Seminars and Colloquia by Series

The Almost Subadditivity of the Entropy on Kac’s Sphere

Series
Math Physics Seminar
Time
Wednesday, August 20, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dr. Amit Einav – University of Cambridge, UK –
It is an interesting well known fact that the relative entropy of the marginals of a density with respect to the Gaussian measure on Euclidean space satisfies a simple subadditivity property. Surprisingly enough, when one tries to achieve a similar result on the N-sphere a factor of 2 appears in the right hand side of the inequality (a result due to Carlen, Lieb and Loss), and this factor is sharp. Besides a deviation from the simple ``equivalence of ensembles principle'' in equilibrium Statistical Mechanics, this entropic inequality on the sphere has interesting ramifications in other fields, such as Kinetic Theory.In this talk we will present conditions on a density function on the sphere, under which we can get an ``almost'' subaditivity property; i.e. the factor 2 can be replaced with a factor that tends to 1 as the dimension of the sphere tends to infinity. The main tools for proving this result is an entropy conserving extension of the density from the sphere to Euclidean space together with a comparison of appropriate transportation distances such as the entropy, Fisher information and Wasserstein distance between the marginals of the original density and that of the extension. Time permitting, we will give an example that arises naturally in the investigation of the Kac Model.

Particle Physics and Cosmology from Almost Commutative Manifolds

Series
Math Physics Seminar
Time
Friday, January 31, 2014 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Professor Mairi Sakellariadou – King's College, Physics –

Please Note: This is a joint Seminar School of Mathematics and Center of Relativistic Astrophysics, Georgia Tech

The unification of the four fundamental forces remains one of the most important issues in theoretical particle physics. In this talk, I will first give a short introduction to Non-Commutative Spectral Geometry, a bottom-up approach that unifies the (successful) Standard Model of high energy physics with Einstein's General theory of Relativity. The model is build upon almost-commutative spaces and I will discuss the physical implications of the choice of such manifolds. I will show that even though the unification has been obtained only at the classical level, the doubling of the algebra may incorporate the seeds of quantisation. I will then briefly review the particle physics phenomenology and highlight open issues and current proposals. In the last part of my talk, I will explore consequences of the Gravitational-Higgs part of the spectral action formulated within such almost-commutative manifolds. In particular, I will study modifications of the Friedmann equation, propagation of gravitational waves and the onset of inflation. I will show how current measurements (Gravity Probe, pulsars, and torsion balance) can constrain free parameters of the model. I will conclude with a short discussion on open questions. Download the POSTER

Quantum scissors and single photon states

Series
Math Physics Seminar
Time
Friday, November 15, 2013 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Brian Kennedy – GT Physics –
Sources of single photons (as opposed to sources which produce on average a single photon) are of great current interest for quantum information processing. Perhaps surprisingly, it is not easy to produce a single photon efficiently and in a controlled way. Following earlier progress, recent experimental activity has resulted in the production of single photons by taking advantage of strong inter-particle interactions in cold atomic gases.I will show how the systematic use of the method of steepest descents can be used to understand the dynamics of the single photon source developed here at Georgia Tech and how this describes a kind of quantum scissors effect. In addition to the mathematical results, I will present the background quantum mechanics in a form suitable for a general audience. Joint work with Francesco Bariani and Paul Goldbart.

Existence and Regularity in the Oval Problem

Series
Math Physics Seminar
Time
Tuesday, November 12, 2013 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jochen Denzler – University of Tennessee, Knoxville –
The oval problem asks to determine, among all closed loops in${\bf R}^n$ of fixed length, carrying a Schrödinger operator${\bf H}= -\frac{d^2}{ds^2}+\kappa^2$ (with curvature $\kappa$ andarclength $s$), those loops for which the principal eigenvalue of${\bf H}$ is smallest. A 1-parameter family of ovals connecting the circlewith a doubly traversed segment (digon) is conjectured to be the minimizer.Whereas this conjectured solution is an example that proves a lack ofcompactness and coercivity in the problem, it is proved in this talk(via a relaxed variation problem) that a minimizer exists; it is eitherthe digon, or a strictly convex planar analytic curve with positivecurvature. While the Euler-Lagrange equation of the problem appearsdaunting, its asymptotic analysis near a presumptive singularity givesuseful information based on which a strong variation can excludesingular solutions as minimizers.

Fluctuations in the Wigner Ensemble

Series
Math Physics Seminar
Time
Monday, November 11, 2013 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Anna Maltsev – University of Bristol –
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.

A quantitative Brunn-Minkowski inequality and estimates on the the remainder in the Riesz rearrangement inequality.

Series
Math Physics Seminar
Time
Friday, November 1, 2013 - 16:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Eric Carlen – Rutgers University
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.

Blobbed topological recursion and matrix models

Series
Math Physics Seminar
Time
Thursday, October 31, 2013 - 15:35 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Gaetan Borot – MPI Bonn and MIT –
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.

On Square Root Domains for Non-Self-Adjoint Operators Under Additive Perturbations

Series
Math Physics Seminar
Time
Thursday, October 3, 2013 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Roger Nichols – University of Tennessee, Chattanooga –
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.

Localization for the quasi 1D operators

Series
Math Physics Seminar
Time
Friday, September 27, 2013 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Stanislav Molchanov – UNC Charlotte –
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.

The Kac Model Coupled to a Thermostat

Series
Math Physics Seminar
Time
Thursday, September 19, 2013 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ranjini Vaidyanathan – Georgia Tech –
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.

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