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Series: Math Physics Seminar

I will discuss the fluctuations of the spectral density for the Wigner ensemble on the optimal scale. We study the fluctuations of the Stieltjes transform, and improve the known bounds on the optimal scale. As an application, we derive the semicircle law at the edge of the spectrum. This is joint work with Claudio Cacciapuoti and Benjamin Schlein.

Series: Math Physics Seminar

We prove a quantitative Brunn-Minkowski inequality for sets E and K,one of which, K, is assumed convex, but without assumption on the other set. We are primarily interested in the case in which K is a ball. We use this to prove an estimate on the remainder in the Riesz rearrangement inequality under certain conditions on the three functions involved that are relevant to a problem arising in statistical mechanics: This is joint work with Franceso Maggi.

Series: Math Physics Seminar

The (blobbed) topological recursion is a recursive structure which defines, for any initial datagiven by symmetric holomorphic 1-form \phi_{0,1}(z) and 2-form \phi_{0,2}(z_1,z_2) (and symmetricn-forms \phi_{g,n} for n >=1 and g >=0), a sequence of symmetric meromorphic n-forms\omega_{g,n}(z_1,...,z_n) by a recursive formula on 2g - 2 + n.If we choose the initial data in various ways, \omega_{g,n} computes interesting quantities. A mainexample of application is that this topological recursion computes the asymptotic expansion ofhermitian matrix integrals. In this talk, matrix models with also serve as an illustration of thisgeneral structure.

Series: Math Physics Seminar

Following Kato, we define the sum, $H=H_0+V$, of two linear operators, $H_0$ and $V$, in a fixed Hilbert space in terms of its resolvent. In an abstract theorem, we present conditions on $V$ that guarantee $\text{dom}(H_0^{1/2})=\text{dom}(H^{1/2})$ (under certain sectorality assumptions on $H_0$ and $H$). Concrete applications to non-self-adjoint Schr\"{o}dinger-type operators--including additive perturbations of uniformly elliptic divergence form partial differential operators by singular complex potentials on domains--where application of the abstract theorem yields $\text{dom}(H^{1/2})=\text{dom}((H^{\ast})^{1/2})$, will be presented. This is based on joint work with Fritz Gesztesy and Steve Hofmann.

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.