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Series: Math Physics Seminar

The talk will present several recent results on the singular
and pure point spectra for the (random or non-random) Schrӧdinger
operators on the graphs or the Riemannian manifolds of the “small dimensions”.
The common feature of all these results is the existence in the potential of
the infinite system of the “bad conducting blocks”, for instance, the
increasing potential barriers (non-percolating potentials). The central idea of
such results goes to the classical paper by Simon and Spencer.
The particular examples will include the random Schrӧdinger
operators in the tube (or the surface of the cylinder),
Sierpinski lattice etc.

Series: Math Physics Seminar

We consider a model of randomly colliding particles interacting with a thermal bath. Collisions between particles are modeled via the Kac master equation while the thermostat is seen as an inﬁnite gas at thermal equilibrium at inverse temperature \beta. The system admits the canonical distribution at inverse temperature \beta as the unique equilibrium state. We prove that the any initial distribution approaches the equilibrium distribution exponentially fast both by computing the gap of the generator of the evolution, in a proper function space, as well as by proving exponential decay in relative entropy. We also show that the evolution propagates chaos and that the one-particle marginal, in the large system limit, satisﬁes an effective Boltzmann-type equation. This is joint work with Federico Bonetto and Michael Loss.

Series: Math Physics Seminar

El Soufi will be visiting Harrell for the week leading up to this seminar

We shall survey some of the classical and recent results giving upper bounds of the eigenvalues of the Laplace-Beltrami operator on a compact Riemannian manifold (Yang-Yau, Korevaar, Grigor'yan-Netrusov-Yau, etc.). Then we discuss extensions of these results to the eigenvalues of Witten Laplacians associated to weighted volume measures and investigate bounds of these eigenvalues in terms of suitable norms of the weights.

Series: Math Physics Seminar

Several low dimensional interacting fermionic systems, including g
raphene and spin chains, exhibit remarkable universality properties in the c
onductivity, which
can be rigorously established under certain conditions by combining Renormal
ization Group methods with Ward Identities.

Series: Math Physics Seminar

Lots of attention and research activity has been devoted to partially
hyperbolic dynamical systems and their perturbations in the past few decades;
however, the main emphasis has been on features such as stable ergodicity and
accessibility rather than stronger statistical properties such as existence of
SRB measures and exponential decay of correlations.
In fact, these properties have been previously proved under some specific
conditions (e.g. Anosov flows, skew products) which, in particular, do not
persist under perturbations.
In this talk, we will construct an open (and thus stable for perturbations)
class of partially hyperbolic smooth local diffeomorphisms of the two-torus
which admit a unique SRB measure satisfying exponential decay of correlations
for Hölder observables.
This is joint work with C. Liverani

Series: Math Physics Seminar

The lattice, two dimensional, Coulomb gas is the prototypical model of
Statistical Mechanics displaying the 'Kosterlitz-Thouless' phase
transition. In this seminar I will discuss conjectures, results and
works in progress about this model.

Series: Math Physics Seminar

We study a gas of N hard disks in a box with semi-periodic boundary
conditions. The unperturbed gas is hyperbolic and ergodic (these facts are
proved for N=2 and expected to be true for all N>2). We study
various perturbations by "twisting" the outgoing velocity at collisions with
the walls. We show that the dynamics tends to collapse to various stable
regimes, however we define the perturbations and however small they are.

Series: Math Physics Seminar

In this talk I will discuss a family of lower bounds on the indirect Coulomb energy for atomic and molecular systems in two dimensions in terms of a functional of the single particle density with gradient correction terms

Series: Math Physics Seminar

Abstract: In this talk we will survey some recent developments in the scattering theory of complete, infinite-volume manifolds with ends modeled on quotients of hyperbolic space. The theory of scattering resonances for such spaces is in many ways parallel to the classical case of eigenvalues on a compact Riemann surface. However, it is only relatively recently that progress has been made in understanding the distribution of these resonances. We will give some introduction to the theory of resonances in this context and try to sketch this recent progress. We will also discuss some interesting outstanding conjectures and present numerical evidence related to these.

Series: Math Physics Seminar

I'll discuss two methods for finding bounds on sums of graph eigenvalues (variously for the Laplacian, the renormalized Laplacian, or the adjacency matrix). One of these relies on a Chebyshev-type estimate of the statistics of a subsample of an ordered sequence, and the other is an adaptation of a variational argument used by P. Kröger for Neumann Laplacians. Some of the inequalities are sharp in suitable senses. This is ongoing work with J. Stubbe of EPFL