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Series: Math Physics Seminar

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.

Series: Math Physics Seminar

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).

Series: Math Physics Seminar

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.

Series: Math Physics Seminar

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.

Series: Math Physics Seminar

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.

Series: Math Physics Seminar

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.

Series: Math Physics Seminar

We consider a class of singular ordinary differential equations,
describing systems subject to a quasi-periodic forcing term and
in the presence of large dissipation, and study the existence of
quasi-periodic solutions with the same frequency vector as the forcing term.
Let A be the inverse of the dissipation coefficient.
More or less strong non-resonance conditions on the frequency
assure different regularity in the dependence on the parameter A:
by requiring a non-degeneracy condition on the forcing term,
smoothness and analyticity, and even Borel-summability,
follow if suitable Diophantine conditions are assumed,
while, without assuming any condition, in general no more than a continuous dependence on A is obtained.
We investigate the possibility of weakening the non-degeneracy
condition and still obtaining a solution for arbitrary frequencies.

Series: Math Physics Seminar

Non-compact hyperbolic surfaces serve as a model case for quantum scattering theory with chaotic classical dynamics. In this talk I’ll explain how scattering resonances are defined in this context and discuss our current understanding of their distribution. The primary focus of the talk will be on some recent conjectures inspired by the physics of quantum chaotic systems. I will introduce these and discuss the numerical evidence as well as recent theoretical progress.

Series: Math Physics Seminar

TBA

Series: Math Physics Seminar

We generalize the Lp spectral cluster bounds of Sogge for the Laplace-Beltrami operator on compact Riemannian manifolds to systems of orthonormal functions. We show that these bounds are optimal on any manifold in a very strong sense. These spectral cluster bounds follow from Schatten-type bounds on oscillatory integral operators and their optimality follows by semi-classical analysis.