Chương trình

Thứ Bảy, 26/7/2014

Sáng

-  7h45-8h15: Đăng ký đại biểu

-  8h15-8h30: Tiệc trà

-  8h30-8h45: Khai mạc

-  8h45 – 10h15: Bài giảng của GS. Louis Chen

-  10h15 – 10h30: Tiệc trà giữa buổi

-  10h30 – 12h00: Bài giảng của GS. Vũ Hà Văn

Chiều

-  14h00 – 14h15: Tiệc trà

-  14h15 – 15h45: Bài giảng của GS. Phạm Xuân Huyên

Chủ Nhật, 27/7/2014

Sáng

- 8h30 – 8h45: Tiệc trà

- 8h45 – 10h15: Bài giảng của GS. Vogelius

- 10h15 – 10h30: Tiệc trà giữa buổi

- 10h30 – 12h00: Bài giảng của GS. Szemeredi

Tóm tắt các bài giảng:

1. GS. Louis H. Y. Chen – Đại học Quốc gia Singapore

Tiêu đề: Normal Approximation by using Stein’s Method and Malliavin Calculus

Bản tóm tắt:

Stein’s method is a method of probability approximation which hinges on the solution of a functional equation. For normal approx-imation the functional equation is a first order differential equation. The Malliavin calculus of variations is an infinite-dimensional differ-ential calculus whose operators act on functionals of general Gaus-sian processes. Nourdin and Peccati (2009) established a fundamental connection between Stein’s method for normal approximation and the Malliavin calculus through integration by parts. This connection is ex-ploited to obtain error bounds in central limit theorems for functionals of general Gaussian processes. Of particular interest is the fourth mo-ment theorem which provides error bounds of the order $\sqrt{\mathbb{E}(F^4)-3}$ in the central limit theorem for elements F of Wiener chaos of any order such that E(F^2) = 1. In this talk we will give an exposition of the work of Nourdin and Peccati.

2. GS. Phạm Xuân Huyên – ĐH Paris Diderot (Paris 7), Pháp

Tiêu đề: Feynman-Kac representation of fully nonlinear PDEs and applications

Bản tóm tắt:

The classical Feynman-Kac formula states the connection between linear parabolic partial differential equations (PDEs), like the heat equation, and expectation of stochastic processes driven by Brownian motion.

It gives then a method for solving linear PDEs by Monte Carlo simulations of random processes. The extension to (fully)nonlinear PDEs led in the recent years to important developments in stochastic analysis and the emergence of the theory of backward stochastic differential equations (BSDEs), which can be viewed as nonlinear Feynman-Kac formulas. In this talk, I will review the main results and ideas in this area, and present implications of these probabilistic representations for the numerical resolution of nonlinear PDEs, together with some applications to stochastic control problems in finance.

3. GS. Endre Szemeredi – Đại học Rutgers, Mỹ

Tiêu đề: The “absorbing” method

Bản tóm tắt:

The idea of the method is that we don’t have to work that hard. This roughly means that we first construct a so-called “absorbing” configuration, and after that we only need to solve the problem “almost”. We’re going to give examples from both graph and hypergraph theory. In the first example we will give a tight bound for the “co-degree” of a 3-uniform hypergraph for having a “tight Hamiltonian cycle”. In the second example we give strong bounds for the minimum degree of graphs containing a spanning tree for which the maximum degree is less than n/2log n.

4. GS. Vũ Hà Văn – Đại học Yale, Mỹ

Tiêu đề: Roots of random polynomials

Bản tóm tắt:

Estimating the number of real roots of a polynomial is among the oldest and most basic questions in mathematics.The answer to this question depends very strongly on the structure of the coefficients, of course.

What happens if we choose the coefficients randomly? In this case, the number of real roots become a random variable, whose value is between 0 and the degree of the polynomial. Can one understand this random variable? What is its mean, variance, and limiting distribution?

Random polynomials were first studied by Waring in the 18th century. In the 1930s, Littlewood and Offord started their famous studied which led to the surprising fact that in general a random polynomials with iid coefficients have (with high probability), order log n real roots. Their investigation opened a whole new area of random polynomials and random functions, with deep contributions from several leading mathematicians, including Erdos, Turan, Kac, Kahane, Ibragimov etc.

An exciting turn occurred in the 1990s, when physicists Bogomolny, Bohigas and Leboeuf established a link between roots of random polynomials and quantum chaotic dynamics. Since then, random polynomials and random series are also studied intensively by researchers from analysis, probability, and mathematical physics.

In these studies, the complex roots become also important. One views the set of all roots as a random point process, and a fundamental problem, among others, is to understand the correlation between nearby points in a small region. (This correlation is often use to model the interaction between particles in certain physics problems.)

In this talk, I will try to survey this fascinating area, presenting some of its main questions, results, and ideas. The talk is self-contained, and requires only basic knowledge in analysis and probability.

5. GS. Michael Vogelius – Đại học Rutgers, Mỹ

Tiêu đề: Đang cập nhật.

Bản tóm tắt: Đang cập nhật.