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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

Series: Math Physics Seminar

In this talk we will discuss the definition of chaoticity and entropic chaoticity, as well as the background that led us to define these quantities, mainly Kac's model and the Boltzmann equation. We will then proceed to investigate the fine balance required for entropic chaoticity by exploring situations where chaoticity is valid, but not entropic chaoticity. We will give a general method to construct such states as well as two explicit example, one of which is quite surprising.

Series: Math Physics Seminar

Host: Predrag Cvitanovic

More than 125 years ago Osborne Reynolds launched the quantitative
study of turbulent transition as he sought to understand the conditions
under
which fluid flowing through a pipe would be laminar or turbulent. Since
laminar and turbulent flow have vastly different drag laws, this question is
as important now as it was in Reynolds' day. Reynolds understood how one
should define "the real critical value'' for the fluid velocity beyond
which
turbulence can persist indefinitely. He also appreciated the difficulty in
obtaining this value. For years this critical Reynolds number, as we now
call
it, has been the subject of study, controversy, and uncertainty. Now, more
than a century after Reynolds pioneering work, we know that the onset of
turbulence in shear flows is properly understood as a statistical phase
transition. How turbulence first develops in these flows is more closely
related to the onset of an infectious disease than to, for example, the
onset
of oscillation in the flow past a body or the onset of motion in a fluid
layer
heated from below. Through the statistical analysis of large samples of
individual decay and proliferation events, we at last have an accurate
estimate of the real critical Reynolds number for the onset of turbulence in
pipe flow, and with it, an understanding of the nature of transitional
turbulence.
This work is joint with: K. Avila, D. Moxey, M. Avila, A. de Lozar, and B.
Hof.

Series: Math Physics Seminar

The relative isoperimetric inequality inside an open, convex cone C states that under a volume constraint, the ball intersected the cone minimizes the perimeter inside C. In this talk, we will show how one can use optimal transport theory to obtain this inequality, and we will prove a corresponding sharp stability result. This is joint work with Alessio Figalli.