Monday, February 27, 2012 - 12:00 , Location: Skiles 006 , Gregory Berkolaiko , Texas A&M Univ. , email@example.com , Organizer:
Zeros of vibrational modes have been fascinating physicists for several centuries. Mathematical study of zeros of eigenfunctions goes back at least to Sturm, who showed that, in dimension d=1, the n-th eigenfunction has n-1 zeros. Courant showed that in higher dimensions only half of this is true, namely zero curves of the n-th eigenfunction of the Laplace operator on a compact domain partition the domain into at most n parts (which are called "nodal domains"). It recently transpired that the difference between this "natural" number n of nodal domains and the actual values can be interpreted as an index of instability of a certain energy functional with respect to suitably chosen perturbations. We will discuss two examples of this phenomenon: (1) stability of the nodal partitions of a domain in R^d with respect to a perturbation of the partition boundaries and (2) stability of a graph eigenvalue with respect to a perturbation by magnetic field. In both cases, the "nodal defect" of the eigenfunction coincides with the Morse index of the energy functional at the corresponding critical point. Based on preprints arXiv:1107.3489 (joint with P.Kuchment and U.Smilansky) and arXiv:1110.5373
Monday, February 6, 2012 - 12:05 , Location: Skiles 006 , Rafael Tiedra de Aldecoa , Catholic University of Chile , Organizer: Michael Loss
We present an improved version of commutator methods for unitary operators under a weak regularity condition. Once applied to a unitary operator, the method typically leads to the absence of singularly continuous spectrum and to the local finiteness of point spectrum. Some applications for Floquet operators and for cocycles over irrational rotations will be presented.
Monday, January 30, 2012 - 12:05 , Location: Skiles 006 , Michael Loss , School of Mathematics, Georgia Tech , Organizer: Michael Loss
This talk is concerned with new symmetry results for the extremals of the Caffarelli-Kohn-Nirenberg inequalities in a range of parameters for which no explicit results of symmetry have previously been known. The method proceeds via spectral estimates. This is joint work with Jean Dolbeault and Maria Esteban.
Monday, January 23, 2012 - 12:05 , Location: Skiles 006 , Daniel Blazevski , University of Texas , Organizer: Michael Loss
I will discuss local and nonlocal anisotropic heat transport along magnetic field lines in a tokamak, a device used to confine plasma undergoing fusion. I will give computational results that relate certain dynamical features of the magnetic field, e.g. resonance islands, chaotic regions, transport barriers, etc. to the asymptotic temperature profiles for heat transport along the magnetic field lines.
Tuesday, December 6, 2011 - 11:05 , Location: Skiles 005 , Dr. Lilian Wong , SoM, Georgia Tech , Organizer: Michael Loss
We offer several perspectives on the behavior at infinity of solutions of discrete Schroedinger equations. First we study pairs of discrete Schroedinger equations whose potential functions differ by a quantity that can be considered small in a suitable sense as the index n \rightarrow \infty. With simple assumptions on the growth rate of the solutions of the original system, we show that the perturbed system has a fundamental set of solutions with the same behavior at infinity, employing a variation-of-constants scheme to produce a convergent iteration for the solutions of the second equation in terms of those of the original one. We use the relations between the solution sets to derive exponential dichotomy of solutions and elucidate the structure of transfer matrices.Later, we present a sharp discrete analogue of the Liouville-Green (WKB) transformation, making it possible to derive exponential behavior at infinity of a single difference equation, by explicitly constructing a comparison equation to which our perturbation results apply. In addition, we point out an exact relationship connecting the diagonal part of the Green matrix to the asymptotic behavior of solutions. With both of these tools it is possible to identify an Agmon metric, in terms of which, in some situations, any decreasing solution must decrease exponentially. This is joint work with Evans Harrell.
Tuesday, November 1, 2011 - 11:05 , Location: Skiles 005 , Rupert Frank , Dept. of Math, Princeton University , Organizer: Michael Loss
We describe the first rigorous derivation of the celebrated Ginzburg-Landau (GL) theory, starting from the microscopic Bardeen-Cooper-Schrieffer (BCS) model. Close to the critical temperature, GL arises as an effective theory on the macroscopic scale. The relevant scaling limit is semiclassical in nature, and semiclassical analysis, with minimal regularity assumptions, plays an important part in our proof. The talk is based on joint work with C. Hainzl, R. Seiringer and J. P. Solovej.
Tuesday, October 25, 2011 - 11:05 , Location: Skiles 005 , Rafael De la Llave , SoM Georgia Tech , Organizer: Michael Loss
We consider several models from solid state Physics and consider the problem offinding quasi-periodic solutions. We present a KAM theorem that showsthat given an approximate solution with good condition numbers, onecan find a true solution close by. The method of proof leads tovery efficient algorithms. Also it provides a criterion for breakdown.We will present the proof, the algorithms and some conjectures obtainedby computing in some cases. Much of the work was done with R. Calleja and X. Su.
Wednesday, September 28, 2011 - 15:00 , Location: Marcus Nanotech Conference , Henriette Elvang , Physics Department, University of Michigan , Organizer: Stavros Garoufalidis
Hosted by Predrag Cvitanovic, School of Physics
Particle scattering processes at experiments such as the Large Hadron Collider at CERN are described by scattering amplitudes. In quantum field theory classes, students learn to calculate amplitudes using Feynman diagram methods. This is a wonderful method for a process like electron + positron -> muon^- + muon^+, but it is a highly challenging for a process like gluon+gluon -> 5 gluons, which requires 149 diagrams even at the leading order in perturbation theory. It turns out, however, that the result for such gluon scattering processes is remarkably simple, in some cases it is just a single term! This has lead to new methods for calculating scattering amplitudes, and it has revealed that amplitudes have a surprisingly rich mathematical structure. The applications of these new methods range from calculation of processes relevant for LHC physics to theoretical explorations of quantum gravity. I will give a pedagogical introduction to these new approaches to scattering theory and their applications, not assuming any prior knowledge of quantum field theory or Feynman rules.
Wednesday, July 13, 2011 - 14:05 , Location: Skiles 005 , Jimmy Lamboley , Dauphine , firstname.lastname@example.org , Organizer:
Shape optimization is the study of optimization problems whose unknown is a domain in R^d. The seminar is focused on the understanding of the case where admissible shapes are required to be convex. Such problems arises in various field of applied mathematics, but also in open questions of pure mathematics. We propose an analytical study of the problem. In the case of 2-dimensional shapes, we show some results for a large class of functionals, involving geometric functionals, as well as energies involving PDE. In particular, we give some conditions so that solutions are polygons. We also give results in higher dimension, concerned with the Mahler conjecture in convex geometry and the Polya-Szego conjecture in potential theory. We particularly make the link with the so-called Brunn-Minkowsky inequalities.
Wednesday, May 4, 2011 - 16:30 , Location: Skiles 006 , Maria Westdickenberg , School of Math, Georgia Tech , Organizer: Michael Loss
The logarithmic Sobolev inequality (LSI) is a powerful tool for studying convergence to equilibrium in spin systems. The Bakry-Emery criterion implies LSI in the case of a convex Hamiltonian. What can be said in the nonconvex case? We present two recent sufficient conditions for LSI. The first is a Bakry-Emery-type criterion that requires only LSI (not convexity) for the single-site conditional measures. The second is a two-scale condition: An LSI on the microscopic scale (conditional measures) and an LSI on the macroscopic scale (marginal measure) are combined to prove a global LSI. We extend the two-scale method to derive an abstract theorem for convergence to the hydrodynamic limit which we then apply to the example of Guo-Papanicolaou-Varadhan. We also survey some new results.This work is joint with Grunewald, Otto, and Villani.