Seminars and Colloquia by Series

Resonant tori of arbitrary codimension for quasi-periodically forced systems

Series
Math Physics Seminar
Time
Thursday, March 5, 2020 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Guido Gentile – Universita' di Roma 3 –

Consider a system of rotators subject to a small quasi-periodic forcing which (1) is analytic, (2) satisfies a time-reversibility property, and (3) has a Bryuno frequency vector. Without imposing any non-degeneracy condition, we prove that there exists at least one quasi-periodic solution with the same frequency vector as the forcing. The result can be interpreted as a theorem of persistence of lower-dimensional tori of arbitrary codimension in degenerate cases. This is a joint work with Livia Corsi.

An application of the sharp Caffarelli-Kohn-Nirenberg inequalities

Series
Math Physics Seminar
Time
Thursday, January 16, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael Loss – Georgia Tech –

This talk is centered around the symmetry properties of optimizers for the Caffarelli-Kohn-Nirenberg (CKN) inequalities, a two parameter family of inequalities. After a general overview I will explain some of the ideas on how to obtain the optimal symmetry region in the parameter space and will present an application to non-linear functionals of Aharonov-Bohm type, i.e., to problems that include a  magnetic flux concentrated at one point. These functionals are rotationally invariant and, as I will discuss, depending on the magnitude of the flux, the optimizers are radially symmetric or not.

Freezing of the optical-branch energy in a diatomic nonlinear chain

Series
Math Physics Seminar
Time
Monday, November 18, 2019 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Alberto Maiocchi – Universita di Padova –

We show that the dynamics of nonlinear dynamical systems with many degrees of freedom (possibly infinitely many) can be similar to that of ordered system in a surprising fashion. To this aim, in the literature one typically uses techniques from perturbation theory, such as KAM theorem or Nekhoroshev theorem. Unfortunately they are known to be ill-suited for obtaining results in the case of many degrees of freedom. We present here a probabilistic approach, in which we focus on some observables of physical interest (obtained by averaging on the probability distribution on initial data) and for several models we get results of stability on long times similar to Nekhoroshev estimates. We present the example of a nonlinear chain of particles with alternating masses, an hyper-simplified model of diatomic solid. In this case, which is similar to the celebrated Fermi-Pasta-Ulam model and is widely studied in the literature, we show the progress with respect to previous results, and in particular how the present approach permits to obtain theorems valid in the thermodynamic limit, as this is of great relevance for physical implications.

Quantum fate of classical solitons

Series
Math Physics Seminar
Time
Monday, October 28, 2019 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael Pustilnik – School of Physics, Georgia Tech –
This talk will focus on one-dimensional interacting quantum systems near the classical limit described by the Korteweg–de Vries (KdV) equation. Classical excitations in this regime are the well-known solitons, i.e., localized disturbances with particle-like properties, and delocalized waves of density, or phonons. It turns out, however, that the semiclassical description inevitably breaks down at long wavelengths. In this limit, quantum effects become dominant, the system is best described in terms of weakly interacting fermions, and classical solitons and phonons reach their ultimate quantum fate of being demoted to fermionic particles and holes.
 
We will give simple heuristic arguments in support of this claim and present the exact solution for the spectra of elementary excitations. The results are universally applicable to all quantum one-dimensional systems with a well-defined classical limit described by the KdV equation. This includes identical bosons with a weak short-range repulsion and identical particles, either bosons or fermions, with a strong long-range repulsion.

Proof of Kac's conjecture for the hard sphere gas

Series
Math Physics Seminar
Time
Monday, October 21, 2019 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael Loss – Georgia Tech –
This talk will be about the master equation approach to kinetic theory pioneered by Mark Kac. Specifically, the physically realistic case of three dimensional hard spheres will be considered.  This process describes an ensemble of  hard spheres undergoing binary energy and momentum preserving collisions.  One measure for the speed of approach to equilibrium is the gap which was conjectured by Kac to be bounded below by a positive constant independent of the number of particles. In this talk a proof of this conjecture  will be presented. This is joint work with Eric Carlen and Maria Carvalho.

Efficient Representations of Correlated Data as Tensor Networks

Series
Math Physics Seminar
Time
Monday, October 7, 2019 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Glen Evenbly – School of Physics, Georgia Tech –
Tensors networks are a formalism for expressing high-order tensors as networks of low-order tensors, thus can offer a compact representation of certain high-dimensional datasets. Originally developed in the context of quantum many-body theory, where they are used to efficiently represent quantum wave-functions, tensor networks have since found application in big data analytics, error correction, classical data compression and machine learning.
 
In this talk I will provide a brief introduction to the theory and application of tensor networks, and outline some of the current research directions in the tensor network program.    
 

Sharp diameter bound on the spectral gap for quantum graphs

Series
Math Physics Seminar
Time
Monday, September 30, 2019 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Kenny Jones – Emory –

We establish an upper bound on the spectral gap for compact quantum graphs which depends only on the diameter and total number of vertices. This bound is asymptotically sharp for pumpkin chains with number of edges tending to infinity. This is a joint work with D. Borthwick and L. Corsi.

Periodic Dynamics of a Local Perturbation in the Isotropic XY Model

Series
Math Physics Seminar
Time
Monday, September 16, 2019 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Livia Corsi – Universita' di Roma 3 –

I will consider the isotropic XY chain with a transverse magnetic field acting on a single site, and analyze the long time behaviour of the time-dependent state of the system when a periodic perturbation drives the impurity. I will show that, under some conditions, the state approaches a periodic orbit synchronized with the forcing. Moreover I will provide the explicit rate of convergence to the asymptotics. This is a joint work with G. Genovese.

Exponential decay of quantum conditional information in thermal states of 1D short-ranged gapped Hamiltonians.

Series
Math Physics Seminar
Time
Friday, April 19, 2019 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Pavel Svetlichnyy – School of Physics, GaTeach –

I will talk about a conjecture that in Gibbs states of one-dimensional spin chains with short-ranged gapped Hamiltonians the quantum conditional mutual information (QCMI) between the parts of the chain decays exponentially with the length of separation between said parts. The smallness of QCMI enables efficient representation of these states as tensor networks, which allows their efficient construction and fast computation of global quantities, such as entropy. I will present the known partial results on the way of proving of the conjecture and discuss the probable approaches to the proof and the obstacles that are encountered.

Periodic and quasi-periodic attractors of the spin-orbit dynamics of Mercury

Series
Math Physics Seminar
Time
Tuesday, April 9, 2019 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Guido Gentile – Universita' di Roma 3 –

Please Note: Unusual time.

Mercury is entrapped in a 3:2 resonance: it rotates on its axis three times for every two revolutions it makes around the Sun. It is generally accepted that this is due to the large value of Mercury's eccentricity. However, the mathematical model commonly used to study the problem -- sometimes called the spin-orbit model -- proved not to be entirely convincing, because of the expression used for the tidal torque. Only recently, a different model for the tidal torque has been proposed, with the advantage of both being more realistic and providing a higher probability of capture into the 3:2 resonance with respect to the previous models. On the other hand, a drawback of the model is that the function describing the tidal torque is not smooth and appears as a superposition of peaks, so that both analytical and numerical computations turn out to be rather delicate. We shall present numerical and analytical results about the nature of the librations of Mercury's spin in the 3:2 resonance, as predicted by the realistic model. In particular we shall provide evidence that the librations are quasi-periodic in time, so that the very concept of resonance should be revisited. The analytical results are mainly based on perturbation theory and leave several open problems, that we shall discuss.

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