# University of Virginia Mathematical Physics Seminar 2014-15

 August 27 No meeting September 3 Organizational Meeting September 10 Speaker: John Imbrie Title: What is Many-Body Localization? Abstract: I will explain the nature of the many-body localized phase, in the context of a one-dimensional quantum spin chain. September 17 Speaker: John Imbrie Title: On Many-Body Localization for Quantum Spin Chains Abstract: I will describe a proof of many-body localization, under a physically reasonable assumption that limits the amount of level attraction in the system. The construction uses a sequence of local unitary transformations to diagonalize the Hamiltonian and connect the exact many-body eigenfunctions to the original basis vectors. September 24 No meeting October 1 Speaker: Rajinder Mavi Title: Localization in the non-monotonic Anderson model Abstract: Recently John Imbrie developed a method to demonstrate localization in many body systems. We apply a similar method to a tight binding multichannel alloy model, where the potentials may depend analytically on the random parameters, for example these models can be realized as tight binding model in Z^D with dilute randomness. Such models are only partially understood by standard methods, though it is expected that they behave much the same as the standard (monotonic and single channel) Anderson model. The goal is to obtain representations of the diagonalizing basis in the case of large disorder, we achieve this by multiscale renormalizations by Rayleigh - Schrodinger approximations. We state localization results which can be derived from the derivation. This talk covers joint work with John Imbrie. October 8 Speaker: Rajinder Mavi Title: Localization in the non-monotonic Anderson model Abstract: Continuation of the previous week’s talk October 15 No meeting. October 22 Speaker: Rajinder Mavi Title: Localization in the non-monotonic Anderson model Abstract: Continuation of the previous week’s talk October 29 Speaker: Abdelmalek Abdesselam Title: Counting maps with tridiagonal random matrices Abstract: H. F. Trotter discovered that, as far as spectral properties are concerned, GUE random matrices can be modeled using tridiagonal matrices. I will explain ongoing work about using this idea to count random maps with fixed vertex degree distribution. In particular one can recover, in straightforward way the connection to trees and labeled mobiles that combinatorialists have discovered using very ingenious bijections. This is joint work with G. W. Anderson, O. Bernardi and A. Miller. November 5 No meeting. November 12 Speaker: Abdelmalek Abdesselam Title: Counting maps with tridiagonal random matrices Abstract: Continuation of the talk of Oct 29 November 19 Speaker: Steve Sontz, CIMAT, Mexico Title: Dunkl Operators as Covariant Derivatives in a Quantum Principal Bundle Abstract: A quantum principal bundle is constructed for every Coxeter group acting on a finite-dimensional Euclidean space E, and then a connection is also defined on this bundle. The covariant derivatives associated to this connection are the Dunkl operators, originally introduced as part of a program to generalize harmonic analysis in Euclidean spaces. This gives us a new, geometric way of viewing the Dunkl operators. In particular, we present a new proof of the commutativity of these operators among themselves as a consequence of a geometric property, namely, that the connection has curvature zero. November 26 THANKSGIVING RECESS December 3 Speaker: Gonzalo Bley Title: The Energy of Multi-Polarons Abstract: A polaron is an electron plus the polarization field it generates when it is embedded in a polar crystal, such as common salt. The polaron has been studied extensively for several decades, starting with the pioneering work of Herbert Frohlich, passing through the results derived by T.D. Lee, Richard Feynmann, and others. In this talk I will briefly summarize the story behind the theory of polarons, and will discuss several results concerning the ground-state energy of multipolarons - the case where several electrons are put into a crystal - first obtained by Frank, Lieb, Seiringer, and Thomas, followed by extensions and improvements by Rafael Benguria and myself. I will finalize by sketching recent results obtained by Benguria, Frank and Lieb concerning the non-stability of multipolarons in the context of a semi-classical approximation, known as the Pekar multi-polaron. December 10 Speaker: Title: Abstract: December 17 SEMESTER BREAK January 14 Speaker: Title: Abstract: January 21 Speaker: Title: Abstract: January 28 Speaker: Title: Abstract: February 4 Speaker: Title: Abstract: February 11 Speaker: Title: Abstract: February 18 Speaker: Title: Abstract: February 25 Speaker: Bourama Toni, Virginia State University Title: Quasi Limit Cycles and Applications Abstract: We develop the concept of Quasi Limit Cycles at the interface between the theories of limit cycles and quasi periodicity to include almost and pseudo-almost periodicity, almost automorphy, and p-Adic almost periodicity, in both the usual Archimedean metric spaces and the p-Adic (ultrametric) spaces. We investigate the conditions of existence and uniqueness, stability and bifurcation, as well as the notion of quasi-isochrons. Illustrative examples include perturbations of the Poincaré oscillator and the Liénard systems under various external forcing. Also presented is on-going research on the co-existence of limit cycles and strictly quasi limit cycles, the existence of quasi isochronous limit cycles, and most importantly the transition to chaotic behavior, possibly through coupling/synchronization of quasi oscillators. March 4 Speaker: Title: Abstract: March 11 SPRING RECESS March 18 Speaker: Title: Abstract: March 25 Speaker: Title: Abstract: April 1 Speaker: Title: Abstract: April 8 Speaker: Title: Abstract: April 15 Speaker: Title: Abstract: April 22 Speaker: Larry Thomas Title: Stochastic integral estimates in quantum mechanics Abstract: Abstract: Stochastic integrals can be used to estimate certain action functionals appearing in elementary quantum mechanics and simple quantum field theories. I will review how these action functionals arise, and illustrate in simple examples how the estimates work. April 29 Speaker: Gonzalo Bley Title: A Functional Integral Inequality with Applications in Non-Relativistic Quantum Field Theory Abstract: Here I present an inequality involving functional integrals that provides rigorous lower bounds to the ground state energy of certain Hamiltonians that arise in non-relativistic quantum field theory. In this talk I will focus on the one-particle, massless Nelson model. After performing a renormalization at the level of path-integrals, I will arrive at a lower bound to the energy of the renormalized Nelson Hamiltonian, and will show explicitly how the large coupling behavior of the ground state energy is at most alpha to the power of four plus epsilon, for any positive epsilon, when a constant square root of alpha is put in front of the particle-field interaction term. This is joint work with Larry Thomas.

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