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Series: SIAM Student Seminar

We discuss results from classical weighted theory and give a characterization of the two-weight inequality for a simple vector-valued operator.

Series: SIAM Student Seminar

In 1956 Mark Kac published his paper about the Foundation of Kinetic Theory in which he gave a mathematical, probabilistic description of a system of N particles colliding randomly. An interesting result that was found, though not causing any surprise, was the convergence to the stable equilibrium state. The question of the rate of the L2 convergence interested Kac and he conjectured that the spectral gap governing the convergence is uniformly bounded form below as N goes to infinity. While this was proved to be true, and even computed exactly, many situations show that the time scale of the convergence for very natural cases is proportional to N, while we would hope for an exponential decay. A different approach was considered, dealing with a more natural quantity, the entropy. In recent paper some advancement were made about evaluating the rate of change, and in 2003 Villani conjectured that the corresponding 'spectral gap', called the entropy production, is of order of 1/N. In our lecture we'll review the above topics and briefly discuss recently found results showing that the conjecture is essentially true.

Series: SIAM Student Seminar

In this talk I will outline a topic that has been of interest due to its
applicability in physics and engineering. The so called small noise model is a very
technical subject that lies in the center of probability theory and usually study
thorough a large deviations approach. I will explain this terminology and why is the
correlation with dynamical systems so strong. Recent developments will be given at
the end if time allows.

Series: SIAM Student Seminar

We will discuss about the paper "An efficient algorithm for large-scale detection of protein families" by A Enright, S Van Dongen and C Ouzounis

Series: SIAM Student Seminar

A copula C of n arbitrary random variables X_1, ..., X_n contains all
the information about their dependence. First I will briefly introduce
the definition, basic properties and elementary examples of copulas, as
well as Sklar's Theorem (1959). Then I will present a family of
multivariate copulas whose marginal copula belongs to a family of
extreme copulas. Finally I will discuss a minimization problem related
to copula, which is still open. The talk should be easy to understand
for all level audience who have knowledge of basic probability theory

Series: SIAM Student Seminar

We will start with a brief introduction to the broad area of
machine learning, with the focus on empirical risk minimization
methods and their connection to the theory of empirical processes.
Using some results from our recent work with V. Koltchinskii, I
will explain how sparsity affects the risk bounds.

Series: SIAM Student Seminar

Last semester, I reviewed the relation between dynamical system,
Fokker-Planck equation and thermodynamics (free energy and Gibbs
distribution). This time let's go further. I will review the geometric
properties of a kind of dissipative evolution equations. I will explain
why this kind of evolutionary equations (Fokker-Planck equation,
nonlinear Fokker-Planck equation, Porous medium equation) are the
gradient flow of some energy function on a Riemannian manifold --
2-Wasserstein metric space.

Series: SIAM Student Seminar

This will be an introductory talk about Hardy inequalities. These inequalities are solutions to optimization problems, and their results are well-known. I will survey these results, and discuss some of the techniques used to solve these problems. The applications of Hardy inequalities are broad, from PDE's and mathematical physics to brownian motion. This talk will also serve as a lead-in to my talk at the Analysis seminar next Wednesday in which I discuss some current results that Michael Loss and I have obtained.

Series: SIAM Student Seminar

This is an introductory talk to everyone who wants to learn skills in Latex. We will discuss including and positioning graphics and the beamer document class for presentations. A list of other interesting topics will be covered if time permits.

Series: SIAM Student Seminar

We are dealing with the following minimization problem: inf {I(\mu): \mu
is a probability measure on R and \int f(x)=t_{0}}, where I(\mu) = \int
(x^2)/2 \mu(dx) + \int\int log|x-y|^{-1} \mu(dx)\mu(dy), f(x) is a bounded
continuous function and t is a given real number. Its motivation and its connection to radom matrices theory will be introduced. We will show that the solution is unique and has a compact support. The possible extension of the
class of f(x) will be discussed.