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Series: Math Physics Seminar

Host: Predrag Cvitanovic, School of Physics

Suppose that x(t) is a signal generated by a chaotic system and that the signal has been recorded in the interval [0,T]. We ask: What is the largest value t_f such that the signal can be predicted in the interval (T,T+t_f] using the history of the signal and nothing more? We show that the answer to this question is contained in a major result of modern information theory proved by Wyner, Ziv, Ornstein, and Weiss. All current algorithms for predicting chaotic series assume that if a pattern of events in some interval in the past is similar to the pattern of events leading up to the present moment, the pattern from the past can be used to predict the chaotic signal. Unfortunately, this intuitively reasonable idea is fundamentally deficient and all current predictors fall well short of the Wyner-Ziv bound. We explain why the current methods are deficient and develop some ideas for deriving an optimal predictor. [This talk is based on joint work with X. Liang and K. Serkh]. To view and/or participate in the webinar from wherever you are, click on:EVO.caltech.edu/evoNext/koala.jnlp?meeting=MvM2Ml2M2tDvDn9n9nDe9v

Series: Math Physics Seminar

We study the sum of the negative eigenvalues of the Dirichlet Laplace operatoron a bounded domain in the semiclassical limit. We give a new proof thatyields not only the Weyl term but also the second asymptotic term involvingthe surface area of the boundary of the domain.The proof is valid under weak smoothness assumptions on the boundary and theresult can be extended to non-local, non-smooth operators like fractionalpowers of the Dirichlet Laplacian.(This is joint work with Rupert L. Frank.)

Series: Math Physics Seminar

We study a two dimensional Schrödinger operator for a limit-periodic potential. We prove that the spectrum contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves in the high energy region. Second, the isoenergetic curves in the space of momenta corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure). Third, the spectrum corresponding to these eigenfunctions (the semiaxis) is absolutely continuous.

Series: Math Physics Seminar

The motivation is to compute the spectral properties of the Schrodinger operator describing an electron in a quasicrystal. The talk will focus on the case of the Fibonacci sequence (one dimension), to illustrate the method. Then the Wannier transform will be defined. It will be shown that the Hamiltonian can be seen as a direct integral over operators with discrete spectra, in a way similar to the construction of band spectra for crystal. A discussion of the differences with crystal will conclude this talk.This is joint work with Giuseppe De Nittis and Vida Milani

Series: Math Physics Seminar

An explicit asymptotic expression for the ground-state energy of the Pekar-Tomasevich functional for the N-polaron is found, when the positive repulsion parameter U of the
electrons is less than twice the coupling constant of the polaron. This is joint workwith Gonzalo Bley.

Series: Math Physics Seminar

I will describe recent work with Marina Chugunovaand Ben Stephens on the evolution of a thin-filmequation that models a "coating flow" on a horizontalcylinder. Formally, the equation defines a gradientflow with respect to an energy that controls theH^1-norm.We show that for each given mass there exists aunique steady state, given by a droplet hanging from thebottom of the cylinder that meets the dry region withzero contact angle. The droplet minimizes the energy andattracts all strong solutions that satisfy certain energyand entropy inequalities. (Such solutions exist for arbitraryinitial values of finite energy and entropy, but it is notknown if they are unique.) The distance of any solutionfrom the steady state decays no faster than a power law.

Series: Math Physics Seminar

I'll present a simple model of market where the use of (commodity) money naturally arisefrom the agents interaction. I'll introduce the relevant notion of (Nash) equilibrium and discuss itsexistence and properties.

Series: Math Physics Seminar

I'll describe some connections between identities for commutators and boundson eigenvalues, including Stubbe's proof of classical Lieb-Thirringinequalities and other sharp Lieb-Thirring inequalities for different models(including Schrödinger operators with periodic potentials or on manifolds,and quantum graphs).

Series: Math Physics Seminar

I'll talk about recent work, jointly with J. Baker, F. Klopp, S. Nakamura and G. Stolz concerning the random displacement model. I'll outline a proof of localization near the edge of the deterministic spectrum. Localization is meant in both senses, pure point spectrum with exponentially decaying eigenfunctions as well as dynamical localization. The proof relies on a well established multiscale analysis and the main problem is to verify the necessary ingredients, such as a Lifshitz tail estimate and a Wegner estimate.

About symmetry and symmetry breaking for extremal functions in interpolation functional inequalities

Series: Math Physics Seminar

In this talk I will present recent work, in collaboration with J.Dolbeault, G. Tarantello and A. Tertikas,about the symmetry properties of extremal functions for (interpolation)functional inequalities playing an important rolein the study of long time behavior of evolution diffusion equations.Optimal constants are rarely known,in fact one can write them explicitely only when the extremals enjoymaximal symmetry. This is why the knowledge of the parameters' regionswhere symmetry is achieved is of big importance. In the case of symmetrybreaking, the underlying phenomena permitting it are analyzed.