MTH2007: Introduction to Real Analysis
Maxime Zavidovique
University Pierre and Marie Curie, Paris, France
Email: maxime.zavidovique@imj-prg.fr
Real numbers: basic notions, properties, countable and uncountable sets. Sequences and Series: basic
definitions, Cauchy sequences, Bolzano-Weierstrass Theorem. Continuity: definition, continuous functions
on closed intervals, Intermediate Value Theorem, uniform continuity. Riemann Integral: upper and lower
sums, existence of integral for continuous functions. Differentiation: definition of derivative, Fundamental
Theorem of Calculus, Taylor Theorem. Sequences of functions: pointwise and uniform convergence. Series
of functions: power series, radius of convergence, behavior of series under differentiation and integration,
Taylor series.
References:
1. Bernard Gelbaum and John Olmested, Counterexamples in analysis. Dover, 2003.
2. Michael Reed, Fundamental Ideas of Analysis, Wiley, 1998.
3. Walter Rudin, Principles of mathematical Analysis. Third edition. International Series in Pure and
Applied Mathematics. McGraw-Hill Book Co., 1976.
4. William Trench, Introduction to Real Analysis.
MTH2013: Advanced Algebra
Takayasu KUWATA
Tokai University, Japan
Email: kuwata@tokai-u.jp
The aim of this course is to understand theory of field extensions and Galois theory. We will begin
reviewing basic knowledge of rings and ideals.
References:
1. Abstract Algebra - An Introduction , Thomas W. Hungerford, Harcourt
2. Algebra, Thomas W. Hungerford, Springer
MTH2023: Algebra and Geometry
Michel Waldschmidt
University Pierre and Marie Curie, Paris, France
Email: michel.waldschmidt@upmc.fr
This course has two parts :
1. Algebra : summary of groups, Rings and Fields. Linear Algebra : Vector spaces, Subspaces and Spanning
Sets, Linear independance, Basis, Linear Maps, Finite Dimensional Vector Spaces, Matrices, Linear
equations, Eigentheory.
2. Affine Geometry: Affine Spaces, Barycenters, Affine Maps, Affine Frames, Thales, Pappus and
Desargues Theorem
References:
1. Geometry by M. Audin (Springer Verlag) and lecture notes by M. Jambu
MTH2011: Approximation of Partial Differential Equations I
Brigitte Lucquin
University Pierre and Marie Curie, Paris, France
Email: lucquin@ann.jussieu.fr
In this course of level I, we will focus on the finite difference approximation of some classical Partial
Differential Equations (PDE) : Laplace equation, heat equation and wave equation. We will also present the
finite volume approximation of a transport equation. The finite element approximation will be the object of
the next course of level II.
For any of the examples studied, we will check the convergence of the numerical approximation. When
possible, we will also study these equations at the theoretical level (existence of solutions, uniqueness,
regularity, stability estimates ...); in order to do so, we will recall some basic tools in analysis ($L^p$ spaces,
Fourier series, Fourier transform, ...).
References:
1. Atkinson K., Han W., Theoretical Numerical Analysis, Springer (2001), Texts in Applied
Mathematics, Vol. 39
2. Lascaux P., Lectures on Numerical methods for time dependent equations. Tata Institute of
Fundamental Research, Bombay (1976)
3. Lucquin B., Pironneau O., Introduction to scientific computing, John Wiley and Sons (1998).
MTH2002: Ordinary Differential Equations
Mark S.Gockenbach, Michigan Technological University, USA
Email: msgocken @mtu.edu
Course Description: This course covers both the basic theory of ordinary differential equations (ODEs) and
techniques for solving them.Techniques include separation of variables, integrating factors, eigenvalue
methods for systems of linear ODEs with constant coefficients, reduction of order, variation of parameters,
undetermined coefficients, and series solutions. The basic existence, uniqueness, and continuous
dependence theory will be covered, along with an introduction to phase plane analysis.
Background: Students are expected to have strong calculus and linear algebra skills. Important prerequisites
from calculus include techniques of integration and power series. Prerequisite concepts from linear algebra
include subspace, basis, eigenvalues and eigenvectors, and the diagonalization of square matrices.
References:
1. Elementary Differential Equations by William Boyce and Richard Diprima (9th edition, Wiley,
2008)
2. Ordinary Differential Equations in Theory and Practice by Robert Mattheij and Jaap Molenaar
(SIAM 2002)
MTH2005: Differential Geometry
Naoyuki ISHIMURA
Hitosubashi University, Japan
Email: ishimura@econ.hit-u.ac.jp
This course provides an introduction to the topics of differential geometry n Euclidean spaces. After
recalling the basics of vector calculus, the theory of plane curves and surfaces are lectured. One aim of this
course is the famous Gauss-Bonnet theorem, which clarifies the importance of curvatures. Other contents
include: Frenet-Serret formula, the first and the second fundamental form, the mean and the Gauss
curvature. An application to the problem of curvature evolution may be treated if the schedule is allowed.
References:
1. M.P. Do Carmo; Differential Geometry of Curves and Surfaces, Prentice-Hall, 1976.
MTH2024: Financial Mathematics
Thomas Lim
Evry, France
Email: thomaslim.tl@ gmail.com
This course will cover basics of mathematical finance: actuarial computation, interest rate, bond and
portfolio management.
References:
1. Hull (Prentice Hall) Options, futures and other derivatives Martellini, Priaulet, Priauler (Wiley)
Fixed income securities: valuation; risk management and portfolio strategies.
MTH2001: Topology I
Michel Jambu
University of Nice-Sophia Antipolis, France
Email: jambu@unice.fr
We intend to give a comprehensive introduction to basic topology, with an emphasis towards advanced
calculus. After recalling some basic set theoretic notions, we shall start exploring topology with the real
numbers as our guideline. We expect students to be familiar with standard calculus and basic notions such as
continuity and differentiability of real valued functions as usually taught in early semesters of college.
We shall the extract the relevant concept to introduce the notion of topology and we shall keep the metric
case in mind as one of the key object of study.
As the course develops, we shall introduce more subtle notions. Further reading and references will be given
to students as the course develops.
References:
1. Basic Topology, M.A. Amstrong Undergraduate texts in Maths, Springer
MTH2003: Probability I
Pierre Arnoux
Marseille University, France
Email: arnoux@iml.univ-mrs.fr
Finite probability spaces: Combinatorial analysis, Finite probabilities, Random variables. Examples,
expectation, variance, standard deviation, basic properties. Generating function. Conditioning and
Independence. Weak law of large numbers.
Countable probability spaces: countability, countable probabilities, classical examples, limit theorems.
General probability spaces: general probabiilty measures. Random variables having a density: basic
properties. Examples.
Techniques of simulation.
Convergence of random variables (in distribution, in probability, almost sure, in mean). Characteristic
function and inversion formula.
Laws of large numbers: strong law of large numbers, central limit theorem.
The course will use simulation for all the chapters to give examples of the various properties, and to give
students an intuition of probabilities.
Some useful references:
1. William Feller: An introduction to probability theory and its application, vol. I
2. Jim Pitman: Probability
3. Sheldon Ross: A first course in probability.
MTH2009: Numerical Analysis I
Seam Ngonn
Royal University of Phnom Penh, Cambodia
Email: seamngonn@yahoo.fr
In this course, we will study numerical methods and its implementation with Scilab. The students have to
learn:
1. Introduction to Scilab
2. Linear equation (directed method, iterative method and conjugate gradient)
3. Nonlinear equation (fixed points, Newton)
4. Differential equations
References :
1. David Kincaid and Ward Cheney, Numerical Analysis.
2. Alfio Quarteroni, Numerical Method (application with matlab)
MTH2006: Statistics I
Xavier Bressaud
Univerité Paul Sabatier - Toulouse 3, France
Email: bressaud@math.univ-toulouse.fr
Under construction.