Seminars and Colloquia by Series

Upper bounds on quantum dynamics

Series
Math Physics Seminar
Time
Thursday, March 9, 2023 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles Room 005
Speaker
Mira ShamisQueen Mary University of London

We shall discuss the quantum dynamics associated with ergodic
Schroedinger operators with singular continuous spectrum. Upper bounds
on the transport moments have been obtained for several classes of
one-dimensional operators, particularly, by Damanik--Tcheremchantsev,
Jitomirskaya--Liu, Jitomirskaya--Powell. We shall present a new method
which allows to recover most of the previous results and also to
obtain new results in one and higher dimensions. The input required to
apply the method is a large-deviation estimate on the Green function
at a single energy. Based on joint work with S. Sodin.

Synchronization and averaging in a simple dynamical systems with fast/slow variables

Series
Math Physics Seminar
Time
Thursday, February 9, 2023 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles Room 005
Speaker
Federico BonettoSchool of Mathematics, Georgia Tech

 We study a family of dynamical systems obtained by coupling a chaotic (Anosov) map on the two-dimensional torus -- the chaotic variable -- with the identity map on the one-dimensional torus -- the neutral variable -- through a dissipative interaction. We show that the  two systems synchronize, in the sense that the trajectories evolve toward an attracting invariant manifold, and that the full dynamics is conjugated to its linearization around the invariant manifold. When the interaction is small, the evolution of the neutral variable is very close to the identity and hence the neutral variable appears as a slow variable with respect to the fast chaotic variable: we show that, seen on a suitably long time scale, the slow variable effectively follows the solution of a deterministic differential equation obtained by averaging over the fast  variable.

Sets of non-Lyapunov behaviour for transfer matrices of Schroedinger operators

Series
Math Physics Seminar
Time
Thursday, February 2, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
ONLINE and Skiles room 005
Speaker
Sasha SodinQueen Mary University of London

We shall discuss the asymptotics of singular values of the transfer matrices of ergodic Schroedinger and block-Schroedinger  operators. At a fixed value of the spectral parameter, the logarithmic asymptotics is almost surely given by the Lyapunov exponents; however, this is not, in general, true simultaneously for all the values of the parameter.  We shall try to explain the importance of these sets in various problems of spectral theory, and then review some of the earlier works on the subject and present some new results. Based on joint work with I. Goldsheid.

This talk will be online.  Meeting ID: 919 5236 6315.  Pleas note the unusual time!

Field theory of spatiotemporal chaos

Series
Math Physics Seminar
Time
Thursday, January 26, 2023 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles Room 005
Speaker
Predrag CvitanovićSchool of Physics, Georgia Tech

Gutzwiller semi-classical quantization, Boven-Sinai-Ruelle dynamical zeta functions for chaotic dynamical systems, statistical mechanics partition functions, and path integrals of quantum field theory are often presented in ways that make them appear as disjoint, unrelated theories. However, recent advances in describing fluid turbulence by its dynamical, deterministic Navier-Stokes underpinning, without any statistical assumptions, have led to a common field-theoretic description of both (low-dimension) chaotic dynamical systems, and (infinite-dimension) spatiotemporally turbulent flows. 

I have described the remarkable experimental progress connecting turbulence to deterministic dynamics in the Sept 24, 2023 colloquium (the recoding is available on the website below). In this seminar I will use a lattice discretized field theory in 1 and 1+1 dimension to explain how temporal `chaos', `spatiotemporal chaos' and `quantum chaos' are profitably cast into the same field-theoretic framework.

https://ChaosBook.org/overheads/spatiotemporal/

The talk will also be on Zoom:   GaTech.zoom.us/j/95338851370

Continuity of the Lyapunov exponent for analytic multi-frequency quasi-periodic cocycles

Series
Math Physics Seminar
Time
Thursday, January 12, 2023 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles Room 005
Speaker
Matthew PowellSchool of Mathematics, Georgia Tech

The purpose of this talk is to discuss our recent work on multi-frequency quasi-periodic cocycles, establishing continuity (both in cocycle and jointly in cocycle and frequency) of the Lyapunov exponent for non-identically singular cocycles. Analogous results for one-frequency cocycles have been known for over a decade, but the multi-frequency results have been limited to either Diophantine frequencies (continuity in cocycle) or SL(2,C) cocycles (joint continuity). We will discuss the main points of our argument, which extends earlier work of Bourgain.

Quantum mechanics and diffusion on metric graphs

Series
Math Physics Seminar
Time
Thursday, December 1, 2022 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles Room 005
Speaker
Evans HarrellSchool of Mathematics, Georgia Tech

Quantum mechanics and diffusion on a network, in the sense of a metric graph, are locally one-dimensional, but the way the graph is connected can add multidimensional features and some strange phenomena.  Quantum graphs have been an active area of research since the 1990s.  I’ll review the subject and share some ideas about analyzing Schrödinger and heat equations on metric graphs, through the associated eigenvalue problem and the heat kernel.

This talk is based on a 2022 article with David Borthwick and Kenny Jones, and on work in progress with David Borthwick, Anna Maltsev, and Haozhe Yu. 

New bounds on the excess charge for bosonic systems interacting through Coulomb potentials

Series
Math Physics Seminar
Time
Thursday, November 17, 2022 - 16:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Rafael BenguriaCatholic University of Chile

In this talk, using a technique introduced by P.~T.~Nam in 2012 and the Coulomb Uncertainty Principle, I will present the proof of new bounds on the excess charge for non relativistic  atomic systems, independent of the particle statistics. These new bounds are the best bounds to date for bosonic systems. This is joint work with Juan Manel González and Trinidad Tubino.

Join Zoom Meeting: https://gatech.zoom.us/j/94786316294

A Keller-Lieb-Thirring Inequality for Dirac operators.

Series
Math Physics Seminar
Time
Thursday, November 10, 2022 - 16:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Hanne Van Den BoschUniversity of Chile

The classical Keller-Lieb-Thirring inequality bounds the ground state energy of a Schrödinger operator by a Lebesgue norm of the potential. This problem can be rewritten as a minimization problem for the Rayleigh quotient over both the eigenfunction and the potential. It is then straightforward to see that the best potential is a power of the eigenfunction, and the optimal eigenfunction satisfies a nonlinear Schrödinger equation. 

This talk concerns the analogous question for the smallest eigenvalue in the gap of a massive Dirac operator. This eigenvalue is not characterized by a minimization problem. By using a suitable Birman-Schwinger operator, we show that for sufficiently small potentials in Lebesgue spaces, an optimal potential  and eigenfunction exists. Moreover, the corresponding eigenfunction solves a nonlinear Dirac equation.

This is joint work with Jean Dolbeaults, David Gontier and Fabio Pizzichillo

Join Zoom Meeting:  https://gatech.zoom.us/j/91396672718

On the emergence of a quantum Boltzmann equation near a Bose-Einstein condensate

Series
Math Physics Seminar
Time
Thursday, November 3, 2022 - 16:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Thomas ChenUniversity of Texas, Austin

The mathematically rigorous derivation of nonlinear Boltzmann equations from first principles in interacting physical systems is an extremely active research area in Analysis, Mathematical Physics, and Applied Mathematics. In classical physical systems, rigorous results of this type have been obtained for some models. In the quantum case on the other hand, the problem has essentially remained open. In this talk, I will explain how a cubic quantum Boltzmann equation arises within the fluctuation dynamics around a Bose-Einstein condensate, within the quantum field theoretic description of an interacting Boson gas. This is based on joint work with Michael Hott.

Join Zoom Meeting at https://gatech.zoom.us/j/92873362365

Persistence of periodic orbits in functional perturbations of an ODE

Series
Math Physics Seminar
Time
Thursday, October 27, 2022 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles Room 005
Speaker
Joan GimenoUniversitat de Barcelona & Georgia Institute of Technology

With very minor assumptions, I show that periodic orbits in
an ODE can persist under (singular) perturbations of including a delay
term.  These perturbations change the phase space from finite to
infinite dimensions. The results apply to electrodynamics and give new
approaches to handle state-dependent, small, nested, and distributed
delays.

I will discuss and explain some motivations, the new methods, sketches
of the proofs, and possible applications. I will end the talk giving
some ideas of work in progress and possible future works.

Pages