Senior undergraduate courses

• 4320: Intro to Stochastic Processes
• 4331: Real Analysis
• 4335: Partial Differential Equation
• 4355: Math of Signal Representation
• 4364: Numerical Analysis
• 4377: Advanced Linear Algebra
• 4383: Number Theory

Online course

• 5331 Linear Algebra with Applications
• 5350 Into to Differential Geometry
• 5386 Regression and Linear Models
• 5397 Discrete Mathematics II
• 6376 Numerical Linear Algebra

Graduate course

• 6302: Modern Algebra
• 6320: Func Real Variables
• 6322: Func Complex Variable
• 6326: Partial Diff Equations
• 6342: Topology/Geometry
• 6366: Optimization Theory
• 6370: Numerical Analysis
• 6377: Basic Tools For Applied Math
• 6382: Probability Statistics
• 6384: Discrete Time Model in Finance
• 6395: Select Topics Analysis
• 6395: Abstract Harmonic Analysis
• 6397: Intro To Riemannian Geometry
• 6397: Mathematical Biology
• 6397: Data Mining and Auto.Learning
• 6397: Numerical Methods for Option Pricing in Finance
• 6397: Introduction to Information Theory and Applications
• 6397: Dynamical System and Ergodic Theory
• 7396: Control System Governed By Pde
• 7396: Numerical Analysis for Nonlinear and Bifurcation Problems with Applications

- Introduction to Numerical Analysis, by J. Stoer and R. Bulirsch Springer-Verlag, 3rd edition, 2002
- Numerical Linear Algebra, by Lloyd N . Trefethen and David Bau, SIAM, 1997

Further reference:
- Numerical Analysis in Modern Scientific Computing, by Peter Deuflhard, Andreas Hohmann, Springer-Verlag, 2003
- Numerical Mathematics, by A. Quarteroni, R. Sacco, and F. Saleri. Springer-Verlag, 2000

Recommended Texts:

• Mathematical Statistics with Applications, Sixth Edition, by Wackerly, Mendenhall III and Scheaffer, Duxbury Press
• A First Look at Rigorous Probability Theory by Jeffrey Rosenthal, 2000.
• An Introduction to Stochastic Modeling, 3rd Edition, by H Taylor and S Karlin, Academic Press.
• A First Course in Probability, Sixth Edition by Sheldon Ross, 2002, Prentice Hall
• An Introduction to Probability Theory and Its Applications, Vol1, 3rd edition, 1968 by William Feller (any edition would be fine).
• Probability by Leo Breiman, 1968, Addison- Wesley.

Topics include: Differentiable Manifolds, tangent space, tangent bundle, Riemannian metric, connections, curvatures, geodesics, Jacobi fields, comparison theorems, harmonic forms and Hodge theory.

Mathematical Models in Biology by: Leah Edelstein-Keshet in: SIAM Classics in Applied Mathematics

Other references:

• Dynamical Models in Biology Ellner and Guckenheimer Princeton university Press
• A Course in Mathematical Biology de Vries, Hillen, Lewis, Muller, and Schonfisch SIAM

Topics will be selected from the text and the following list.

• 1) Mathematical neuroscience derivation of Hodgkin-Huxley equations bursting, excitability, Fitzhugh-Nagumo eqn's Wilson-Cowan equations and rate models
• 2) Mathematical epidemiology SIR compartment models (susceptable, infected, recovered)
• 3) Population models discrete time Leslie matrices (largest and next largest eigenvalue issues)
• 4) Pattern formation Turing bifurcations, phylotaxis
• 5) Genetics and evolution
• 6) Networks theory scale-free, etc. models; gene transcription networks
• 7) Synchrony (fire-flies) a la Mirolla-Strogatz

Automatic Learning of unknown functional relationships Y = f(X) between multidimensional inputs X and outputs Y, involves algorithms dedicated to the intensive analysis of large finite "training sets" of "examples" of inputs/outputs pairs (X,Y). In numerous applications such as artificial vision, shape recognition, sound identification, handwriting recognition, text classification, Automatic Learning has become a major tool to emulate many perceptive tasks. Hundreds of machine learning algorithms have been emerged in the last 10 years, as well as powerful mathematical concepts, focused on key learning features : generalisation, accuracy, speed, robustness. The mathematics involved rely on information and probability theory, non parametric statistics, complexity, functional approximation. The course will present major learning architectures : Support Vector Machines and Artificial Neural Networks, as well as Clustering techniques such as the Kohonen networks . Concrete applications will be presented.

Homework and exams : Students familiar with Mathlab or equivalent scientific softwares will have the possibility to replace a large part of the homework assignments and of the exams by applied projects involving computer simulations and implementation of algorithms taught in the course

not required, but for reference

• R. Seydel; Tools for Computational Finance. 2nd Edition. Springer, Berlin-Heidelberg-New York, 2004;
• Y. Achdou, O. Pironneau; Computational Methods for Option Pricing. SIAM, Philadelphia, 2005

Brief Description: The course gives a brief overview on the foundations of financial markets and financial derivatives and then focuses on the following topics:

* Binomial methods and the discrete Black-Scholes formula.
*The Black-Scholes equation and its numerical evaluation.
* Monte-Carlo methods and numerical solution of stochastic differential equations.
* Pricing of European options, numerical solution of parabolic PDEs.
* Pricing of American options, numerical solution of free boundaryproblems.
* Pricing of exotic options.

Shannon's information theory is arguably one of the most important achievements of 20th century mathematics, and it has found a wide range of applications.

The purpose of this course is to offer a mathematically rigorous introduction to information theory. This introduction will be followed by a discussion of the uses of information theory in practice which will include guest lectures by specialists who work in ergodic theory, neuroscience, data mining and physics.

Some reference books:

• "Introduction to Ergodic Theory" by Peter Walters,
• "Dynamical Systems and Ergodic Theory" by Mark Pollicott and Michiko Yuri
• the first chapters of "Zeta functions and the periodic orbit structure of hyperbolic dynamics" by William Parry and Mark Pollicott

The course will discuss dynamical systems and measure-theoretic properties used to characterize them, in particular entropy (a number measuring the complexity of a system) and mixing (which characterizes how correlated successive observations are).

Topics include: basic examples (circle and interval maps, toral automorphisms, subshits of finite type); ergodicity, mixing and their spectral characterization; recurrence, transitivity, minimality; the Ergodic Theorems of Birkhoff and von Neumann; topological entropy, measure theoretic entropy and the relation between them; construction of invariant measures. We will illustrate these concepts with examples and applications.

We will then use functional analysis (operators on Banach spaces) to describe a class of invariant measures for subshits of finite type, and the mixing properties of these measures. Through Markov partitions, subshits of finite type model a wide class of dynamical systems, as we will describe.

Other topics can also be presented. The material and level of the course will depend on the background of the audience. Notes will be provided.