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Series: Math Physics Seminar

Note nonstandard day and time.

Consider an N by N matrix X of complex entries with iid real and imaginary parts
with probability distribution h where h has Gaussian decay. We show that the local density of
eigenvalues of X converges to the circular law with probability 1. More precisely, if we let a
function f (z) have compact support in C and f_{\delta,z_0} (x) = f ( z-z^0 / \delta ) then the sequence of densities
(1/N\delta^2) \int f_\delta d\mu_N
converges to the circular law density (1/N\delta^2) \int f_\delta d\mu
with probability 1. Here we show
this convergence for \delta = N^{-1/8}, which is an improvement on the previously known results
with \delta = 1. As a corollary, we also deduce that for square covariance matrices the number of
eigenvalues in intervals of size in the intervals [a/N^2 , b/N^2] is smaller than log N with probability
tending to 1.

Series: Math Physics Seminar

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

Series: Math Physics Seminar

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.

Series: Math Physics Seminar

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.

Series: Math Physics Seminar

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.

Series: Math Physics Seminar

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.

Series: Math Physics Seminar

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.

Series: Math Physics Seminar

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.

Series: Math Physics Seminar

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.

Series: Math Physics Seminar

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.