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.

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

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.

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.

Introduction to mathematical modelling in physiology and medicine

The 4 lessons are devoted to describe the derivation of some famous mathematical models in the fields of neuroscience and metabolism, by emphasizing how the scientists built the model. To this purpose is crucial to introduce some relevant concepts of physiology.

The topics we will consider are the following:

Lessons 1 and Lesson 2: Introduction to mathematical modelling of the biological neuron

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. Then the McCulloch-Pitts model is presented, aimed to propose a different approach to describe the behaviour of neuron. Finally, we will introduce the trion model, a mathematical description of Mountcastle's model of neocortex, with particular reference to the utilization of the trion model in the explanation of the so-called “Mozart effect”

Lesson 3: Introduction to mathematical modelling of sleep

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

Lesson 4: Introduction to mathematical modelling of insulin secretion

In the last lesson, after a description of the main actions of insulin, the Grodsky's model and the Sturis's model will be introduced.