## Seminars and Colloquia by Series

Friday, April 19, 2019 - 16:00 , Location: Skiles 005 , Pavel Svetlichnyy , School of Physics, GaTeach , , Organizer: Federico Bonetto

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.

Tuesday, April 9, 2019 - 12:00 , Location: Skiles 005 , Guido Gentile , Universita&#039; di Roma 3 , , Organizer: Federico Bonetto

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.

Monday, April 8, 2019 - 10:00 , Location: Skiles 005 , L.A.Bunimovich , School of Mathematics, Georgia Tech , , Organizer: Federico Bonetto

Unusual time.

In standard (mathematical) billiards a point particle moves uniformly in a billiard table with elastic reflections off the boundary. We show that in transition from mathematical billiards to physical billiards, where a finite size hard sphere moves in the same billiard table, virtually anything may happen. Namely a non-chaotic billiard may become chaotic and vice versa. Moreover, both these transitions may occur softly, i.e. for any (arbitrarily small) positive value of the radius of a physical particle, as well as by a ”hard” transition when radius of the physical particle must exceed some critical strictly positive value. Such transitions may change a phase portrait of a mathematical billiard locally as well as completely (globally). These results are somewhat unexpected because for all standard examples of billiards their dynamics remains absolutely the same after transition from a point particle to a finite size (”physical”) particle. Moreover we show that a character of dynamics may change several times when the size of the particle is increasing. This approach already demonstrated a sensational result that quantum system could be more chaotic than its classical counterpart.

Friday, April 5, 2019 - 16:00 , Location: Skiles 005 , Alexander Grigo , Department of Mathematics, University of Oklahoma , , Organizer: Federico Bonetto

In this talk I will discuss a particular fast-slow system, and describe an averaging theorem. I will also explain how this particular slow-fast system arises in a certain problem of energy transport in an open system of interacting hard-spheres. The technical aspect involved in this is how to deal with singularities present and the fact that the dynamics is fully coupled.

Friday, March 15, 2019 - 16:00 , Location: Skiles 005 , Peter Nandori , University of Maryland , , Organizer: Federico Bonetto

We present a convenient joint generalization of mixing and the local
central limit theorem which we call MLLT. We review results on the MLLT
for hyperbolic maps and present new results for hyperbolic flows. Then
we apply these results to prove global mixing properties of some
mechanical systems. These systems include various versions of the
Lorentz gas (periodic one; locally perturbed; subject to external
fields), the Galton board and pingpong models. Finally, we present
applications to random walks in deterministic scenery. This talk is
based on joint work with D. Dolgopyat and partially with M. Lenci.

Friday, March 15, 2019 - 14:45 , Location: Skiles 005 , Mihai Ciucu , Mathematics Department, Indiana University , , Organizer: Federico Bonetto

We consider a triangular gap of side two in a 90 degree angle on the triangular lattice with mixed boundary conditions: a constrained, zig-zag boundary along one side, and a free lattice line boundary along the other. We study the interaction of the gap with thecorner as the rest of the angle is completely filled with lozenges. We show that the resulting correlation is governed by the product of the distances between the gap and its three images in the sides of the angle. This, together with a few other results we worked out previously, provides evidence for a unified way of understanding the interaction of gaps with the boundary under mixed boundary conditions, which we present as a conjecture. Our conjecture is phrased in terms of the steady state heat flow problem in a uniform block of material in which there are a finite number of heat sources and sinks. This new physical analogy is equivalent in the bulk to the electrostatic analogy we developed in previous work, but arises as the correct one for the correlation with the boundary.The starting point for our analysis is an exact formula we prove for the number of lozenge tilings of certain trapezoidal regions with mixed boundary conditions, which is equivalent to a new, multi-parameter generalization of a classical plane partition enumeration problem (that of enumerating symmetric, self-complementary plane partitions).

Friday, February 22, 2019 - 16:00 , Location: Skiles 005 , Ian Jauslin , Princeton University , , Organizer: Federico Bonetto

Consider a metallic field emitter shaped like a thin needle, at the tip of which a large electric field is applied. Electrons spring out of the metal under the influence of the field. The celebrated and widely used Fowler-Nordheim equation predicts a value for the current outside the metal. In this talk, I will show that the Fowler-Nordheim equation emerges as the long-time asymptotic solution of a Schrodinger equation with a realistic initial condition, thereby justifying the use of the Fowler Nordheim equation in real setups. I will also discuss the rate of convergence to the Fowler-Nordheim regime.

Friday, November 30, 2018 - 16:00 , Location: Skiles 006 , Jake Fillman , Virginia Polytechnic Institute , Organizer: Michael Loss

A limit-periodic function on R^d is one which lies in the L^\infty closure of the space of periodic functions. Schr\"odinger operators with limit-periodic potentials may have very exotic spectral properties, despite being very close to periodic operators. Our discussion will revolve around the transition between thick'' spectra and thin'' spectra.

Wednesday, November 14, 2018 - 16:00 , Location: Skiles 005 , Rohan Ghanta , SoM Georgia Tech , Organizer: Michael Loss

We shall consider a three-dimensional Quantum Field Theory model of an electron
bound to a Coulomb impurity in a polar crystal and exposed to a homogeneous
magnetic field of strength B > 0. Using an argument of Frank and Geisinger
[Commun. Math. Phys. 338, 1-29 (2015)] we can see that as B → ∞ the ground-
state energy is described by a one-dimensional minimization problem with a delta-
function potential. Our contribution is to extend this description also to the ground-
state wave function: we shall see that as B → ∞ its electron density in the direction
of the magnetic field converges to the minimizer of the one-dimensional problem.
Moreover, the minimizer can be evaluated explicitly.

Wednesday, November 7, 2018 - 16:00 , Location: Skiles 005 , Fan Yang , Georgia Tech , Organizer: Michael Loss

I will talk about what happens on the spectral transition lines for the almost Mathieu operator. This talk is based on joint works with Svetlana Jitomirskaya and Qi Zhou. For both transition lines \{\beta(\alpha)=\ln{\lambda}\} and \{\gamma(\alpha,\theta)=\ln{\lambda}\} in the positive Lyapunov exponent regime, we show purely point spectrum/purely singular continuous spectrum for dense subsets of frequencies/phases.