COURSE DESCRIPTIONS

Applied Probability and Statistics

Ishimura Naoyuki, Japan (Email : ishimur at econ.hit-u.ac.jp)

This course provides basic techniques on the modelling and the analysis of uncertainty. Topics covered include: the review on random variables and probability distributions, simple random processes, Markovian models, elements of statistical inference, and the application to financial modelling (mainly Cox-Ross-Rubinstein model). A part of the course is devoted to the corresponding computer skills.

Commutative Algebra

Takayasu Kuwata, Tokai University (Email : kuwata at tokai-u.jp )

The course is an introduction to commutative algebra. The goal is to understand Hilbert's Nullstellensatz. We will study : Commutative rings and ideals ; Quotient rings and ring holomorphisms ; Prime and maximal ideals ; Affine algebraic sets and coordinate rings ; Euclidean domain, PID and UFD ; Natherian rings ; Radical of an ideal; Primary ideals ; Modules over a ring ; Integral dependence ; Noether's normalization lemma ; Hilbert's Nullstellensatz

References:

1. Miles Ried, Undergraduate Commutative Algebra, Cambridge
2. M.F.Atiyah & I.G.MacDonald, Introduction to Commutative Algebra, Perseus Books

Discrete mathematics

Pierre Arnoux, Marseille University, France (Email : arnoux at iml.univ-mrs.fr)

Course Description:

1. Elementary group theory and cryptography.Finite groups. Examples: Cyclic groups, ring Z/nZ, multiplicative group of the invertible elements of the ring. Small Fermat theorem, Euler-Fermat theorem, Cauchy and Lagrange theorems. Application: the RSA cryptography. Some primality tests.
2. Vector spaces on a finite field and error-correcting codes. Finite fields, vector spaces on finite fields. Explicit constructions. Basic notions on error-correcting codes: Hamming's distance, weparation distance. Code, weight of a code. Examples of codes with good properties.
3. Matrices and graphs.Adjacency matrix of a graph and path counting. Elementary theory of positive matrices. Irreducible and primitive matrices. Perron-Frobenius theorem and applications. Incidence matrix of an oriented graph. Flows and tensions on a graph. Cyclomatic and co-cyclomatic numbers. First notions of homology.

Functional Analysis

Franck Sueur, Pierre and Marie Curie University, France (E-mail : fsueur at ann.jussieu.fr)

In these lectures we will study the essentials of functional analysis with a special attention towards potential use in mathematical physics, probability theory and calculus of variations. The following subjects will be tackled : Integration and measure theory, Lebesgues spaces, Linear functionals and duality, Duality in Lebesgue spaces and finite Radon measures, Hilbertian analysis, Fourier series, Fourier transform, Distributions.

References :

1. J.-F. Babadjian, D. Smets and F. Sueur : Basic Functional Analysis, Lecture Notes for the Paris Graduate School of Mathematical Science program. http://www.ann.jussieu.fr/~fsueur/bfa.html, 2010.
2. H. Brézis : Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2010
3. E. Di Benedetto : Real analysis, Birkhaüser, Boston (2002)
4. W. Rudin : Real and complex analysis, McGraw-Hill Int. Ed., third edition (1987)
5. W. Rudin : Functional analysis, McGraw-Hill Int. Ed., second edition (1991)

Group theory

Helene R. Tyler, Manhattan College, USA

Course Description: An introduction to algebraic structures with an emphasis on groups, this course covers symmetric groups and Cayley's Theorem, normal subgroups, cosets, Lagrange's Theorem, the Isomorphism Theorems, and Sylow's Theorem.

Reference : A Book of Abstract Algebra, by Charles C. Pinter, 2nd edition, McGraw-Hill

Introduction to Partial Differential Equations and their approximations

Brigitte Lucquin, Pierre and Marie Curie University, France (E-mail : lucquin at ann.jussieu.fr)

This course is an introduction to the theoretical and numerical study of boundary value problems leading to partial differential equations of different types. After the introduction of Sobolev spaces, we will first focus on the variational formulation of elliptic boundary value problems in order to solve them, thank's to the Lax Milgram theorem. In a second step, we will approximate them by use of the finite element approximation.

We will finally introduce the finite difference method and use it for the approximation of elliptic, but also parabolic (heat equation) or hyperbolic (wave equation) equations.
References :

1. Atkinson K., Han W., Theoretical Numerical Analysis, Springer (2001), Texts in Applied Mathematics, Vol. 39
2. Dautray R., Lions J.L., Mathematical Analysis and Numerical Methods for Science and Technology, Springer Verlag (1988), Vol. 2 Functional and Variational Methods.
3. Lucquin B., Pironneau O., Introduction to scientific computing, John Wiley and Sons (1998).

Introduction to Stochastic Processes

Sonia Fourati, INSA Rouen, France (E-mail : fourati at insa-rouen.fr)

1. Markov Chains on Denumerable spaces. General definitions. Matrix Transition. Stopping times, Strong Markov Properties. Reccurent/trensient states. Irreductibility. Invariant measures. Convergence theorem. Applications and Examples : Random walks. Galton-Watson Chains.
2. Poisson Processes.Definition of (compound, compensated) Poisson processses. Strong Markov Property. Point Poisson Processes. Birth and Death Processes. Waiting Lines.
3. Brownian Motion. Definition. Existence. Brownian Motion as a Markov and a L\'evy process. Brownian Motion as a Martingale. Irregularity of the paths, behavior at small times and at large times. Distribution of first passage times.

References :

1. W. Feller : An Introduction to Probability Theory and Its Applications. Vol. 1 & 2. Wiley
2. S.Karlin : A first course in stochastic processes. Acad. Press.
3. M. Lefebvre : Applied Stochastic Processes. Universitext Springer
4. J. Medhi : Stochastic Processes. Wiley.
5. J.R. Norris : Markov chains. Cambridge University Press.

Number theory

Fidel R. Nemenzo, Diliman University of the Philippines (E-mail : fidel at math.upd.edu.ph) and Michel Waldschmidt, Pierre and Marie Curie University, France (E-mail : miw at math.jussieu.fr)

This course serves as an introduction to algebraic number theory. It will begin with a review of elementary number theory and abstract algebra. The arithmetic of algebraic number fields will be discussed, including examples such as quadratic and cyclotomic fields, with the study of Diophantine equations as motivation. An application to Fermat’s Last Theorem will be discussed.

Reference

1. Andre Weil, Number Theory for Beginners. Springer Verlag, 1979
2. Daniel Marcus, Number Fields. Springer Verlag, 1979
3. Kenneth Ireland and Michael Rosen, A Classical Introduction to Modern Number Theory, Springer Verlag (GTM 84), 1990

Optimization

Bernard Rousselet, University of Nice Sophia-Antipolis, France (Email: Bernard.Rousselet at math.unice.fr)

The course is an introduction to the optimization of functions of n variables. Necessary and sufficient conditions will be introduced. The case without constraints will yield non linear systems of n equations which may be solved with numerical algorithms of optimisation. the case with equality and inequality constraints will yield systems of a particular structure for which some algorithms will be considered. These techniques will be used to solve optimal design of discrete mechanical structures or of optimization problems of economics . If the audience is interested, the extension of these results to situations in infinite dimensions involving differential equations may be considered.

Software : scilab (www.scilab.org) for numerical computations and wims (wims.unice.fr) for interactive exercises

Projective Geometry

Michel Jambu, University of Nice-Sophia Antipolis, France (Email : jambu at unice.fr)

In this course, we will introduce a non-Euclidean geometry. After some summary of the basic knowledge of affine and Euclidean geometries, we will define the projective spaces as quotient of a vector space or the space of all the vector lines in a vector space and their properties in relation with affine and Euclidean geometry. This course is a basic introduction to projective geometry.

Reference : Geometry, by Michèle Audin, Universitext, Springer

Topology II

Michel Jambu, University of Nice-Sophia Antipolis, France (Email : jambu at unice.fr)

In the first part of this course, the aim is to provide the students with a foundation with general topology in continuation of the course of Topology I. We will recall compactness , we will introduce local compactness and one-point compactification as well as complete metric spaces. In the second part, we will first focus on limit of functions, uniform convergence and Ascoli theorem. Then we will introduce normed spaces and Banach spaces to finish with asymptotic series.