This is a series of lectures on "Geometric Perspectives for Fluid Dynamic Limit Theory" by the following speakers:
[DAY 1: Aug 31] Dr. Makiko Sasada (University of Tokyo)
[DAY 2: Sept 1] Prof. Kenichi Bannai (Keio University)

Abstract:
One of the fundamental problems in the natural and social sciences is to explain macroscopic phenomena that we can observe from the rules governing the microscopic system giving rise to the phenomena. Hydrodynamic limit provides a rigorous mathematical method to derive the deterministic partial differential equations describing the time evolution of macroscopic parameters, from the stochastic dynamics of a microscopic large scale interacting system.

In the article "Topological Structures of Large Scale Interacting Systems via Uniform Locality" joint with Yukio Kametani, we introduce a general framework encompassing a wide variety of interacting systems in order to systematically investigate various microscopic stochastic large scale interacting systems in a unified fashion. In particular, we introduced a new cohomology theory called the uniformly local cohomology to investigate the underlying geometry of the interacting system. Our theory gives a new interpretation of the macroscopic parameters, the role played by the group action on the microscopic system, and the origin of the diffusion matrix associated to the macroscopic deterministic partial differential equation obtained via the space-time scaling limit of the microscopic system.

The purpose of the series of lectures is to introduce to the audience the theory of hydrodynamic limits, especially the relation between the macroscopic observables and the microscopic interacting system. We then explain our new perspective of how geometry comes into play in investigating the interacting system, and introduce the ideas and results of our article.

*Detailed information about the seminar refer to the email.

The theory of maximal regularity is a powerful tool to get solutions having the best regularity to linear partial differential equations (PDEs) of parabolic type. The theory is also applicable to show well-posedness of various non-linear PDEs.

In the first part, We introduce the history of the development of the theory of maximal regularity and the way to apply non-linear PDEs.

In the second part, We give some applications to PDEs, e. g. the primitive equations, the Navier-Stokes equations, and elliptic equations with dynamic boundary conditions.

*Please contact Keita Mikami's mail address to get access to the Zoom meeting room.

We introduce the Uchuu suite of large high-resolution cosmological N-body simulations. The largest simulation, named Uchuu, consists of 2.1 trillion dark matter particles in a box of 2.0 Gpc/h. The highest resolution simulation, called Shin-Uchuu, consists of 262 billion particles in a box of 140 Mpc/h. Combining these simulations we can follow the evolution of dark matter haloes (and subhaloes) spanning from dwarf galaxies to massive galaxy cluster hosts. We present basic statistics, dark matter power spectra and halo (subhalo) mass function, to demonstrate the huge dynamic range and superb statistics of the Uchuu simulations. From the analysis of the evolution of the power spectra we conclude that our simulations are accurate enough from the Baryon Acoustic Oscillations up to very small scales. We also provide parameters of a mass-concentration model, which describes the evolution of halo concentrations, that reproduces our simulation data within 5% error for haloes with masses spanning nearly eight orders of magnitude at redshift 0