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Series: Math Physics Seminar

Two-point symmetrizations are simple rearrangementsthat have been used to prove isoperimetric inequalitieson the sphere. For each unit vector u, there is atwo-point symmetrization that pushes mass towardsu across the normal hyperplane.How can full rotational symmetry be recovered from partialinformation? It is known that the reflections at d hyperplanes in general position generate a dense subgroup of O(d);in particular, a continuous function that is symmetric under thesereflections must be radial. How many two-point symmetrizationsare needed to verify that a function which increases under thesesymmetrizations is radial? I will show that d+1 such symmetrizationssuffice, and will discuss the ergodicity of the randomwalk generated by the corresponding folding maps on the sphere.(Joint work with G. R. Chambers and Anne Dranovski).

Series: Math Physics Seminar

We build a family of
spectral triples for a discrete aperiodic tiling space, and derive the
associated Connes distances. (These are non commutative geometry
generalisations of Riemannian structures, and associated geodesic
distances.) We show how their metric properties lead to a characterisation
of high aperiodic order of the tiling. This is based on joint works with
J. Kellendonk and D. Lenz.

Series: Math Physics Seminar

In this talk I will begin with our recent results on non-equilibrium steady states (NESS) of a microscopic heat conduction model, which is a stochastic particle system coupled to unequal heat baths. This stochastic model is derived from a mechanical chain model (Eckmann and Young 2006) by randomizing certain quantities while retaining the other features. We proved various results including the existence and uniqueness of NESS and the exponential rate of mixing. Then I will follow with an energy dependent Kac-type model that is obtained from an improved version of randomization of the “local" dynamics. We rigorously proved that this Kac-type model has a mixing rate $\sim t^{-2}$. In the end, I will show that slow (polynomial) mixing rates appear in a large class of statistical mechanics models.

Series: Math Physics Seminar

I propose a generalization of Hopf fibrations to quotient the streamwise translation symmetry of water waves and turbulent pipe flows viewed as dynamical systems. In particular, I exploit the geometric structure of the associated high dimensional state space, which is that of a principal fiber bundle.
Symmetry reduction analysis of experimental data reveals that the speeds of large oceanic crests and turbulent bursts are associated with the dynamical and geometric phases of the corresponding orbits in the fiber bundle. In particular, in the symmetry-reduced frame I unveil a pattern-changing dynamics of the fluid structures, which explains the observed speed u ≈ Ud+Ug of intense extreme events in terms of the geometric phase velocity Ug and the dynamical phase velocity Ud associated with the orbits in the bundle. In particular, for oceanic waves Ug/Ud~-0.2 and for turbulent bursts Ug/Ud~0.43 at Reynolds number Re=3200.

Series: Math Physics Seminar

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.

Series: Math Physics Seminar

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

Series: Math Physics Seminar

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.

Series: Math Physics Seminar

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.

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.