21st Century COE Program at Keio
Integrative Mathematical Sciences:
Progress in Mathematics Motivated by Natural and Social Phenomena

Our Center Of Excellence (COE) program has started in August, 2003. The Integrative Mathematical Sciences is to open a new horizon to mathematical sciences by challenging to model various natural and social phenomena through data. The core of our program is fundamental mathematics, and it is wrapped with data science and experimental mathematics. The data science plays a role of an interface to various phenomena by means of data. The experimental mathematics supports experimental aspect of mathematical sciences. The main objective of our program is to promote the construction of international educational and research bases through integrating such three different aspects of mathematical sciences.

Pathways Lecture Series in Mathematics, Keio

November 15, Monday 2004   Place:  Keio University, Yagami Campus (Room 12-211) Speaker:  Prof. Benjamin Weiss (Hebrew University of Jerusalem) Lecture 1 15:30 - 16:00   Pre-Lecture   Prof. Hitoshi Nakada (Keio University) 16:30 - 18:00   Amenable groups and Ergodic theorems

Amenable groups will be defined and illustrated and I will explain why they are the natural setting for ergodic theory. For this purpose I will sketch a proof of von Neumann's ergodic theorem and show how to extend it to amenable groups. The more recent generalizations of Birkhoff's pointwise ergodic theorem will also be described.

November 16, Tuesday 2004   Place:  Keio University, Yagami Campus (Room 12-104) Speaker:  Prof. Benjamin Weiss (Hebrew University of Jerusalem) Lecture 2 16:30 - 18:00   The Rokhlin lemma and of its users

Probably the most useful tool in measurable ergodic theory is the lemma originally established by Rokhlin which says roughly that modulo a small set all ergodic actions look alike. I will explain what form this lemma takes in the setting of amenable groups and give some illustrations of its power.

November 17, Wednesday 2004   Place:  Keio University, Yagami Campus (Room 14-203) Speaker:  Prof. Benjamin Weiss (Hebrew University of Jerusalem) Lecture 3 16:30 - 18:00   Entropy and mixing properties

Entropy was introduced by Shannon in his theory of communication and was extended to the dynamical setting by Kolmogorov. After defining the entropy several applications will be given. Entropy is also closely related to a very strong kind of mixing - called uniform mixing. This and several other kinds of mixing will be discussed. The extensions to actions of amenable groups will also be given

November 18, Thursday 2004   Place:  Keio University, Yagami Campus (Room 14-203) Speaker:  Prof. Benjamin Weiss (Hebrew University of Jerusalem) Lecture 4 16:30 - 18:00   On the isomorphism problem in ergodic theory

The central theoretical problem in ergodic theory is the problem of classifying all ergodic actions up to isomorphism. After defining this key notion I will demonstrate its intimate connection with entropy. In particular I will discuss a new result (joint with D. Ornstein) that says that all FINITELY OBSERVABLE functions of stochastic processes that are isomorphism invariants are continuous functions of the entropy.

November 1, Monday 2004   Place:  Keio University, Yagami Campus (Room 12-211) Speaker:  Prof. Pierre Schapira (Université Pierre et Marie Curie) Lecture 1 15:30 - 16:00   Pre-Lecture   Prof. Nobuyuki Tose (Keio University) 16:30 - 18:00   Sheaves

The six operations Microsupport Constructible sheaves Generalized solutions of holomorphic functions

November 2, Tuesday 2004   Place:  Keio University, Yagami Campus (Room 12-211) Speaker:  Prof. Pierre Schapira (Université Pierre et Marie Curie) Lecture 2 16:30 - 18:00   D-modules

The ring D Characteristic variety of D-modules, Microsupport and characteristic variety

November 4, Thursday 2004   Place:  Keio University, Yagami Campus (Room 14-203) Speaker:  Prof. Pierre Schapira (Université Pierre et Marie Curie) Lecture 3 15:00 - 16:00   Holomorphic solutions of D-modules

Elliptic and hyperbolic systems Integral transforms Solutions of holonomic D-modules and perverse sheaves

November 4, Thursday 2004   Place:  Keio University, Yagami Campus (Room 14-203) Speaker:  Prof. Pierre Schapira (Université Pierre et Marie Curie) Lecture 4 16:30 - 18:00   Temperate Solutions

The subnalytic (Grothendieck) topology The sheaf of temperate holomorphic functions Open problems

October 4, Monday 2004   Place:  Keio University, Mita Campus (8th Floor Lecture Hall, East Research Bldg.) Speaker:  Prof. John Ball (University of Oxford) Lecture 15:00 -  Some Open Problems in the calculus of variations.

Some outstanding open problems of nonlinear elasticity are described. The problems range from questions of existence, uniqueness, regularity and stability of solutions in statics and dynamics to issues such as the modelling of fracture and self-contact, the status of elasticity with respect to atomistic models, and the passage from the three-dimensional elasticity to models of rods and shells.

October 5, Tuesday 2004   Place:  Keio University, Yagami Campus (Room 12-109) Speaker:  Prof. John Ball (University of Oxford) Lecture 1 14:45 - 16:15   Pre-Lecture   Prof. Norio Kikuchi (Keio University) 16:30 - 18:00   Global Attractors for Semiflows without Uniqueness.

The talk will describe an abstract framework for semiflows with possibl non-unique solutions for given initial data. The measurability and continuity properties of such 'generalized semiflows' will be discussed. Necessary and sufficient conditions are given for the exisence of a global attractor. As applications, the existence of a global attractor is proved (a) for damped semilinear hyperbolic equations under weak growth hypotheses (b) for the 3D Navier-Stokes equations under the (unproved) hypothesis that all weak solutions are continuous in time with values in L²

October 6, Wednesday 2004   Place:  Keio University, Yagami Campus (Room 14-203) Speaker:  Prof. John Ball (University of Oxford) Lecture 2 14:45 - 16:15  Mathematical Models of Crystal Microstructure (I).  (suitable for a general (scientific) audience) Lecture 3 16:30 - 18:00  Mathematical Models of Crystal Microstructure (II)  (more mathematical than Lecture 2)

Materials undergoing solid phase transformations (involving a change of shape at a critical temperature) typically form patterns of microstructure, in the simplest case involving fine layering of regions in which the deformation gradient alternates between approximately constant values. The scale of such microstructure is intermediate between atomic scales and macroscopic ones. The microstructure has a profound affect on macroscopic material properties. The tools that are currently used to analyse the formation and morphology of microstructure, and the relationship between microstructure and macroscopic properties, come from nonlinear analysis - specifically weak convergence, Young measures, and the calculus of variations. A particularly important role is played by consciderations of compatibility, and several key questions reduce to what the possible mappings are having gradients in a specified set of matrices, or to approximate versions of this. The talk will describe these tools, and applications of them to understanding austenite/martensite interfaces, shape-memory alloys, hysteresis and the like. The lectures describe joint work with R.D. James, C. Carstensen and D. Schryvers.

October 7, Thursday 2004   Place:  Keio University, Yagami Campus (Room 14-203) Speaker:  Prof. John Ball (University of Oxford) Lecture 4 16:30 - 18:00  Some Open Problems in Elasticity.

Some outstanding open problems of nonlinear elasticity are described. The problems range from questions of existence, uniqueness, regularity and stability of solutions in statics and dynamics to issues such as the modelling of fracture and self-contact, the status of elasticity with respect to atomistic models, and the passage from the three-dimensional elasticity to models of rods and shells.

Pathways Lecture Series in Mathematics, Keio

May 31, Monday 2004   Place:  Keio University, Yagami Campus (Room 12-209) Speaker:  Prof. David Elworthy (Mathematics Insitute, University of Warwick) Lecture 1 15:00 - 16:00   Pre-Lecture   Prof. Atsushi Atsuji (Keio University) 16:30 - 18:00   Introduction to Wiener space and geometric analysis on it.

Classical Wiener space is the space of continuous paths on a Euclidean space, starting at the origin, and furnished with the so called Wiener measure. As with other Gaussian measures a rich structure is induced on the space. We will consider special vector fields on this space and their divergences, indicating how these analogues of classical constructions relate to constructions in stochastic analysis such as the Ito integral and martingale representation. 'Malliavin Calculus' will be briefly introduced.

June 1, Tuesday 2004   Place:  Keio University, Yagami Campus (Room 12-109) Speaker:  Prof. David Elworthy (Mathematics Institute, University of Warwick) Lecture 2 16:30 - 18:00  Stochastic differential equations on manifolds and path integral solutions to heat equations.

Stochastic differential equations can be thought of in many ways. One is as a model for random perturbations of deterministic dynamical systems, another is as a way of construction solutions of linear diffusion equations, such as the heat equation for differential forms, by integrating over all paths on the manifold. We will describe different ways to do this, including the construction of Brownian motion by the stochastic version of Cartan's development (rolling without slipping), and show the insights they give for vanishing theorems.

June 2, Wednesday 2004   Place:  Keio University, Yagami Campus (Room 12-109) Speaker:  Prof. Robert Devaney (Boston University) Lecture 3 16:30 - 18:00  Geometry of stochastic flows.

On a compact manifold smooth stochastic differential equations have solution flows of diffeomorphisms. As with deterministic dynamical systems there is a corrresponding 'smooth ergodic theory` with Lyapunov exponents, stable manifolds etc.describing the sensitive dependence of the solution to changes in the initial condition. If we look at 'moment exponents' the corresponding versions of stability have strong implications for the topology of the manifold. An important concept here is the 'LeJan-Watanabe' connection induced on the manifold by the flow. We will briefly describe some of the main ideas and some relations between stability and the geometry of the manifold.

June 3, Thursday 2004   Place:  Keio University, Yagami Campus (Room 14-204) Speaker:  Prof. Robert Devaney (Boston University) Lecture 4 16:30 - 18:00  Functions of finite energy and generalizations.

Most of this talk will be self contained and fairly elementary, but at the end I hope to draw in some points discussed in the earlier talks. We will start with the question: if the derivative of a function is square integrable is the function itself square integrable (up to addition of a constant)? and show how this leads to some geometry and analysis when considered on Euclidean spaces, compact and non-compact manifolds, classical Wiener space, and the space of paths and loops on a Riemannian manifolds.

May 19, Wednesday 2004   Place:  Keio University, Yagami Campus (Room 12-101) Speaker:  Prof. Robert Devaney (Boston University) Lecture 1 15:00 - 16:00   Pre-Lecture   Prof. Hitoshi Nakada (Keio University) 16:30 - 17:30   Chaos Games and Fractal Images (Appropriate for undergrads with no real much background)

In this lecture we will describe some of the beautiful images that arise from the "Chaos Game". We will show how the simple steps of this game produce, when iterated millions of times, the intricate images known as fractals. We will describe some of the applications of this technique used in data compression as well as in Hollywood. We will also challenge students present to "Beat the Professor" at the chaos game and maybe win his computer.

May 20, Thursday 2004   Place:  Keio University, Yagami Campus (Room 12-101) Speaker:  Prof. Robert Devaney (Boston University) Lecture 2 16:30 - 17:30  The Mandelbrot Set, the Farey Tree, and the Fibonacci Sequence (Appropriate for undergrad math majors who know calculus)

In this lecture we will describe several folk theorems concerning the Mondelbrot Set. While this set is extremely complicated from a geometric point of view, we will show that, as long as you know how to add and how to count, you can understand this geometry completely. We will encounter many famuos mathematical objects in the Mandelbrot set, like the Farey tree and the Fibonacci sequence. And we will find many soon-to-be-famous objects as well, like "Devaney"sequence. There might even be a joke or two in the talk.

May 24, Monday 2004   Place:  Keio University, Yagami Campus (Room 12-101) Speaker:  Prof. Robert Devaney (Boston University) Lecture 3 16:30 - 17:30  Exponential Dynamics and Topology (Appropriate for beginning grad students)

In this talk we describe some of the rich topological stductures and bifurcations that arise in the dynamics of the complex exponential (and other entire and meromorphic) maps. In particular, we show how much such objects as Cantor bouquets and indecomposable continua arise and bifurcate as various parameters are changed.

May 25, Tuesday 2004   Place:  Keio University, Yagami Campus (Room 12-101) Speaker:  Prof. Robert Devaney (Boston University) Lecture 4 16:30 - 17:30  Sierpinski Curve and Triangle Julia Sets (Appropriate for beginning grad students)

In this lecture we show how such fractals as the Sierpinski curve and triangle appear as the Julia sets of rational maps of the complex plane of the form z^n + A / z^d. There are some very rich dynamics associated with these sets. For example, it is known that any two Sierpinski curves are always homeomorphic. However, we shall show that in any neighborhood of A=0, there are always infinitely many parameters for which the Julia set is a Sierpinski curve, but the dynamics on any pair of these sets is very different.

April 4, 14, 21, and 28, 2004, 14:45 - 16:15   Place:  Keio University, Yagami Campus (Room 14-203) Speaker:  Prof. Shoshichi Kobayashi (University of California at Berkeley) Introduction to hyperbolic complex analysis/ geometry

In this lecture, I will explain how to exten, by geometric means, such results as Liouville`s thorem and Picard`s theorems in functions of one complex variable to higher dimension. Altough the lectures are aimed at beginning graduate students, they could be understood by undergraduates who have already studied function theory.

	Inquiries
	21st Century COE Program at Keio Integrative Mathematical Sciences Liaison Officer: Miyuki SAITO Tel 81-45-566-1442 Fax 81-45-566-1768 E-mail coe-admin@math.keio.ac.jp URL http:/coe.math.keio.ac.jp
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