I will talk about the relation between the classical models of phase transitions with some variational functionals. First I will cover the classical work of Modica-Mortola model on phase transitions (that analizes the interface in between); and then the generalization by Alberti-Bouchitte-Seppecher, that includes the interaction with the wall of the container. Both can be formulated as a minimization problem for suitable functionals.

In this talk, we will focus on homogenization problem for linear elliptic operators. We will use asymptotic expansions in order to derive the homogenization formula, and then prove a convergence theorem for the solution.

I will describe the explicit formulas for the projections of a loop algebra splitting and discuss some consequences and applications. This is a thesis defense.

We provide a sketch of the proof of one half of the (simple) Birch and Swinnerton-Dyer conjecture for elliptic curves over function fields. This proof has a very geometric flavor, resting on the cohomology and arithmetic of the associated elliptic surface. We attempt to keep the geometry at a minimum but must discuss the N\'eron-Severi group of a variety and prove it is finitely generated in the case of elliptic surfaces. We will also discuss the famous Weil conjectures as well as touching upon -adic cohomology. A little bit of hand-waving will be needed, but we hope to demonstrate the power of the underlying geometry inherent in function fields.

Backward stochastic differential equations (BSDE) is a system of stochastic equations with both initial and terminal data. These equations appear naturally in optimal stochastic control and in mathematical finance and it has been studied extensively in the last decade. There is also an interesting conncetion with nonlinear, parabolic equations through a simple use of stochastic calculus. In this talk, I will outline this connection and then use it to develop a Monte-Carlo type numerical scheme for the nonlinear PDE's.

This is a Ph.D thesis defense.

A way of doing '2-dimensional continued fractions' for a 1) ray in with positive rationally independent coordinates, and 2) a metric in the normal subspace, $L^*,$ leads to a finite 2dimensional complex; the 2-cells are labeled with a set of integral coordinates $z \in Z^3.$ A subset of with either 3 or 4 elements has as a positive span. This implies that is the span of $B.$ In fact, has four elements only in the case of a 'cross': $\{a,b,c,d\}$ satisfy the equation $a+d=b+c.$ It follows that $a,b,c$ forms a basis of $Z^3.$ It is an definitive fact that these 'crosses' occur only if is very near a rational subspace of dimension 2, e.g. with basis $a,b.$

Today I will talk about the derived category of projective space and use Beilenson's resolution of the identity in order to explicitly write down generators. This is an extremely nice situation where we have an explicit way writing the derived category of a variety.

In this lecture, I will present some results in the Theory of Regularizing Free This type of problems appears as model in Combustion Theory to describe laminar flames as an asymptotic limit for high energy activation. We use the Least supersolution approach to construct regularized solution such that its level surfaces satisfies some geometric properties. This allow us to prove that free boundary of the limit function has the right weak geometry, in the geometric measure theoretical sense. By the construction of some barriers with uniformly curved free boundaries, the classification of global solutions are achieved, and the free boundary condition is proven. Finally, the limit function is proven to be a viscosity solution of some elliptic free boundary problem and the free boundary is proven to be a C^{1,\alpha} surface around almost every point with respect to the (n-1) dimensional Hausdorff Measure. This lecture is a Thesis Defense.

"Weigted inequalities for the Fourier transform provide a natural measure to characterize uncertaniy. On R^n the question is to determine sharp quantitative comparisons between the relative size of a function and it s Fourier transfrom at infinity. The purpose of this talk is to formulate the uncertanity principle by using logarithmic estimates obtained from a sharp form of Pitt's inequality" This is an optional prelim talk.

In incomplete financial markets not every contingent claim can be replicated by a self-financing strategy. The risk of the resulting shortfall can be measured by convex risk measures. We consider the dynamic optimization problem of finding a self-financing strategy that minimizes the convex risk of the shortfall. This problem can be split into a static optimization problem and a representation problem. We show that the optimal strategy consists in superhedging a modified claim $\tilde{\varphi}H$, where $H$ is the payoff of the claim and $\tilde{\varphi}$ is the solution of the static optimization problem, the optimal randomized test. In this talk, we provide necessary and sufficient optimality conditions for the static problem using convex duality methods. The optimal randomized test will have a typical $0-1$-structure.

We study smooth families of diffeomorphisms that have an invariant manifold arising from a perturbation of a saddle connection. We develop a general method to detect the transverse intersection of stable and unstable manifolds. We relate this intersection to the set of zeros of a canonical Melnikov displacement that is defied to be a section of the normal bundle of the saddle connection. The geometric content of Melnikov's method is exploited in order to show that a generalized Melnikov function can be written as a canonical formula. In addition, we show how our definition reproduces the classical methods of Poincar'eand Melnikov, and specializes to the methods previously used for exact-symplectic and volume-preserving maps.

For centuries, many mystics and some carpenters believed the value of pi to be 3. In today's modern dog-eat-dog world, we know that this is generally false. In fact, some might even go so far to say that pi is never 3 and perhaps also never rational. Come for the food but stay for the awkwardly communicated mathematics in a talk that promises to transcend the rational and even the algebraic. I will begin with a general overview of algebraic/transcendental extensions and then launch into Lindemann's proof of the transcendence of pi.

In this talk, we will report recent advances on the study of fully nonlinear obstacle problems in infinite dimensional Hilbert spaces. We will discuss the appropriate set-up of the problem and will show existence, uniqueness and optimal regularity of solutions. Some questions regarding geometric measure properties of the free boundary shall be raised. This is a joint work with A. Swiech

The generalized Waldhausen Conjecture asserts that a closed orientable irreducible atoroidal 3-manifold has only finitely many Heegaard splittings of any given genus up to isotopy. Tao Li has given a proof of the conjecture, using the theory of measured laminations and branched surfaces. We'll be looking at the main ideas of his argument, including the requisite background material.