## 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 its spectrum. I will give an example of how...

Read More## 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 closed formulas for these invariants in dimension four....

Read More## 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 at exactly the curve c(q) at a finite small time....

Read More## 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.

Read More## Periodic Homogenization for Nonlocal Hamilton-Jacobi Equations at a Critical Diffusive Regime.

Abstract. In this talk, I will report periodic homogenization results for integro-differential Hamilton-Jacobi equations in which the nonlocal elliptic operator have “order” one. This makes the nonlocal operator to have the same scaling property of the gradient, which can be understood as a critical regime among the diffusion and the transport terms. This is a joint work with Adina Ciomaga (Paris Diderot University, Paris) and Daria Ghilli (Karl-Franzens-Universität Graz, Austria).

Read More## Lp-estimates for nonlocal in time heat kernels

Abstract: In [1] and [2], the authors study independently the so called fully nonlocal diffusion equation, which is of fractional order both in space and time. In both papers, the authors need several technical results about the so-called Mittag-Leffler functions and Fox H-functions to obtain Lp-estimates of the solutions. This approach seems not to be very helpful (or easy) to derive the Lp-estimates of solutions to equations with other nonlocal in time operators (for example sums of fractional derivatives). In this talk, we present a method to obtain Lp-estimates of fundamental solutions...

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