Diophantine equations are equations where only integer solutions are accepted. There are many types of Diophantine equations and many results are known. Our Diophantine equation is of the form x^n+y^n=m!. Erdös and Obláth showed that the Diophantine equation x^2+y^2=m! has only two positive integer solutions (x,y,m)=(1,1,2),(12,24,6).
In this talk, the factorial function m! is replaced with a generalized factorial function Π(m) over number fields. Then whether there are infinitely many solutions or not depends on the number field. We give necessary and sufficient condition for existence of infinitely many solutions of x^2+y^2=Π(m). More generally, we introduce an observation for higher degree equation x^n+y^n=Π(m).

In this talk we give estimates for the dimensions of sets in real numbers which uniformly avoid finite arithmetic progressions. More precisely, we say that $F$ uniformly avoids arithmetic progressions of length $k\geq 3$ if there is an $\epsilon>0$ such that one cannot find an arithmetic progression of length $k$ and gap length $\Delta>0$ inside the $\epsilon\Delta$ neighbourhood of $F$. Our main result is an explicit upper bound for the Assouad (and thus Hausdorff) dimension of such sets in terms of $k$ and $\epsilon$. In the other direction, we give examples of sets which uniformly avoid arithmetic progressions of a given length. We also consider higher dimensional analogues of these problems, where arithmetic progressions are replaced with arithmetic patches lying in a hyperplane. As a consequence, we obtain a discretised version of a `reverse Kakeya problem': we show that if the dimension of a set in $\mathbb{R}^d$ is sufficiently large, then it closely approximates arithmetic progressions in every direction. The above is a joint work with Fraser and Yu. Finally we show that the converse of `reverse Kakeya problem' does not hold. This is a single-author work.

We propose a theory of the evolution of a minimal food web by sequential invasion of new species [1]. Our theory is based on the standard generalized Lotka-Volterra equations, where basal species compete through resource depletion [2]. The considered food webs are “minimal”, as each species only feeds on a single resource, leading to a hierarchical, tree-like food web [1,3]. We prove that at each invasion step there is one uniquely determined outcome: either the invader peacefully coexists with the residents and resources are re-distributed; the invader is eliminated; or one or several of the resident species are removed in a uniquely defined extinction cascade.
At the end of either of these processes the resulting food web relaxes to a globally stable (and feasible) steady state. We break down the essence of our theory in the conceptual “invasion extinction model” (IEM), which allows us to analytically compute the persistence time and the extinction size distribution.

Program:
10:00-10:15　T. Hatsuda (RIKEN) “Opening”
10:15-10:45　S. Maekawa (RIKEN) “Introduction to Spintronics”.
10:45-11:45　M. Matsuo (KITS, Beijing) “Spintronics phenomena in non-inertial frames” (Review)
13:30-14:30　M. Hongo (Riken) “Introduction to chiral transport phenomena” (Review)
14:30-15:00　Y. Hidaka (RIKEN) “Non-Equilibrium Quantum Transport of Chiral Fluids”
15:30-16:00　Y. Tsutsumi (RIKEN) “Spin-Vorticity Coupling in Liquid He”.
16:00-16:30　T. Takiwaki (NAOJ) “Chiral magnetohydrodynamic turbulence in core-collapse supernovae”
16:30-17:30　Discussion (and Closing by T. Hatsuda)

Title: Application of a gradient flow method to thermodynamics of QCD with dynamical quarks
Language: Japanese or English

The aim of this workshop is to gather researchers of high-energy and condensed-matter physics working on chiral Matter and Topology, to exchange ideas and establish collaborations to tackle unsolved issues and carry out future extensions. The workshop expects to welcome 40-60 participants who are interested in the aforementioned topics.

Organizers:
Tomoki Ozawa, Tetsuo Hatsuda (RIKEN iTHEMS)
Di-Lun Yang (RIKEN Nishina Center; YITP, Kyoto)
Chang-Tse Hsieh (Kavli IPMU / ISSP, the Univ. of Tokyo)
Jiunn-Wei Chen, Guang-Yu Guo (National Taiwan Univ.)
Hsiang-Nan Li (Academia Sinica)

Title: An introduction to operator algebras

Abstract: Operators are linear maps from a (usually an infinite dimensional) linear space (most frequently the Hilbert space) to itself, which is like matrices of infinite degree. Operators form an algebra by obvious addition and multiplication. Operators appear in most of the fields in mathematics, in algebra, in geometry, in analysis, ... Some of the key words at the beginning of these lectures are "spectral theory" "operator algebras" "Tomita-Takesaki theory". These lectures are for non-professional people.

The symposium will focus on the cross-cutting issues shared in broad applications of data assimilation from geoscience to various physical and biological sciences. In particular, the symposium will enhance discussions among researchers with various background on, for example, non-Gaussian and nonlinear data assimilation problems, Big Data Assimilation (BDA), high-performance computation (HPC), Uncertainty Quantification (UQ), advanced intelligence (AI) and machine learning, multi-scale and multi-component treatments, observational issues, and mathematical problems.

Abstract Submission Deadline: October 14, 2018 at 11:59 p.m. UTC
Registration Deadline: December 16, 2018 at 11:59 p.m. UTC
Submission/Registration Fee: Free