Mathematics Department - Mathematical Physics Seminar - Spring 2015

Mathematical Physics Seminar - Spring 2015


Joel Lebowitz, Michael Kiessling



Upcoming Talks

Thursday, February 5th

Haim Brezis, Rutgers University and Technion

"From the characterization of constant functions to isoperimetric inequalities"

Time: 12:00 PM
Location: Hill 705
Abstract: I will present a "common roof" to various, seemingly unrelated, known statements asserting that integer-valued functions satisfying some kind of mild regularity are constant. For this purpose I will introduce a new function space B which is so large that it contains many classical spaces, such as BV (=functions of bounded variation) , BMO (=John-Nirenberg space of functions of bounded mean oscillation) and some fractional Sobolev spaces. I will then define a fundamental closed subspace B_0 of B contaning in particular W^{1,1}, VMO--- and thus continuous functions---- H^{1/2} etc. A remarkable fact is that integer-valued functions belonging to B_0 are necessarily constant. I will also discuss connections of the B-norm to geometric concepts, such as the perimeter of sets.

This is joint work with L. Ambrosio, J. Bourgain, A. Figalli and P. Mironescu.


Thursday, February 5th

Alessandro Giuliani , University of Roma Tre

"Height fluctuations in interacing dimers"

Time: 2:00 PM
Location: Hill 705
Abstract: Perfect matchings of Z^2 (also known as non-interacting dimers on the square lattice) are an exactly solvable 2D statistical mechanics model. It is known that the associated height function behaves at large distances like a massless gaussian field, with the variance of height gradients growing logarithmically with the distance. As soon as dimers mutually interact, via e.g. a local energy function favoring the alignment among neighboring dimers, the model is not solvable anymore and the dimer-dimer correlation functions decay polynomially at infinity with a non-universal (interaction-dependent) critical exponent. We prove that, nevertheless, the height fluctuations remain gaussian even in the presence of interactions, in the sense that all their moments converge to the gaussian ones at large distances. The proof is based on a combination of multiscale methods with the path-independence properties of the height function.

Joint work with V. Mastropietro and F. Toninelli.

Thursday, February 12th

Roger Nussbaum, Rutgers University

"Perron-Frobenius Operators, Positive C^m Eigenvectors and the Computation of Hausdorff Dimension"

Time: 12:00 PM
Location: Hill 705
Abstract: We shall discuss a class of linear Perron-Frobenius operators L which, under added assumptions, arise in the computation of Hausdorff dimension for invariant sets of iterated function systems or graph directed iterated function systems. We shall describe theorems which insure the existence of a strictly positive, C^m eigenvector of L. In important cases it is possible to obtain explicit bounds on second order (and higher) partial derivatives of v. We shall indicate how (joint work with Richard Falk) information about partial derivatives of v can be used to obtain rigorous estimates of Hausdorff dimension (at least three to four decimal point accuracy) for some previously intractable examples like the set of complex continued fractions.


Thursday, February 12th

Peter Nandori , New York University

"Local thermal equilibrium for certain stochastic models of heat transport "

Time: 2:00 PM
Location: Hill 705
Abstract: This talk is about nonequilibrium steady states (NESS) of a class of stochastic models in which particles exchange energy with their 'local environments' rather than directly with one another. The physical domain of the system can be a bounded region of R^d for any dimension d. We assume that the temperature at the boundary of the domain is prescribed and is nonconstant, so that the system is forced out of equilibrium. Our main result is local thermal equilibrium in the infinite volume limit. We also prove that the mean energy profile of NESS satisfies Laplace's equation for the prescribed boundary condition. Our method of proof is duality: by reversing the sample paths of particle movements, we convert the problem of studying local marginal energy distributions at x to that of joint hitting distributions of certain random walks starting from x.

This is a joint work with Yao Li and Lai-Sang Young.

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