# PISA-HOKKAIDO Summer School on Mathematics and Its Applications

## Cofinanced by Hokkaido University (Leading Program) and partially supported by the Department of Mathematics of the University of Pisa and by the Scuola Normale Superiore

30 August 2016 - 13 September 2016

# Planned Activities

MINI COURSES

Topic: Algebra - Geometry
Lecturers: Masahiko Yoshinaga, Giovanni Gaiffi

Topic: PDEs
Lecturers: Hideo Kubo, Tommaso Pacini

Topic: Mathematics in Biology and Neuroscience
Lecturers: Toshiyuki Nakagaki, Maria Laura Manca

KUBO-PACINI CLASSES

• Hideo Kubo

Title: "Wave equations with metric perturbation"

In this lecture a systematic approach to wave equations with metric perturbation will be discussed based on the Minkowski null-frame. More precisely, our Lorentz metric is supposed to be a perturbation by the unknown function from the Minkowski metric, and we will examine which kind of assumption on the perturbation is necessary to guarantee the global existence result for small initial disturbance. Since this course is intended as an introductory one for undergraduate students, the only pre-requisites will be (i) the calculus (ii) basic functional analysis.

• Tommaso Pacini

Title: "Ricci flow on Riemann surfaces"

Given a surface or a manifold, the Ricci flow is a well-known technique for deforming its metric, trying to reach a new metric with better properties. Specifically, it is a system of Partial Differential Equations. We will examine it in the simplest case, metrics on a surface, and discuss its main properties from both the analytic and the geometric perspective. This is intended as an introductory course for undergraduates, so the only pre-requisites will be (i) the geometry of curves and surfaces, and (ii) basic complex analysis.

YOSHINAGA-GAIFFI CLASSES

• Masahiko Yoshinaga

Title: "Eulerian polynomials, lattice points counting, and arrangements."

Abstract: One of the most important combinatorial invariant of an arrangement is the so-called characteristic polynomial. Recently, Kamiya-Takemura-Terao introduced the notion of "characteristic quasi-polynomial" which is a refinement of characteristic polynomials, and has close relationships with Ehrhart quasi-polynomials of rational polytopes. In this course, I would explain these materials together with Eulerian polynomials and then apply to "Riemann hypothesis for Linial arrangements" by Postnikov-Stanley.

(Reference: arXiv:1501.04955 and references in it.)

Contents: 1. Characteristic quasi-polynomials of integral arrangements (due to Kamiya-Takemura-Terao). 2. Ehrhart theory. (Ehrhart quasi-polynomials for rational polytopes.) 3. Eulerian polynomials and root system generalizations (along the work by Lam-Postnikov). 4. Location of zeros of characteristic polynomials.

• Giovanni Gaiffi

Title: "Configuration spaces and representations of the symmetric group"

- Orlik solomon algebras - action of the symmetric group on the cohomology of the complement of the braid arrangement - compactifications of complements of arrangements (in particular De Concini- Procesi models) - action of the symmetric group on the cohomology of the models of the braid arrangement - some information on results for other reflection groups and open questions

NOTES: http://www.dm.unipi.it/~gaiffi/Teorapp2016/

NAKAGAKI-MANCA CLASSES

• Toshiyuki Nakagaki

Title: "Introduction to mathematical ethology"

Mathematical ethology is proposed as a new direction of mathematical life science: the idea is to bring equations of motion into conventional ethology (ethology is study of animal behavior). Here we primarily focus on single-celled organisms since cell behaviors are elementary and basic in full range of organisms. In this lecture, we'd like to present some of current topics in mathematical ethology of cell and lower animal. We emphasize how standard methods of applied mathematics are used there. The aim of lecture is to show an example of how mathematical methods develop a new direction of science.

The topics we will consider are listed below.

(1) Adaptive Optimization of Foraging Network. A giant amoeba of Physarum (a single celled organism) optimizes its body shape of network form that connects spatially distributed multiple locations of food source. We consider the equations of motion for self-organizations of the optimal shape of network, and a new bio-inspired method of optimal design. In this topic, some of standard methods in applied mathematics are used.

(2) Capacity of Space Memory . Ciliates like Paramecium and Tetrahymena (single-celled swimmer by many hair called cilia emerged from surface of cell) have capacity of memorizing a shape of swimming arena. It is well known that swimming behaviors in ciliates can depend on electrical potential across the membrane, whose dynamics obeys so-called Hodgkin-Huxley type equations (originally proposed for excitation of squid neuron). Based on this knowledge, we will consider the mathematical model for the space memory. Some standard methods of nonlinear dynamics are introduced. This might be partially complimentary to the lecture by Prof. Maria Laura Manca.

(3) Basic mechanics of Crawling Locomotion. Crawling locomotion of lower organisms is often adaptable to a wide variety of ground conditions. Basic and general mechanics of crawling locomotion is considered. You will see the mathematical tools playing a pivotal role of understanding legless and legged crawling although they look different.

• Maria Laura Manca

Title: "Brain math models"

Firstly, PhD and Master Degree students will learn that neurons are the functional units of the brain, and that they convey information using electrical and chemical signals. This is crucial for understanding the basis of the Hodgkin - Huxley mathematical model that describes how action potentials in neurons are initiated and propagated.

Furthermore, students will study the main biological mechanisms of the sleep focused to explain the most important mathematical models of sleep regulation.

Finally, students can learn about alternative math models describing the phenomena of synchronization for Kuramoto type systems, Schroedinger - Kuramoto and fractional Schroedinger - Kuramoto systems.

PRATELLI CLASSES

• Maurizio Pratelli

Title: "Stochastic Processes and Stochastic Calculus with application to Economics and Finance"

San Miniato, September 12-16, 2016