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 ONLINE https://gatech.zoom.us/j/96285037913
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.

The talk will be online at https://gatech.zoom.us/j/96285037913

Long-time dynamics of the sine-Gordon equation

Series
Math Physics Seminar
Time
Thursday, March 2, 2023 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Gong ChenSchool of Mathematics, Georgia Tech

 I will discuss the soliton resolution and asymptotic stability problems for the sine-Gordon equation. It is known that the obstruction to the asymptotic stability for the sine-Gordon equation in the energy space is the existence of small breathers which is also closely related to the emergence of wobbling kinks. Our stability analysis gives a criterion for the weight which is sharp up to the endpoint so that the asymptotic stability holds. This is joint work with Jiaqi Liu and Bingying Lu.

Stability for Sobolev and Log-Sobolev inequalities

Series
Math Physics Seminar
Time
Thursday, February 16, 2023 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael LossSoM Georgia Tech

I discuss a sharp quantitative stability result for the Sobolev inequality with explicit constants. Moreover, the constants have the optimal behavior in the limit of large dimensions, which allows us to deduce an optimal quantitative stability estimate for the Gaussian log-Sobolev inequality with an explicit dimension-free constant.

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, and online zoom link: Meeting ID: 961 2577 3408
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.

The seminar can also be accessed online via zoom link: Meeting ID: 961 2577 3408

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

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