Mathematics Department - Mathematical Physics Seminar - Spring 2016

Mathematical Physics Seminar - Spring 2016



Organizer(s)

Joel Lebowitz, Michael Kiessling

Archive

Website

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




Past Talks


Thursday, April 28th

Ovidiu Costin , Ohio State University

"New developments in the summation of divergent series "

Time: 2:00 PM
Location: Hill 705
Abstract: I will speak about a development, to a good extent surprising, in asymptotics: factorially divergent asymptotic series of special functions can be, surprisingly, rearranged to become asymptotic and geometrically convergent expansions in powers of the variable --suitably shifted. The new expansions are valid throughout (and slightly beyond) the region of applicability of the original power series.

Work in collaboration with Sir Michael Berry, Rodica Costin and Chris Howls.


Thursday, April 28th

Dmitry Ioffe , Technion, Israel Institute of Technology

"Low temperature interfaces, ordered random walks and Ferrari-Spohn diffusions"

Time: 12:00 PM
Location: Hill 705
Abstract: I shall discuss scaling limits for a class of ordered random walks subject to positivity constraint and self-potentials, which look like generalized area tilts. Such polymers arise, for instance, as effective models for:

(a) Phase segregation lines in 2D Ising model with negative b.c. and positive magnetic fields.

(b) Level lines of 2+1 discrete SOS models coupled with Bernoulli bulk fields.

The limiting objects happen to be ergodic Ferrari-Spohn diffusions, conditioned on non-intersection. Invariant measures for n such diffusions are given in terms of Slater determinants constructed from n first eigenfunctions of appropriate Sturm-Liouville operators.

Based on joint works with S. Shlosman, Y.Velenik and V.Wachtel ______________________________________ BROWN BAG LUNCH FROM 1-2PM


Thursday, April 21st

Stefano Olla , Universite Paris-Dauphine

"Diffusive macroscopic transport in non-acoustic chains"

Time: 2:00 PM
Location: Hill 705
Abstract: We consider a non acoustic chain of harmonic oscillators with the dynamics perturbed by a random local exchange of momentum, such that energy and momentum are conserved. The macroscopic limits of the energy density, momentum and the curvature (or bending) of the chain satisfy, in a diffusive space-time scaling, an autonomous system of equations. The curvature and momentum evolve following a linear system that corresponds to a damped Euler-Bernoulli beam equation. The macroscopic energy density evolves following a non linear diffusive equation. In particular the energy transfer is diffusive in this dynamics. This provides a first rigorous example of a normal diffusion of energy in a one dimensional dynamics that conserves the momentum. This is also in contrast with the macroscopic behavior in acoustic chains, where there is a clear separation of scale between the relaxation to mechanical equilibrium governed by Euler equations, and to the thermal equilibrium, governed by a super diffusive fractional heat equation.

(Work in collaboration with T. Komorowski).


Thursday, April 21st

Lev Vaidman , Tel Aviv University

"In favor of the many-worlds interpretation of quantum mechanics"

Time: 12:00 PM
Location: Hill 705
Abstract: I will first argue that the picture of quantum mechanics according to which the only ontology is the text-book quantum wave collapsing at measurements and evolving unitarily between them provides a satisfactory explanation of our experiences. Then I will show how the collapse can be removed at the price of admitting multiple parallel experiences. I also will present controversial views regarding Aharonov-Bohm effect and the past of quantum particles.

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


Friday, April 15th

Special Mathematical Physics Seminar

Franca Hoffmann , Cambridge University (SPECIAL SEMINAR!)

"Asymptotic behaviour of diffusing and self-attracting particles"

Time: 3:00 PM
Location: Hill 423
Abstract: We study interacting particles behaving according to a reaction-diffusion equation with non-linear diffusion and non-local attractive interaction. This class of equations has a very nice gradient flow structure that allows us to make links to variations of well-known functional inequalities (Hardy-Littlewood-Sobolev inequality, logarithmic Sobolev inequality). Depending on the non-linearity of the diffusion, the choice of interaction potential and the dimensionality, we obtain different regimes. Our goal is to understand better the asymptotic behaviour of solutions in each of these regimes, starting with the fair-competition regime where attractive and repulsive forces are in balance.

This is joint work with Jose A. Carrillo and Vincent Calvez.


Thursday, April 14th

Alex Kontorovich , Rutgers University

"Reciprocal geodesics"

Time: 2:00 PM
Location: Hill 705
Abstract: Reciprocal geodesics are a special family of closed geodesics on the modular surface which are invariant under the time reversal involution. These correspond to elements of order 4 in a real quadratic ideal class group (as will be defined from first principles), and a basic question is how they are distributed. We will discuss some recent results on this family.


Thursday, April 14th

Sagun Chanillo , Rutgers University

"The Fundamental Theorem of Calculus, Generalizations and Applications"

Time: 12:00 PM
Location: Hill 705
Abstract: This talk has two parts, in the first part we discuss various extensions of the fundamental theorem of Calculus to higher dimension and other geometric structures besides Euclidean space. In the second part we present applications of the first part of our lecture to some equations of Fluid Mechanics and Electromagnetism.

This is joint work with J. Van Schaftingen and Po-lam Yung, a former Hill instructor at Rutgers.

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


Thursday, April 7th

Thomas Chen , University of Texas

"The time-dependent Hartree-Fock-Bogoliubov equations for Bosons"

Time: 2:00 PM
Location: Hill 705
Abstract: In this talk, we discuss the use of quasifree reduction to derive the time-dependent Hartree-Fock-Bogoliubov (HFB) equations describing the dynamics of quantum fluctuations around a Bose-Einstein condensate in R^d . We prove global well-posedness for the HFB equations for sufficiently regular pair interaction potentials, and establish key conservation laws. Moreover, we relate the HFB equations to the HFB eigenvalue equations encountered in the physics literature. Furthermore, we construct the Gibbs states at positive temperature associated with the HFB equations, and establish criteria for the emergence of Bose-Einstein condensation.

This is based on joint work with V. Bach, S. Breteaux, J. Frohlich, and I.M. Sigal.


Thursday, April 7th

Zeev Rudnick , Tel Aviv University and IAS

"Quantum chaos, eigenvalue statistics and the Fibonacci sequence"

Time: 12:00 PM
Location: Hill 705
Abstract: One of the outstanding insights obtained by physicists working on "Quantum Chaos" is a conjectural description of local statistics of the energy levels of simple quantum systems according to crude properties of the dynamics of classical limit, such as integrability, where one expects Poisson statistics, versus chaotic dynamics, where one expects GOE statistics. I will describe in general terms what these conjectures say and discuss recent joint work with Valentin Blomer and Jean Bourgain in which we study the size of the minimal gap between the first N eigenvalues for one such simple integrable system, a rectangular billiard having irrational squared aspect ratio. For certain quadratic irrationalities, such as the golden ratio, we show that the minimal gap is about 1=N, consistent with Poisson statistics. In the case of the golden ratio, the argument involves some curious properties of the Fibonacci sequence. We also find deviations from Poisson statistics.

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


Thursday, March 31st

Joel Lebowitz , Rutgers University

"Fluctuations, Large Deviations and Rigidity in Superhomogeneous Systems"

Time: 12:00 PM
Location: Hill 705
Abstract: Superhomogeneous (a.k.a. hyperuniform) particle systems are point processes on R^d (or Z^d) that are translation invariant (or periodic) and for which the variance of the number of particles in a region V grows slower than the volume of V. Examples include Coulomb systems, determinantal processes with projection kernels and certain perturbed lattice models. I will first review some old work on superhomogeneous systems and and then describe some new work (with Subhro Ghosh) providing sufficient conditions (involving decay of pair correlations) for number rigidity in such systems in dimension d=1,2. A particle system is said to exhibit number rigidity if the probability distribution of the number of particles in a bounded region R, conditioned on the particle configuration in R^c, is concentrated on a single integer N. All known (to us) examples in which number rigidity has been established in d=1,2 satisfy our conditions, and we conjecture that superhomogeneity is also a necessary condition for number rigidity in all dimensions d. On the other hand, it follows from the work of Peres and Sly on perturbed lattice systems that in d>2 there are no such sufficiency conditions involving the decay of correlations.


Thursday, March 24th

Eduardo Sontag, Rutgers University

"Qualitative features of transient responses A case study: scale-invariance"

Time: 2:00 PM
Location: Hill 705
Abstract: An ubiquitous property of sensory systems is "adaptation": a step increase in stimulus triggers an initial change in a biochemical or physiological response, followed by a more gradual relaxation toward a basal, pre-stimulus level. Adaptation helps maintain essential variables within acceptable bounds and allows organisms to readjust themselves to an optimum and non-saturating sensitivity range when faced with a prolonged change in their environment. It has been recently observed that some adapting systems, ranging from bacterial chemotaxis pathways to signal transduction mechanisms in eukaryotes, enjoy a remarkable additional feature: scale invariance or "fold change detection" meaning that the initial, transient behavior remains approximately the same even when the background signal level is scaled. I will review the biological phenomenon, and formulate a theoretical framework leading to a general theorem characterizing scale invariant behavior by equivariant actions on sets of vector fields that satisfy appropriate Lie-algebraic nondegeneracy conditions. The theorem allows one to make experimentally testable predictions, and I will discuss the validation of these predictions using genetically engineered bacteria and microfluidic devices, as well their use as a "dynamical phenotype" for model invalidation. I will conclude by briefly engaging in some wild and irresponsible speculation about the role of the shape of transient responses in immune system self/other recognition and in evaluating the initial effects of cancer immunotherapy.


Thursday, March 24th

Hector Sussmann , Rutgers University

"Analyticity properties of subriemannian geodesics"

Time: 12:00 PM
Location: Hill 705
Abstract: Sub-Riemannian geodesics (i.e., locally length-minimizing paths for s Carnot-Caratheodory metric) are similar to Riemannian geodesics. except that, instead of satisfying a differential equation that implies that they are smooth, they satisfy a differential inclusion that implies smoothness for "most" geodesics, but not for all of them. And for that reason the question whether all sub-Riemannian geodesics are smooth (and analytic for real analytic metrics) has been open for several decades. We prove that that for real analytic sub-Riemannian metrics every geodesic parametrized by arc-length is real analytic on an open dense subset of its interval of definition. We discuss a research program that might lead to stronger regularity results.

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


Thursday, March 10th

Juerg Frohlich , Institute for Theoretical Physics, Zurich

"The Arrow of Time - Images of irreversible behavior, continued"

Time: 2:00 PM
Location: Hill 705
Abstract:


Thursday, March 10th

Juerg Frohlich , Institute for Theoretical Physics, Zurich

"The Arrow of Time - Images of irreversible behavior"

Time: 12:00 PM
Location: Hill 705
Abstract: Examples of irreversible behavior in quantum-mechanical systems are discussed. After a brief recapitulation of important properties of relative entropy, I sketch a derivation of the Second Law of Thermodynamics. I then briefly discuss the origin of diffusive motion in unitary quantum dynamics and present a mechanism for friction. I conclude with an outline of the fundamental irreversibility in quantum theory.

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


Thursday, March 3rd

Federico Bonetto , Georgia Institute of Technology

"Time Evolution and Nonequilibrium Stationary State in a Weakly Driven System"

Time: 2:00 PM
Location: Hill 705
Abstract: We study a system of N particles moving in a chaotic billiard under the influence of an external electric field E. The particles interact only via a mean field friction term that keeps the total kinetic energy exactly constant (a Gaussian thermostat). We are interested in the long time evolution and steady state of this system when the electric field E is small and/or the number of particles N is large.

I will present rigorous and heuristic results we have obtained in the past years in collaboration with E. Carlen, J.Lebowitz, N. Chernov et al.


Thursday, March 3rd

Mykhaylo Shkolnikov , Princeton University

"Edge of beta ensembles and the stochastic Airy semigroup"

Time: 12:00 PM
Location: Hill 705
Abstract: Beta ensembles arise naturally in random matrix theory as a family of point processes, indexed by a parameter beta, which interpolates between the eigenvalue processes of the Gaussian orthogonal, unitary and symplectic ensembles (GOE, GUE and GSE). It is known that, under appropriate scaling, the locations of the rightmost points in a beta ensemble converge to the so-called Airy(beta) process. However, very little information is available on the Airy(beta) process except when beta=2 (the GUE case). I will explain how one can write a distribution-determining family of observables for the Airy(beta) process in terms of a Brownian excursion and a Brownian motion. Along the way, I will introduce the semigroup generated by the stochastic Airy operator of Ramirez, Rider and Virag.

Based on joint work with Vadim Gorin.

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


Thursday, February 25th

Konstantin Mischaikow , Rutgers University

"Dynamic Signatures Generated by Regulatory Networks"

Time: 2:00 PM
Location: Hill 705
Abstract: Consider a regulatory network presented as a directed graph with annotated edges that indicate if the first node is up-regulating or down-regulating the second node. What kind of dynamics can this network generate? While this may seem to be an inadequately posed question it arises fairly often in biological contexts. Our motivation for addressing it arises from gene regulatory networks where we assume that the nodes represent genes and act as switches. However, we do not assume that we know the appropriate parameter values let alone the nonlinear reactions that govern the switches. Nevertheless, as I will describe in this talk, for moderate sized networks we can give a mathematically justifiable, computationally tractable, description of the global dynamics over all possible parameter values.

This is a continuation from last weeks talk (2/18/16) on motivation and mathematical foundations. In this talk we will move from mathematical theory to applications.


Thursday, February 25th

Michael Loss , Georgia Institute of Technology

"The phase diagram of the Cafarelli-Kohn-Nirenberg inequalities"

Time: 12:00 PM
Location: Hill 705
Abstract: The Cafarelli-Kohn-Nirenberg inequalities form a two parameter family of inequalities that interpolate between Sobolev's inequality and Hardy's inequality. The functional whose minimization yields the sharp constant is invariant under rotations. It has been known for some time that there is a region in parameter space where the optimizers for the sharp constant are not radial. In this talk I indicate a proof that, in the remaining parameter region, the optimizers are in fact radial. The proof will proceed via a well chosen flow that decreases the functional unless the function is a radial optimizer.

This is joint work with Jean Dolbeault and Maria Esteban.

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


Thursday, February 18th

Fioralba Cakoni, Rutgers University

"Transmission Eigenvalues in Scattering Theory"

Time: 2:00 PM
Location: Hill 705
Abstract: The transmission eigenvalue problem is a non-self adjoint and non-linear eigenvalue problem that is a relatively late arrival to the spectral theory of partial differential equations. It arises in the study of the injectivity of the far field operator corresponding to the scattering of incident plane waves by an inhomogeneous media of compact support. More specifically, transmission eigenvalues are related to so-called non-scattering frequencies for which it is possible to construct an incident wave that, for a given inhomogeneity, does not scatter. These eigenvalues are observable from the scattering operator and hence can be used to obtain information about the scattering media.

In this lecture we describe how the transmission eigenvalue problem arises in scattering theory, how transmission eigenvalues can be determined from multi-frequency scattering data and what is known mathematically about these eigenvalues.


Thursday, February 18th

Konstantin Mischaikow , Rutgers University

"Dynamic Signatures Generated by Regulatory Networks"

Time: 12:00 PM
Location: Hill 705
Abstract: Consider a regulatory network presented as a directed graph with annotated edges that indicate if the first node is up-regulating or down-regulating the second node. What kind of dynamics can this network generate? While this may seem to be an inadequately posed question it arises fairly often in biological contexts. Our motivation for addressing it arises from gene regulatory networks where we assume that the nodes represent genes and act as switches. However, we do not assume that we know the appropriate parameter values let alone the nonlinear reactions that govern the switches. Nevertheless, as I will describe in this talk, for moderate sized networks we can give a mathematically justifiable, computationally tractable, description of the global dynamics over all possible parameter values.

The focus of the first talk will be on motivation and mathematical foundations. In the second talk (on February 25th) we will move from mathematical theory to applications.

BROWN BAG LUNCH FROM 1-2PM!!


Thursday, February 11th

Roger Nussbaum , Rutgers University

"Analyticity Versus Infinite Differentiability: Case Studies and Open Questions"

Time: 12:00 PM
Location: Hill 705
Abstract: This talk considers differential-delay equations, but no prior knowledge will be assumed.

The topics to be discussed represent joint work with John Mallet-Paret.

If x: (-(infinity), T) --> (complex numbers) is infinitely differentiable and bounded, let N denote the subset of the domain of x(.) where x is not real analytic and A denote the set of points t in the domain such that x(.) is real analytic on an open neighborhood of t. We shall describe a simple-looking class of differential-delay equations, defined by real analytic functions. The equations in question have periodic solutions x(.) for which the set N is uncountable and the set A is open and nonempty. Under slightly more restrictive assumptions, N has empty interior. Time permitting, we shall also consider a class of differential delay equations which have bounded, infinitely differentiable solutions x(.) defined on (-(infinity), (infinity)). The solutions in question also have a nonzero limit as t approaches (-infinity). We conjecture that these solutions are nowhere real analytic, but no proofs have been given even for simple equations like x'(t) = exp(it^2)x(t-1)or x'(t) = sin(t^2)x(t-1).


Thursday, February 4th

Eugene Speer, Rutgers University

"Translation invariant extensions of finite volume measures"

Time: 2:00 PM
Location: Hill 705
Abstract: Given a probability measure on the set of particle configurations on a finite subset of a lattice, can it be extended to a translation invariant measure on configurations on the entire lattice? When the answer is yes, what are the properties, e.g., the entropy, of such an extension? We give reasonably complete answers for the one-dimensional case; in higher dimensions we can say much less.

This is joint work with S. Goldstein, J. L. Lebowitz, and T. Kuna.


Thursday, February 4th

Haim Brezis , Rutgers University

"Sobolev inequalities: the work of Sobolev, Nirenberg and beyond"

Time: 12:00 PM
Location: Hill 705
Abstract: The lecture is concerned with various aspects of the Sobolev inequalities on R^N in the border line cases p=1 and p=N, starting with the contributions of S. Sobolev and L. Nirenberg. I will explain how new results when p=N are connected to the solvability of linear elliptic equations with data in L^1 (based joint work with J. Bourgain). I also plan to discuss elliptic equations involving critical nonlinearities (revisiting a joint work with L. Nirenberg)

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


Thursday, January 28th

Denys Bonder , Princeton University

"Measurement inspired modeling of quantum and classical dynamical systems"

Time: 2:00 PM
Location: Hill 705
Abstract: In this talk, I will provide an answer to the question:

"What kind of observations and assumptions are minimally needed to formulate a physical theory?" Our answer to this question leads to the new systematic approach of Operational Dynamical Modeling (ODM), which allows to deduce equations of motions from time evolution of observables. Using ODM, we are not only able to re-derive well-known physical theories (such as the Schrodinger and classical Liouville equations), but also infer novel physical dynamics (and solve open problems) in the realm of non-equilibrium quantum statistical mechanics.


Thursday, January 28th

Roderich Tumulka , Rutgers University

"Probability Distribution of the Time at Which an Ideal Detector Clicks"

Time: 12:00 PM
Location: Hill 705
Abstract: We consider a non-relativistic quantum particle surrounded by a detecting surface and ask how to compute, from the particle's initial wave function, the probability distribution of the time and place at which the particle gets detected. In principle, quantum mechanics makes a prediction for this distribution by solving the Schrodinger equation of the particle of interest together with the 10^23 (or more) particles of the detectors, but this is impractical to compute. Is there a simple rule for computing this distribution approximately for idealized detectors? I will argue in favor of a particular proposal of such a rule, the "absorbing boundary rule," which is based on a 1-particle Schrodinger equation with a certain "absorbing" boundary condition on the detecting surface. The mere existence of such a rule may seem surprising in view of the quantum Zeno effect. Time permitting, I may also be able to explain extensions of this rule to the cases of several particles, moving detectors, particles with spin, Dirac particles, curved space-time, and discrete space (a lattice).

Some of the results are based on joint work with Abhishek Dhar and Stefan Teufel.

BROWN BAG LUNCH BETWEEN 1-2PM


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