Mathematics Department - Mathematical Physics Seminar - Spring 2015

Mathematical Physics Seminar - Spring 2015



Organizer(s)

Joel Lebowitz, Michael Kiessling

Archive

Website

http://www.math.rutgers.edu/~lebowitz/Fall2014seminars.html



Upcoming Talks


Thursday, April 2nd

Subhro Ghosh, Princeton University

"Large deviations and random polynomials"

Time: 12:00 PM
Location: Hill 705
Abstract: We obtain a large deviations principle (in the space of probability measures on $C$) for the empirical measure of zeroes of random polynomials with i.i.d. exponential coefficients. One of the key challenges here is the fact that the coefficients are a.s. all positive, which enforces a growing number of non-linear constraints on the locations of the zeroes. En route, we will discuss a recent characterization theorem of Bergweiler and Eremenko, and its application in the proof of our main theorem.

Based on joint work with Ofer Zeitouni.

BROWN BAG LUNCH FROM 1-2PM


Thursday, April 2nd

Almut Burchard , University of Toronto

"Geometric Stability results for the Coulomb Energy"

Time: 2:00 PM
Location: Hill 705





Past Talks


Thursday, March 26th

Gael Raoul , Ecole Polytechnique

"Dynamics of a species structured by a space variable and a phenotypic trait"

Time: 2:00 PM
Location: Hill 705
Abstract: In many current ecological problems, both the spacial structure and the genetic diversity of species have to be taken into account. Combining those two aspects leads to challenging mathematical questions. We will present two such problems:

- Evolutionary epidemiology. Microbial population have typically a high mutation rate, and a large population size. As a consequence, evolutionary effects can affect the spacial dynamics of the population.

- Effect of climate change. We consider a population of trees that reproduces sexually, and we wonder whether the high dispersion of pollen will help the population to survive global warming.


Thursday, March 26th

Natan Andrei , Rutgers University

"Quench evolution of quantum integrable many body systems"

Time: 12:00 PM
Location: Hill 705
Abstract: Recently, with the appearance of experimental systems ranging from nano-devices to optically trapped cold atom gases, much progress was in our understanding of many fundamental aspects of nonequilibrium processes in isolated quantum systems. Quench evolutions, where a Hamiltonian is suddenly applied to a system and its evolution is followed in time, provide a means of studying the dynamics of these systems and to reveal their intrinsic relaxation mechanisms and time scales, whether the system thermalizes, the dynamic spectrum of states driving the evolution, the energy exchange among the various modes and the phase transitions in time it may cross as it evolves, among many other. In this talk I will describe the quench dynamics of isolated interacting systems in 1-d, governed by integrable Hamiltonians. I shall study the time evolution of a gas of interacting bosons moving on the continuous infinite line and interacting via a short range potential - the Lieb-Liniger model. For a system with a finite number of bosons we find that independently of the initial state the system asymptotes towards a strongly repulsive gas for any value of repulsive coupling, while for any value of attractive coupling the system is dominated by the maximal bound state. I shall then discuss the boson system in the thermodynamic limit and consider several circumstances: quench from a Mott insulator, quench in a box, quench from a domain wall. I will discuss the appearance of a GGE (generalized Gibbs ensemble) in the long time limit and why it usually fails. If time permits I shall discuss also the quench dynamics of the XXZ Heisenberg chain in different parameter regimes.

THERE WILL BE A BROWN BAG LUNCH FROM 1-2PM!!


Thursday, March 5th

Senya Shlosman , University of Marseille, Luminy

"How the Ising Crystal Grows, and Where to Look for Airy Diffusion"

Time: 12:00 PM
Location: Hill 705
Abstract: I will describe the process of the crystal growth, as the crystal acquires mode and more atoms. I will also explain where one should look in order to see the (fashionable nowadays) N^{1/3} exponent, where N is the linear size of the crystal.


Thursday, February 26th

David Huse , Princeton University

"Many-body localization"

Time: 2:00 PM
Location: Hill 705
Abstract: I will discuss some aspects of our present understanding of the physics of the many-body localized phase, and of the quantum phase transition between many-body localization and quantum thermalization. The many-body localized phase can, in certain cases, be understood as a new type of integrable system, where the emergent conserved quantities are localized operators (H., Nandkishore, Oganesyan, PRB 2014). The low frequency dynamics in this phase differs in important ways from that of noninteracting Anderson localization due to rare many-body "resonances" (Gopalakrishnan, H., et al, in progress).

The delocalization phase transition is due to the proliferation of these resonances (Vosk, Altman, H., arXiv:1412.3117).


Thursday, February 26th

Michael Kiessling , Rutgers University

"A novel quantum-mechanical interpretation of Dirac's equation"

Time: 12:00 PM
Location: Hill 705
Abstract: A novel interpretation is given of Dirac's ``wave equation for the relativistic electron'' as a quantum-mechanical one-particle equation in which electron and positron are merely the two different ``topological spin'' states of a single more fundamental particle, not distinct particles in their own right.

This is joint work with A.Shadi Tahvildar-Zadeh

THERE WILL BE A BROWN BAG LUNCH 1-2PM!


Thursday, February 19th

Dan Pirjol , National Institute for Physics and Nuclear Engineering, Romania

"Phase transition in a stochastic growth process with multiplicative noise"

Time: 2:00 PM
Location: Hill 705
Abstract: The talk will discuss the random linear recursion x(i+1) = a(i)*x(i)+b(i) where a(i) > 1 are stochastic multipliers related to the exponential of a standard Brownian motion, and b(i) are positive uncorrelated noise. This is a growth process, which is motivated by problems in mathematical finance related to interest rate modeling and numerical simulation of stochastic volatility models. Under certain conditions x(i) develops heavy tailed distributions, which are manifested as numerical explosions of the positive integer moments <(x(i))^q>, q=1,2,.... This phenomenon can be studied by mapping the problem to a one-dimensional lattice gas with linear attractive potentials, which can be solved exactly. The moment explosions can be related to a phase transition in the equivalent lattice gas.


Thursday, February 19th

Luca Peliti , Istituto Nazionale di Fisica Nucleare

"Beneficial mutations in a range-expansion wave"

Time: 12:00 PM
Location: Hill 705
Abstract: Many theoretical and experimental studies suggest that range expansions can have severe consequences for the gene pool of the expanding population. Due to strongly enhanced genetic drift at the advancing frontier, neutral and weakly deleterious mutations can reach large frequencies in the newly colonized regions, as if they were surfing the front of the range expansion. These findings raise the question of how frequently beneficial mutations successfully surf at shifting range margins, thereby promoting adaptation towards a range-expansion phenotype. We studied this problem by means of individual-based simulations, as a function of two strongly antagonistic factors, the probability of surfing given the spatial location of a novel mutation and the rate of occurrence of mutations at that location. We find that small amounts of genetic drift increase the fixation rate of beneficial mutations at the advancing front, and thus could be important for adaptation during species invasions.

Joint work with R. Lehe (Paris) and O. Hallatschek (now at Berkeley).

BROWN BAG LUNCH FROM 1-2PM!


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.


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.

THERE WILL BE A BROWN BAG LUNCH BETWEEN 1-2PM!


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 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.

THERE WILL BE A BROWN BAG LUNCH FROM 1-2PM!


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