**INTRODUCTION TO MATHEMATICAL ANALYSIS METHODS IN IMAGE PROCESSING**

Instructors:

Lê Minh Triết, Dept of Math, Univ of Pennsylvania, USA .

Faculdo Memoli, Computational Topology group, Stanford Univ, USA.

Nicolas Privault, Professor of Mathematics at the Nanyang Technological University (Singapore) will offer a short course on "Partial Differential Equations and the Monte Carlo Method" during the last week of July 2011.

Partial Differential Equations and the Monte Carlo Method

Nicolas Privault

Summary: Partial differential equations (PDEs) are ubiquitous in the modeling of physical and biological systems, in engineering, as well as in economics and finance.

The aim of this course is to give a presentation of the deep connections existing between partial differential equations and stochastic diffusion processes, and to introduce the audience to some recent research results in the nonlinear case. Applications to the numerical resolution of PDEs by the Monte Carlo method and to mathematical finance will also be discussed.

Proposed schedule: 8 lectures of 90 minutes.

Outline:

1. Fundamental solutions of PDEs (90 minutes)

In this part we recall some results on the analytical solution of PDEs by kernel methods, in relation with potential theory. Here, no probabilistic background is needed and we will essentially rely on integral calculus and partial differentiation.

(a) Green kernels

(b) Solution by kernel methods - examples

(c) Existence criteria by kernel methods

2. Brownian motion and diffusion processes (2 × 90 minutes)

This section is a rapid introduction to stochastic calculus. Our main goal will be to enable the student to effectively compute the solution of simple stochastic differential equations (SDEs) based on Itˆ’s formula.

(a) Stochastic integrals

(b) Itoˆ formula

(c) Explicit solutions of simple SDEs

3. The Feynman-Kac representation formula (90 minutes)

This section is the probabilistic counterpart of Section 1. Using stochastic calculus we will show how the solution of a PDE with potential can be represented as the expectation of a random variable with terminal condition. This will generalize the kernel representations of Section 1 from one-dimensional integrals to path integrals.

(a) Proof of the Feynman-Kac formula

(b) Relation with kernel methods

(c) Interpretation in terms of heat diffusion

4. Explosion and stability of semilinear PDEs (2 × 90 minutes)

Here we will show how the Feynman-Kac representation formula can be used to derive stability and explosion results for semilinear partial differential equations, as a first application of the results of Section 3.

(a) Comparison of subsolutions

(b) First iteration - linear case

(c) Second iteration - semilinear case

5. Numerical solution by the Monte Carlo method (2 × 90 minutes)

In this section we will present another important application of the Feynman-Kac formula to the numerical solution of PDEs, with examples taken from mathematical finance.

(a) The Black-Scholes PDE

(b) Bond pricing equations

(c) Comparison by numerical examples

Prerequisites:

In analysis: integral calculus, partial differentiation, Gaussian convolution.

In probability: conditional probabilities, expectations, conditional expectations.

The following list of references is only indicative.

References

[1] R.F. Bass. Diffusions and elliptic operators. Probability and its Applications (New York). Springer-Verlag, New York, 1998.

[2] M. Birkner, J.A. L ́pez-Mimbela, and A. Wakolbinger. Blow-up of semilinear PDE’s at the critical dimension. A probabilistic approach. Proc. Amer. Math. Soc., 130(8):2431–2442 (electronic), 2002.

[3] M. Freidlin. Functional integration and partial differential equations, volume 109 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1985.

[4] T. Mikosch. Elementary stochastic calculus—with finance in view, volume 6 of Advanced Series

on Statistical Science & Applied Probability. World Scientific Publishing Co. Inc., River Edge,

NJ, 1998.

[5] N. Privault. Potential theory in classical probability. In Quantum potential theory, volume 1954 of Lecture Notes in Math., pages 3–59. Springer, Berlin, 2008.

[6] N. Privault. Notes on stochastic finances. Course notes, 2010.

Chung Nhân Phú (PhD candidate, State University of New York, Buffalo, USA) will give a short course on "Introduction to Ergodic Theory" during 25/07 to 19/08/2011, on Tuesdays & Fridays, 9:30-12:00.

ERGODIC THEORY

SUMMER 2011

NHAN-PHU CHUNG

Prerequisites: Measure theory. The background in other ﬁelds such as functional analysis, harmonic analysis, probability and group theory is not a prerequisite, but will be introduced as needed.

Time&Place: From 25/07 to 19/08, Tuesday&Friday 9:30am-12:00pm.

Room: to be announced.

Course description: Ergodic theory is the study of the qualitative properties of actions of groups on spaces. It is a very active area with many applications in physics, harmonic analysis, probability, dynamical systems and number theory. Recently, in 2010, Elon Lindenstrauss received Fields medal for his results on measure rigidity in ergodic theory, and their applications to number theory. In this course, I will introduce some notations, examples about ergodicity, mixing, measure-theoretic entropy, topological entropy, topological pressure and varia-tional principle theorem.

Recommended reading:

(1) Keller, Equilibrium states in Ergodic Theory. London Mathe-matical Society Student Texts, 42. Cambridge University Press, Cambridge, 1998.

(2) Petersen, Ergodic theory. Corrected reprint of the 1983 original. Cambridge Studies in Advanced Mathematics, 2. Cambridge University Press, Cambridge, 1989.

(3) Walters, An Introduction to Ergodic Theory. Graduate Texts in Mathematics, 79. Springer-Verlag, New York, Berlin, 1982.

Tentative plan: The course will follow, though not very closely, chapters 1,4,7,8,9 of the Walters’ book.

Starting from 18/5/2011 at 9:30, Dr. Nguyễn Hoài Minh (University of Minnesota, USA) will give a series of lectures, with the topics for the first 4 lectures will be:

1) New estimates for the topological degree.

2) Characterizations of Sobolev spaces and related inequalities.

3) Cloaking for the Helmholtz equations and the wave equation.

4) Scaling laws of energy and patterns in some elastic thin film problems.

Title: New estimates for the topological degree

Abstract: In this talk, I will present some new estimates for the topological degree of maps from a sphere into itself and discuss the optimality of a constant which appears naturally in these estimates. This is partly joint with Jean Bourgain (Institute for Advanced Study) and Haim Brezis (Rutgers University).