Courses

1. History of Mathematics

Sonja Brentjes (Aga Khan University, London) and José Ferreirós (Universidad de Sevilla)

The course will analyse facets of the history of mathematics that relate directly to the general theme of the conference: key episodes in the transmission and appropriation of mathematical knowledge across cultures; the configuration of ancient and modern mathematics; local patterns vs. internationalism in the mathematical community; mathematics in peace and war; and the emergence of a world-wide net of mathematics practitioners. The course will thus be designed to strike a balance between episodes and analyses covering large periods in the development of the discipline, allowing participants to form a global picture of mathematics in its contribution to cultural and scientific development.

This course will be delivered by two specialists whose areas of expertise are mathematics in the Ancient and Medieval world, and modern mathematics, respectively. Brentjes has published widely on the transmission and reworking of Greek mathematics through Arabic manuscripts, and edited recently (with S. Gissis) a special issue of the journal Science in Context [12 , 1999, no. 3] devoted to the topic, Crossing boundaries: New approaches to the history of "pre-modern" science and technology . Ferreirós has written on set theory and its role in the emergence of modern mathematics (Birkhäuser, 1999) and just edited (with J. J. Gray) a collective volume, The Architecture of Modern Mathematics: Essays in history and philosophy (Oxford UP, 2006).

Day 1: Mathematics and cross-cultural exchange

Day 2: Mathematics and beliefs

Day 3: Mathematical practices, philosophical foundations and methodological variations

Day 4: Mathematics in cooperation and conflict

Day 5: Mathematics and institutions

2. Mathematical Education

Luis Rico (Granada), and Olimpia Figueras (CINVESTAV, México).

Plenary speaker: Jeremy Kilpatrick (University of Georgia, USA).

"Teaching Relevant, Useful Mathematics"

Abstract

School mathematics reflects what a society values and what it wants for its young people. If students are to become mathematically literate citizens capable of using mathematics to improve themselves and their world, the compulsory mathematics curriculum they study needs to be both relevant and useful. How can mathematics be taught so that students see its relevance and can use what they have learned?

3. Statistics for Human Welfare

Mary Gray (American University) and Nora Donaldson (Kings College School of Medicine, London UK).

Plenary speaker: Fritz Scheuren (National Opinion Research Center Chicago and Washington , D.C. )

This course is intended for graduate students in the mathematical sciences who have had little exposure to applications of statistics to real-world problems whether or not they have had some course work in probability or mathematical statistics.

We begin with fundamentals of exploratory data analysis, including an introduction to the use of SPSS. A quick review of distributions and descriptive statistics will be followed by an introduction to sampling. Standard statistical inference techniques including t-tests, regression, ANOVA, and non-parametric methods will be explored. The emphasis will be on statistics as the art and science of learning from data. Realistic examples illustrating the three main aspects of statistics will be featured:

Design: Planning how to obtain data to answer questions of interest

Description: Summarizing the data that are obtained.

Inference: Making decisions and predictions based on the data.

Among the application areas to be addressed are biostatistics, finance and economics, social sciences, and environmental science. The goal of the course is to equip students with the ability to make substantial professional contributions as statisticians to the educational, economic, health, and public policy arenas in their home countries.

4. Mathematics and Economics

José Luis Fernández (UAM) and Luis A. Seco (Toronto).

Plenary speaker: Paul embrecht (ETM).

Financial markets reached a level of complexity in the seventies that have created a path for risk management, which has flourished in the nineties. These issues in risk management have, in turn, opened up the road for risk mitigation in other areas, such as industrial operations, sustainable growth and disaster insurance. The selection of themes have been done with the objective to present a diverse, all-encompassing coverage of many of these topics.

5. Mathematics and Biology: Biological Motion

Jair Koiller (FGV/RJ and Millenium Institute for Brazilian Mathematics) and John Bush (MIT)

Plenary speaker: Ron Elber (Cornell University).

This course aims to introduce students to a theme that has fascinated mankind since pre-historical times. Aristotle already had some interesting insights on animal locomotion, but true scientific thinking started with da Vinci and Galileo´s school. For a general panorama on Biological Motion, see for instance the special issue of Science (vol 288, number 5463, 7 Apr 2000 ,  "Movement: molecular to robotic"). This minicourse will focus primarily on motion in fluids.  First we discuss the two extreme regimes, high Reynolds (fishes and birds), by John, vs. low Reynolds numbers(from microorganisms to tiny invertebrates), by Jair. Next, John will lecture about the  fuzzy region where these two regimes meet (insects), and in which intense research is going on presently. Finally, Jair will discuss (in a sketchy way) the new research area of intracellular motion -  where the machinery of life resides. In the afternoons the students will work in groups, reading materials and solving problems and puzzles.  As pre-requisites, we expect that student to have done basic courses in ODEs and PDEs, and has no prejudice against some bold approximations and physical arguments.

6. Mathematics and Physics: Moduli spaces in geometry and physics

Oscar García Prada (CSIC, Madrid) and S. Ramanan (Chennai, India)

Plenary speaker: Simon Donaldson (Imperial College, UK)

The aim of this course is to introduce the students to the theory of moduli spaces and to some of their applications.The moduli spaces to be considered include vector bundles and Higgsbundles on Riemann surfaces, as well as  Calabi-Yau threefolds together with stable bundles on them. These have applications to  topology and string theory. The methods used combine techniques from algebraic and differential geometry and global analysis.

7. An introduction to dynamical systems via continuous fractions

Lorenzo Díaz (Catholic University, Rio de Janeiro) and Carlos Gustavo Moreira (Instituto de Matemática Pura e aplicada, Brazil).

Plenary speaker: Maria Jose Pacifico ( UFRJ, Brazil)

We begin the course by introducing the division algorithm and stating the basic properties of the expansion of real numbers via continuos fractions. We next derive some approximation properties of real numbers by continuous fractions (Diophantine Approximations). We proceed by studying the so-called Gauss map which is closely related to the expansion via continuous fractions (we will explain the link). Using the Gauss map we study some notions of symbolic dynamics. We next introduce the Gauss measure (an invariant measure of the Gauss map equivalent to the Lebesgue one) and study its ergodic properties. From such properties, we will derive some properties about the distribution of digits in the expansion via continuous fractions of "typical" numbers. Finally, we interpret some approximation properties in terms of measure (statistically). We close the course by studying dynamically defined Cantor sets. These sets can be used to study geometric and topological properties of the Lagrange spectrum (and of the Markov spectrum). The Lagrange spectrum is a set of positive real numbers associated to the best approximation of real numbers by rationals. The relation between regular Cantor sets and the Lagrange spectrum becomes clear after a work of M. Hall published in 1947.

8. PDEs and applications

Hatem Zaag (ENS, París) and Nader Masmoudi (Courant Institute).

Plenary speaker: Edriss Titi (Irvine and Weizmann).

The first part of the course is dedicated to kinetic equations. A particular attention will be devoted to Boltzmann equation and some hydrodynamic limits will be presented. In a second part, we address the question of blow-up in PDEs. Taking the semilinear heat equation as a lab model, we will introduce several notions and methods to study the blow-up behavior.

Program:

PART 1: Nader Masmoudi

1) Kinetic models (Boltzmann equation, Vlasov Poisson) 2) properties of Boltzmann equations 3) Averaging lemma 4) Renormalized solutions 5) Hydrodynamic limits 1 and 2

PART 2: Hatem Zaag

1) fundamental solutions 2) fixed point method for the Cauchy problem 3) Lyapunov functional methode 4)"Always blow-up" for subcritical (in the Fujita sense) exponent. 5) Occurence of global solutions for supercritical exponent. 6) Selfsimilar variables and no selfsimilar blow-up