European Women in Mathematics

CONTENTS.

PREFACE.

INTRODUCTION.

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The paper presents the main ideas of the Black, Scholes and Merton theory as a simplified introduction for non specialist. One explains what kind of financial product are options, how to price them and how to build a hedging portfolio that allows, according to the theory, to get rid of the risk.

Students's works about modeling and fitting are exposed. A first try to fit the Black-Scholes model is done with estimations of trend and volatility, test of normality, estimations of Hölder exponent in case of a driving fractional Brownian motion. Otherwise, the stationnarity of the return logarithm is studied. Finally, in the Diffusion-Jumps mixed model, an algorithm is done to detect jumps in a financial series and tests are made for the mixing of two (or more) Gaussian laws hypothesis.

Data are these of “Bourse de Paris” (Lyonnaise des eaux, AGF, Peugeot) of the year 1997 and French stocks index (1802-1993).

Let us consider an economic agent, who invests in the assets of a financial market. The agent is facing optimal consumption and investment decisions, and wants to optimize his strategy. The optimization problem is usually dealt with one of the following methods : the mean-variance analysis, or the concept of utility functions. We recall the main tools, such as the method of Lagrange multipliers and the dynamic optimization (Bellman's equation). Then we give explicit expressions for the optimal strategy in some usual cases.

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These notes offer a short presentation of de Rham cohomology on differentiable manifolds with a proof of the isomorphism between de Rham cohomology and singular cohomology. We also recall some duality results, show how Betti numbers relate to Morse theory and shortly present the relation between de Rham and Čech cohomologies with coefficients in the sheaf of real locally constant functions for de Rham cohomology.

In this paper, we discuss the McKay correspondence as a relation between geometry of a resolution of quotient singularity and algebra of the group which acts. To see an example of the generalized McKay correspondence, we consider concrete constructions of crepant resolutions for any finite subgroup of SL(3, ℂ). Moreover, we draw an atlas of the generalized McKay correspondence to look around inside this world.

The aim of this contribution is to give an introduction to some aspects of quantum cohomology rings of flag varieties especially explaining their relationship with total positivity. The results outlined here are written up in greater generality elsewhere¹. In this paper we give a simplified exposition focusing on the variety of complete flags. We explain how the quantum cohomology ring for the full flag variety can be used to study totally positive uni-triangular Toeplitz matrices. The special case considered here contains most of the main ideas but avoids some of the technical difficulties which come up in the more general setting of partial flag varieties and total nonnegativity.

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In this lecture we discuss an intuitive approach to mirror symmetry for Calabi–Yau manifolds. We explain how the generating function for the number of rational curves for a quintic three-fold was originally found using mirror symmetry as well as the geometry of the moduli space of certain String Theories compactified on Calabi–Yau manifolds.

We discuss how some crucial features of chaotic dynamical systems are manifested in certain classical examples from hyperbolic geometry. These examples have been very influential in the development of aspects of dynamical systems theory.

In this talk we explain how certain kinds of subgroups of Coxeter groups can be used to derive regular and semi-regular tessellations. Furthermore, we show that matrices whose entries belong to certain rings of algebraic integers are associated with symmetry groups of tessellations of hyperbolic spaces. We give several examples and summarize some joint work [5,6] with Norman Johnson.

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We discuss some of the properties of the Bernoulli series

We first discuss here some classical results on the number of points with integral coordinates in convex rational polytopes P ⊂ Rⁿ starting with Ehrhart's theorem. Then, following Baldoni-Vergne and Szenes-Vergne, we present some recent results giving number of points with integral coordinates in P in terms of multidimensional residues. In particular, this allows to recover relations established by Khovanskii-Pukhlikov, Cappell-Shaneson and Brion-Vergne between volumes and number of points with integral coordinates in families of polytopes with parallel facets.

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The importance of raising public awareness in mathematics is pointed out. Different ways to achieve this goal are listed with many examples coming from the operation “World Mathematical Year 2000”. The main conclusion is that it is necessary to work closely with the school teachers.

During the World Mathematical Year 2000 various actions to improve the public image of mathematics have been initiated. Many problems and obstacles are now easier to name. The chief problems can be divided into global (i.e. not country specific) and local ones. In this article the local ones are considered for Croatia, as well as popularization actions. Suggestions for possible further actions as well for a more target-group-oriented approach in “mathematical public relations” are also made.

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The European Union report on Women and Science (“ETAN report”, 1999), which gives the status-quo on this issue for Western Europe, is first presented. Then different actions that are being initiated in this field, in particular in France, are described.

In this paper we report and analyze the composition and activities either of different Italian communities which are engaged in promoting positive actions or groups carrying out studies and projects on women and science. During the last years we have been following the Italian women’s effort in overcoming isolation, organizing themselves in official groups and gaining institutional visibility. This paper wants to contribute to enrich information and analysis on women organizations and their public presence in Italy. We provide an overview of active groups and their official roles both in Italian Universities and Public Research Institutions and a few data on the access of women to positions. We will see a large, rich and composite map of subjects which work to promote positive actions or develop precious data collection and analyses. We discuss networking and visioning, along with new technologies exploitation, as necessary and powerful strategies for mainstreaming gender equality.

We report on the “Questionnaire project” of the Committe Women and Mathematics of the European Mathematical Society.

A generalized Bergman-Hardy space is a space of holomorphic functions which includes among many other examples, the Hardy space, the Bergman space on the polydisc as well as the Fock space on ℂⁿ. This article gives a short review of results about compactness characterizations for Hankel operators on generalized Bergman-Hardy spaces.

We report on a recent project that offers multimedia support for classical topics in undergraduate mathematics.

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