Modern variational methods have a wide range of theories; the direct method, the critical point theory represented by the minimax method and the mountain pass theorem, Morse-Conley theory, and so on.
These theories have been extensively studied not only in the field of partial differential equations, such as finding solutions to nonlinear elliptic partial differential equations as the stationary point of some energy functional, but also in connection with various mathematical fields, such as Riemannian geometry, symplectic geometry, mathematical physics, and so on.

We carry out this project based on the following two subjects:
(A) Deepening the insight of the lack of compactness appearing in non-compact variational problems;
(B) Applications of variational methods to evolution equations.

“Non-compact variational problem” in subject (A) is the variational problem such that the approximate sequence (such as minimizing sequence, or Palais-Smale sequence) is not always compact by itself, and it often appears in the application to the fields mentioned in the introduction.
By virtue of the concentration-compactness principle, introduced by Prof. Pierre-Louis Lions in 1980s, the mechanism of the lack of compactness is generally understood and this theory has been widely applied to the study of non-compact variational problems.
Subject (A) aims to refine the understanding of the mechanism and give new applications for non-compact variational problems.

Besides, by the use of the concentration-compactness principle, there are various studies of the asymptotic behavior of global-in-time solutions to nonlinear evolution equations, such as nonlinear parabolic, wave, dissipative equations.
The purpose of subject (B) is to develop the study of evolution equations using variational methods, represented by these applications.

In order to achieve the two subjects, we will invite Prof. Bernhard Ruf (University of Milan) and Prof. Luca Martinazzi (University of Rome ``La Sapienza") for a long-term to give a series of lectures and hold three workshops.
For subject (B), in particular, we will hold a workshop in order to identify current issues and open problems based on the progress since the RIMS workshop “Progress in Variational Problems -Variational Methods in the Study of Evolution Equations-”, which was held under the same theme.
In addition, we will hold a school mainly for young researchers. At these workshops and schools, we will invite young researchers from abroad for a short-term.
This project will be carried out in close collaboration with the Osaka Central Advanced Mathematics Institute (OCAMI).

The singularity theory of smooth maps describes how the singular points of such maps behave. Originating with the pioneering works of Thom, Morse, and Whitney up until about 1960, the basic tools were then developed by Mather in the late 1960s. It continued to develop since then, and continues to be actively researched today. In addition, both inside and outside of mathematics, it has been applied to various areas, particularly in geometric contexts, by considering singular phenomena as singular points of smooth mapsand analyzing them in detail using singularity theory.

A basic tool of singularity theory and its applications is the classification of singularities. For this, the structures of quotient rings of various commutative rings play important roles, but these structures can become very complicated, and investigating them cannot be done by hand, so the use of software becomes necessary. However, well-known general-purpose software is of limited help for this. In recent years, in addition to the significant development of computer performance, a new tool for classification using computer algebra has been developed by Japanese researchers, and the above limits have been greatly exceeded and great progress has been made.
Additionally, there has been remarkable development of criteria for singularities that should be studied hand-in-hand with classification. Furthermore, new kinds of applications have been discovered, such as the analytic Hamiltonian appearing in topological insulators and continuum mechanics, with possibilities for these applications to develop rapidly.

These recently emerging new theories are multidisciplinary and worthy of earnest study. In view of this situation, meetings will be held to present the latest results of singularity theory and to provide lectures by experts in these new theories. Holding such meetings over the course of several months and interacting with young researchers, singularity theory researchers, and researchers who are looking to apply singularity theory as well, we can mutually develop both pure singularity theory and its applications. We can also promote training the next generation of researchers.

Designating October to December 2022 as "Singularity Theory Special Months", and inviting two leading researchers from Brazil and Spain, and representative Japanese researchers as well, we will regularly hold schools and research meetings. In addition, the Mathematical Society of Japan Seasonal Institute (MSJ-SI2022) "Deepening and developing applied singularity theory" will be held during this period.
This project will be linked with this event, with each complementary to the other.

Since the founding of the mathematical theory of stochastic differential equations by Kiyosi Itô in 1942, research on stochastic calculus in Japan has had a great influence on the development of probability theory throughout the world. Probability theory itself has grown remarkably over the last century, and now encompasses an extremely wide range of research topics. Within mathematics, it has strong interactions with fields such as partial differential equations, potential theory and geometry. Moreover, the vast expansion of probability research is directly linked to applications across the sciences, in areas such as statistical mechanics, biology, economics and the analysis of big data. As a result of probability theory now having so many different aspects, it is difficult for individual researchers to grasp the subject in its entirety.

With the keywords "stochastic processes and stochastic analysis", this RIMS Research Project aims to further promote international joint research in probability by disseminating results in the area that originate in Japan. Centering on the following three themes, the program will provide a bird's-eye view of modern probability theory. (For each theme, a conference is planned, the title of which is shown in parentheses.)
(i) Stochastic Processes Related to Stochastic Partial Differential Equations (Stochastic Partial Differential Equations and Stochastic Calculus)
(ii) Analysis of Stochastic Models Motivated by Statistical Mechanics (Stochastic Analysis on Large Scale Interacting Systems)
(iii) Random Matrix Theory, Combinatorial Probability and Quantum Information (Random Matrices and Their Applications)
Furthermore, to inspire exchanges between these themes, in September 2023, we will organize a large-scale conference at the Research Institute for Mathematical Sciences, inviting researchers in stochastic processes and stochastic analysis who are active at the front line of probability research internationally.

Mid-career researchers will take responsibility for the practical organization of conferences on each theme and, in doing so, will promote the participation of young researchers, including as speakers, so that the next generation of researchers plays an active role. We encourage the participation of female researchers in all aspects of the program, seeing it as an opportunity to promote gender diversity in research on probability theory in Japan.