Analysis, classically, refers to the field dealing with limits and differential and integral calculus,but as times have changed, the themes covered by analysis have expanded greatly. The research areas of our analysis section are diverse, including the theory of differential equations,operator algebras, representation theory, real analysis, complex analysis, and probability theory. Among these, the theory of differential equations can be further divided into the theory oflinear partial differential equations, nonlinear partial differential equations, integrable systems,inverse problems, and others. Even within differential equations alone, there are many different research methods; for example, complex analytic methods, real analytic methods, functional analytic methods in the tradition of Tosio Kato and Kôsaku Yosida, methods of nonlinear analysis including variational methods and fixed-point theorems, dynamical systems methods, microlocal analysis methods including Mikio Sato's algebraic analytic methods, probabilistic methods, and algebraic methods used in integrable systems and representation theory. Such diversity can also be seen in other areas of analysis, and sometimes even transcends the boundaries of analysis itself.
　Our specific research themes include the following. In the field of complex analysis, in addition to several complex variables centered on topics such as the Bergman kernel and CR manifolds, research is conducted on complex algebraic varieties, which are also objects of algebraic geometry. In the field of ordinary differential equations, research on the Painlevé equations, originating from the study of special functions, is carried out using algebraic methods and exact WKB analysis. In the field of partial differential equations, research is conducted on equations of major practical importance, including the Navier-Stokes equations describing fluid motion, reaction-diffusion equations appearing in mathematical biology, fractional-order partial differential equations describing anomalous diffusion phenomena, and dispersive equations such as the Schrödinger equation. In real analysis, topics such as the Fourier restriction conjecture, the Kakeya conjecture, and geometric inequalities are studied. In operator algebra theory, research is conducted on areas such as algebraic quantum field theory, the K-theory of C^∗-algebras, and minimal dynamical systems. In the field of probability theory, in addition to stochastic differential equations and stochastic process theory, research is carried out on stochastic analysis on infinite-dimensional spaces and stochastic statistical mechanics models. Research is also pursued with applications in mind not only within mathematics but also in physics, materials science, life sciences, and industrial technology. Other topics include studies of infinite-dimensional representations and the Penrose transform in representation theory, as well as research related to discrete groups and ergodic theory, demonstrating the broad scope of research in this area.