Seminars appear in decreasing order in relation to date. To find an activity of your interest just go down on the list. Normally seminars are given in english. If not, they will be marked as **Spanish Only.**

## Operator algebras, compact operators, and the essential spectrum of the N-body problems

Abstract: I will begin by reviewing a general method to determine the essential spectrum of Schrodinger-type operators. The method is based first on the fact that an operator is Fredholm if, and only if, it is inversible modulo the compacts (Atkinson’s theorem). This reduces the study of certain quotients by the compact operators. To study the invertibility in these quotients, one uses, following Georgescu, Mantoiu, and others, a determination of the spectrum of a suitable operator algebra and of the action of the translation group on...

## Sensitive dependence of geometric Gibbs measures at positive temperature

ABSTRACT: In this talk we give the main ideas of the construction of the first example of a smooth family of real and complex maps having sensitive dependence of geometric Gibbs states at positive temperature. This family consists of quadratic-like maps that are non-uniformly hyperbolic in a strong sense. We show that for a dense set of maps in the family the geometric Gibbs states diverge at positive temperature. These are the first examples of divergence at positive temperature in statistical mechanics or the thermodynamic...

## The singular Yamabe problem and a fully nonlinear generalization & Vortex desingularization for the 2D Euler equations

(16:00 hrs.) Title: The singular Yamabe problem and a fully nonlinear generalization Abstract: I will begin with an overview of the work of Loewner-Nirenberg on constructing complete conformal metrics of constant negative scalar curvature on domains in Euclidean space, and its extension to Riemannian manifolds with boundary. I will then describe some fully nonlinear generalizations. Finally, I will discuss a certain geometric invariant of solutions, called the renormalized volume, and some recent work with Robin Graham on computing...

## Accuracy of the Time-Dependent Hartree-Fock Approximation.

Abstract: We study the time evolution of a system of N spinless fermions in which interact through a pair potential, e.g., the Coulomb potential. We compare the dynamics given by the solution to Schrödinger’s equation with the time-dependent Hartree–Fock approximation, and we give an estimate for the accuracy of this approximation in terms of the kinetic energy of the system. This leads, in turn, to bounds in terms of the initial total energy of the system.

## Singularity formation for the harmonic map flow from a volume into S^2.

Abstract: Consider a volume V ⊂ R3 generated by rotating around the Z axis a bounded smooth domain Ω ⊂ R2 that lives in the XZ plane. We construct a finite time blow-up solution to the harmonic map flow from volume V into the sphere S2, the problem is vt = ∆v+|∇v|2vinV×(0,T) v = v∂V in∂V×(0,T) v(·,0) = v0inV, wherev:V ×[0,T)→S2,v0 :V →S2 issmoothandv∂V = v0|∂V :∂V →S2. Given a point q ∈ Ω we define the circumference c(q) generated by the rotation of q around the Z axis. We find initial and boundary data so that the solution v blows up...

## Analogies between the geodesic flow on a negatively curved manifold and countable Markov shifts

ABSTRACT: By the work of Bowen and Ratner it is known that the geodesic flow on a compact negatively curved manifold can be modeled as a suspension flow over a subshift of finite type. Unfortunately, a symbolic representation is not available if the manifold is non-compact. In this talk I will briefly explain some recent developments on the study of the thermodynamic formalism of the geodesic flow on non-compact negatively curved manifolds. Surprisingly some of the methods used to understand the geodesic flow have consequences to the theory...

## Morse theory for the action functional and a Poincare-Birkhoff theorem for flows

ABSTRACT: The goal of this talk is twofold. Firstly I would like to explain how pseudo-holomorphic curves can be used to study Morse theory of the action functional from classical mechanics. Then I will move to applications, focusing on a generalization of the Poincare-Birkhoff theorem for Reeb flows on the three-sphere.

## Norm and pointwise averages of multiple ergodic averages and applications

ABSTRACT: Via the study of multiple ergodic averages for a single transformation, Furstenberg, in 1977, was able to provide an ergodic theoretical proof of Szemerédi’s theorem, i.e., every subset of natural numbers of positive upper density contains arbitrarily long arithmetic progressions. We will present some recent developments in the area for more general averages, e.g., for multiple commuting transformations with iterates along specific classes of integer valued sequences. We will also get numerous applications of the...