higher order asymptotic theory etc. Optimality of various procedures is often shown by use of local asymptotic normality (LAN), which is due to LeCam. Applications of the theory are enormous.