Mathematics Department - Mathematical Physics Seminar - Fall 2016

# Mathematical Physics Seminar - Fall 2016

### Organizer(s)

Joel Lebowitz, Michael Kiessling

### Website

http://www.sas.rutgers.edu/cms/math/news-events-cmsr/mathematical-physics-seminar/range.listevents/-

## Thursday, December 8th

### "Uniform statistical properties of chaotic dynamical systems"

Time: 2:00 PM
Location: Hill 705
Abstract: I am going to talk about problems which come from the study of deterministic fast-slow systems: statistical limit laws for Birkhoff sums in families of dynamical systems. For example, the central limit theorem or the weak invariance principle. I plan to present our recent results with Zemer Kosloff and Ian Melbourne which allow us to treat nonuniformly hyperbolic maps such as Pomeau-Manneville, logistic and Viana maps, and externally forced Sinai billiards.

## Thursday, December 8th

### "Non-local functionals and Image Processing"

Time: 12:00 PM
Location: Hill 705
Abstract: I will present new results concerning the convergence of non-local, non-convex functionals to the total variation. The mode of convergence is extremely delicate and numerous problems remain open. De Giorgi's concept of Gamma convergence illuminates the situation but also introduces mysterious novelties. The original motivation comes from Image Processing.

This is joint work with Hoai-Minh Nguyen.

## Thursday, December 1st

### "Unveiling Navier-Stokes equation properties from computer simulations"

Time: 12:00 PM
Location: Hill 705
Abstract: We use extensive computer simulations of hard disks under the action of a temperature gradient and a constant gravity force to understand some basic elements of the definition of NS equations. Namely, the local equilibrium hypothesis or the structure of the constitutive relations. Moreover, we study the behavior of several observables and fields at the crossing from a non-convective to a convective state. In particular, we see that the temperature, density and velocity fields in convective states can be, each of them, rescaled into a universal field that only depends on the gravity value.

## Tuesday, November 22nd

### "On the Quantum Mechanics of a Single Photon"

Time: 2:00 PM
Location: Hill 705
Abstract: I will report on some recent joint work with Michael Kiessling, in which we show that a Dirac-type equation for a rank-two bi-spinor field furnishes a Lorentz-covariant quantum-mechanical wave equation for a single free photon in position-space representation. This equation does not encounter any of the roadblocks that obstructed previous attempts by other authors to formulate such a photon wave equation. After vindicating our equation by showing that it leads to the correct dispersion relation and the absence of longitudinal modes, I will derive all the relevant conservation laws, and identify among them a nonnegative quantity that we believe represents the conditional probability density of finding a photon at a given point on a given null hypersurface in physical spacetime. I will end by discussing some preliminary work we have done towards incorporating interactions into our model.

## Tuesday, November 22nd

### "New developments in the summation of divergent series"

Time: 12:00 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.

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

## Thursday, November 17th

### "Applications of modular forms to sphere packing"

Time: 2:00 PM
Location: Hill 705
Abstract: I'll describe some recent joint work with Henry Cohn, Abhinav Kumar, Danylo Radchenko, and Maryna Viazovska which use modular form constructions to get sharp bounds on point configuration problems in R^n, such as the solution to the sphere packing problem in n=24 dimensions. Time permitting, I'll also discuss some recent results about energy minimization in n=8 dimensions for a class of potential functions.

## Thursday, November 17th

### "COMPACTNESS AND LARGE DEVIATIONS"

Time: 12:00 PM
Location: Hill 705
Abstract: In a reasonable topological space, large deviation estimates essentially deal with probabilities of events that are asymptotically (exponentially) small, and in a certain sense, quantify the rate of these decaying probabilities. In such estimates, upper bounds for such small probabilities often require compactness of the ambient space, which is often absent in problems arising in statistical mechanics (for example, distributions of local times of Brownian motion in the full space Rd). Motivated by such a problem, we present a robust theory of translation-invariant compacti cation" of probability measures in Rd. Thanks to an inherent shift-invariance of the underlying problem, we are able to apply this abstract theory painlessly and solve a long standing problem in statistical mechanics, the mean- eld polaron problem.

This talk is based on joint works with S. R. S. Varadhan (New York), as well as with Erwin Bolthausen (Zurich) and Wolfgang Koenig (Berlin).

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

## Thursday, November 10th

### "The Formation of Shock Waves in the Presence of Vorticity"

Time: 12:00 PM
Location: Hill 705
Abstract: *** --- Please note that this is part of "A Day of Revolutionary Thinking" which is a university-wide showcase of alumni success and scholarship marking Rutgers’ 250th birthday.

Learn about new results on the formation of shock singularities in vorticity-containing solutions to the compressible Euler equation.of the formation of shock waves, starting from small, regular initial conditions, in solutions to the relativistic Euler equations.

In 2014, Christodoulou-Miao extended the result to the non-relativistic compressible Euler equations.

In both works, the assumptions on the initial conditions caused the shock to form in the acoustic wave zone, far from the region where vorticity is present. Consequently, Christodoulou and Miao were able to use the potential formulation of the Euler equations to study the shock formation.

In my talk, I will describe my recent joint work with J. Luk, in which we prove a similar shock formation result for the compressible Euler equations, but allowing for small, non-zero vorticity in a neighborhood of the shock. To control the vorticity up to the shock, we rely on a coalition of new geometric and analytic insights that complement the ones used by Christodoulou and Miao. In particular, since the potential formulation is not available, we rely on our new formulation of the compressible Euler equations. The new formulation exhibits remarkable nonlinear null structures, reminiscent of the type found in geometric field theories. By exploiting these structures, we are able to prove that the vorticity remains uniformly bounded up to the shock and does not interfere with the singularity formation mechanisms. Thus, our work yields the first constructive description of the behavior of vorticity in a neighborhood of a singularity formed from compression. Moreover, our work provides the first constructive proof of stable blowup without symmetry assumptions for a quasilinear hyperbolic system featuring multiple speeds of propagation: the speed of sound and the speed associated to vorticity propagation.

## Thursday, November 3rd

### "Longing for independence"

Time: 2:00 PM
Location: Hill 705
Abstract: I'll mainly discuss one simple combinatorial example of a common state of affairs: it would be nice to be able to say that some quantities of interest behave (more or less) as if they were independent. While we still can't show the behavior we think should hold for the question in question, what we do know is at least enough to settle an old conjecture of Alon and Spencer.

## Thursday, November 3rd

### "Some Remarks on the Asymptotic Behavior of Singular Solutions to Some Geometric PDEs"

Time: 12:00 PM
Location: Hill 705
Abstract: Many geometric problems or their resolution involve the study of singular solutions to some relevant geometric equations, even if the problems themselves do not directly involve singular solutions, as singular solutions may arise in the analysis of limits of regular solutions. One basic question is whether there is a precise description of the behavior of the solution near its singular set. The study of this kind of question also leads to, through zooming in near the singular set --- a typical procedure for such problems, the study of singular solutions defined on the entire Euclidean space with some lower dimensional Euclidean space removed; in particular, whether such limiting solutions are unique, or are from a finite parameter of family of known solutions. There are no good answers to such general questions, but exploitation of covariance properties of certain geometric PDEs has led to some good progress to these questions for the relevant PDEs. I will use a few examples to illustrate how the questions are formulated and analyzed.

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

## Thursday, October 27th

### "DETAILED BALANCE IN NONEQUILIBRIUM STAT MECH"

Time: 12:00 PM
Location: Hill 705
Abstract: Detailed balance relates the transition probabilities J -> K and K -> J between two states J,K of a system M in equilibrium with a bath at a certain temperature. This relation is based on time-reversal invariance of physical laws. It can give very specific answers to certain questions. We discuss detailed balance in the case of an “active” bath (containing chemical with which our system M can react). This situation has been considered by J. England, and remains intriguing and interesting.

## Thursday, October 20th

### "Non-equilibrium Dynamics of Quantum Integrable Systems"

Time: 2:00 PM
Location: Hill 705
Abstract: The study of non-equilibrium dynamics of interacting many body systems is currently one of the main challenges of modern condensed matter physics, driven by the spectacular progress in the ability to create experimental systems - trapped cold atomic gases are a prime example - that can be isolated from their environment and be highly controlled. Many of the system so studied are integrable. In this talk I will describe nonequilibrium quench and Floquet dynamics in some integrable quantum systems. I'll discuss the time evolution of the Lieb-Liniger system, a gas of interacting bosons moving on the continuous infinite line and interacting via a short range potential. Considering a finite number of bosons on the line we find that for any value of repulsive coupling the system asymptotes towards a strongly repulsive gas for any initial state, while for an attractive coupling, the system forms a maximal bound state that dominates at longer times. In the thermodynamic limit -with the number of bosons and the system size sent to infinity at a constant density and the long time limit taken subsequently- I'll show that the density and density-density correlation functions for strong but finite positive coupling are described by GGE for translationally invariant initial states with short range correlations. As examples I’ll discuss quenches from a Mott insulator initial state or a Newton’s Cradle. Then I will show that if the initial state is strongly non translational invariant, e.g. a domain wall configuration, the system does not equilibrate but evolves into a nonequilibrium steady state (NESS). A related NESS arises when the quench consists of coupling a quantum dot to two leads held at different chemical potential, leading in the long time limit to a steady state current. I will also present some results on Floquet dynamics for interacting bosons. Time permitting I will discuss the quench dynamics of the XXZ Heisenberg chain.

## Thursday, October 20th

### "Steady States in the non local Luttinger Model"

Time: 12:00 PM
Location: Hill 705
Abstract: We study transport in a one-dimensional system of interacting fermions described by the Luttinger model with finite range interactions, with a domain wall initial state with different densities or temperatures on its left and right sides. Asymptotically in time the system approaches a translation invariant steady state carrying a non vanishing current showing universality properties. The non-locality of interaction, acting as an ultraviolet cut-off, breaks Lorentz and scale invariance and leads to dispersive effects manifested by the shape of the fronts changing with time.

Based on joint work with E.Langmann, J.Lebowitz, P. Moosavi

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

## Thursday, October 13th

### "Universality of Transport Coefficients in the Haldane-Hubbard Model"

Time: 2:00 PM
Location: Hill 705
Abstract: In this talk I will review some selected aspects of the theory of interacting electrons on the honeycomb lattice, with special emphasis on the Haldane-Hubbard model: this is a model for interacting electrons on the hexagonal lattice, in the presence of nearest and next-to-nearest neighbor hopping, as well as of a transverse dipolar magnetic field. I will discuss the key properties of its phase diagram, most notably the phase transition from a standard insulating phase to a Chern insulator, across a critical line, where the system exhibits semi-metallic behavior. I will also review the universality of its transport coefficients, including the quantization of the transverse conductivity within the gapped phases, and that of the longitudinal conductivity on the critical line. The methods of proof combine constructive Renormalization Group methods with the use of Ward Identities and the Schwinger-Dyson equation.

Based on joint works with Vieri Mastropietro, Marcello Porta, Ian Jauslin.

## Thursday, October 13th

### "Ergodicity: an early paper by Boltzmann and its relevance"

Time: 12:00 PM
Location: Hill 705
Abstract: A little known 1868 paper by Boltzmann illustrates the ergodic hypotesis in the sense of Boltzmann and Maxwell, after having a little earlier derived the microcanonical distribution. The property should be illustrated by a simple dynamical system: his results ends up connecting to modern KAM theory and to the ancient theory of conics. They seem to indicate the need for a revisitation of the properties of the Hamiltonian system considered.

BROWN BAG LUNCH FROM 1-2PM

## Thursday, October 6th

### "Relativistic zero-range interactions for multi-time wave functions"

Time: 2:00 PM
Location: Hill 705
Abstract: Multi-time wave functions, a concept introduced by Dirac in 1932, are quantum-mechanical wave functions with N space-time arguments for N particles. In this way, they naturally extend the non-relativistic (single-time) Schrödinger picture to the relativistic domain. However, because of the many time coordinates, the nature of time evolution changes. Setting up an interacting multi-time theory has proven a major challenge; a recent no-go theorem by Petrat and Tumulka e.g. excludes interaction by potentials. In this talk, I will present a multi-time model which achieves interaction in a different way, namley by zero-range (or delta) interactions. After briefly introducing the general multi-time formalism, I will give an overview of the main results and an idea of the novel techniques required to implement zero-range interactions in this setting.

## Thursday, October 6th

### "The truncated moment problem"

Time: 12:00 PM
Location: Hill 705
Abstract: Let K be a subset of the real numbers. The (one-dimensional) truncated moment problem on K is to find, for given numbers m_1,...,m_n, a random variable X which takes values in K and whose moments are given by the m_k: E[X^k]=m_k. More accurately, one wants to find necessary and sufficient conditions, in term of the m_k, for the existence of such a random variable. The multi-dimensional version of this problem, in which K is a subset of a Euclidean space of higher dimension, is surprisingly hard and is far from being resolved; we give a short introduction to the problem and to the state of the art.

Finally, we describe a recent result concerning the truncated moment problem for a discrete set in one dimension; this is work in collaboration with M. Infusino, J. Lebowitz, and E. Speer.

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

## Thursday, September 29th

### "Almost surely recurrent motions in the Euclidean space"

Time: 2:00 PM
Location: Hill 705
Abstract: We will show that measure-preserving transformations of$R^n$ are recurrent if they satisfy a certain growth condition depending on the dimension $n$. Moreover, it is also shown that this condition is sharp. Examples will include non-autonomous Hamiltonian systems $dot{z}=Jnabla_z H(t, z)$ of one degree of freedom and $T$-periodic in $t$, for which our result will imply the existence of a periodic solution, provided that $nabla_z H(t, z) ={cal O}(|z|^{-alpha})$ as $|z|toinfty$ for some $alpha>1$ uniformly in $t$.

This is joint work with Rafael Ortega (Granada).

## Thursday, September 29th

### "Macroscopic temperature profiles in non-equilibrium stationary states"

Time: 12:00 PM
Location: Hill 705
Abstract: Systems that have more than one conserved quantity (i.e. energy plus momentum, density etc.), can exhibit quite interesting temperature profiles. I will present some numerical experiment and mathematical result.

THERE WILL BE A BROWN BAG LUNCH FROM 1-2:00PM

## Thursday, September 22nd

### "Emergence of a nematic phase in a system of hard plates in three dimensions with discrete orientations"

Time: 2:00 PM
Location: Hill 705
Abstract: We consider a system of hard parallelepipedes, which we call plates, of size 1 by k^a by k in which a is larger than 5/6 and no larger than 1. Each plate is in one of six orthogonal allowed orientations. We prove that, when the density of plates is sufficiently larger than k^(2-5a) and sufficiently smaller than k^(3-a), the rotational symmetry of the system is broken, but its translational invariance is not. In other words, the system is in a nematic phase. The argument is based on a two-scale cluster expansion, and uses ideas from the Pirogov-Sinai construction.

## Thursday, September 22nd

### "A Gaussian Gibbs variational principle and geometric inequalities"

Time: 12:00 PM
Location: Hill 705
Abstract: The Gibbs variational principle has been a cornerstone of statistical mechanics since at least J. W. Gibbs' seminal 1902 treatise. It has also proved to be remarkably useful in other areas of mathematics, such as in the study of geometric inequalities of Brunn-Minkowski and Brascamp-Lieb type. This fundamental connection, pioneered by C. Borell, is however not sufficiently powerful to obtain the sharp isoperimetric and Brunn-Minkowski inequalities for Gaussian measures. In this talk, I will describe an unexpected Gaussian refinement of the Gibbs variational principle that makes it possible to recover these sharp inequalities. I will aim to explain how this gives rise to new Gaussian inequalities---in particular, a Gaussian improvement of Barthe's reverse Brascamp-Lieb inequality---and why the apparent duality between the Prekopa-Leindler and Holder inequalities is manifestly absent in the Gaussian setting.

BROWN BAG LUNCH.....1-2:00pm

## Thursday, September 15th

### "Relative Entropy Principles with Complex Measures"

Time: 2:00 PM
Location: Hill 705
Abstract: The notion of relative entropy for probabilitiy measures relative to a given a-priori probability measure is generalized to signed and complex measures relative to a given a-priori signed measure. This generalization is motivated by some problems at the intersection of statistical mechanics and differential geometry. Computer algebra-produced evidence for the utility of this new notion is presented using the example of complex random polynomials.

## Thursday, September 15th

### "Eigenvalues of sums and products of matrices"

Time: 12:00 PM
Location: Hill 705
Abstract: Given two Hermitian real nxn matrices, given their eigenvalues what can one say about the eigenvalues of the sum? Similar, given two unitary nxn matrices what can one say about the eigenvalues of the product?

I will survey some results on this, joint with Agnihotri and Knutson-Tao, and talk about some more recent results by others on generalizing these results to other kinds of matrices.

(Joint with Agnihotri and separately Knutson and Tao)

BROWN BAG LUNCH FROM 1-2PM

## Thursday, September 8th

### "A `liquid-solid' phase transition in a simple model for swarming"

Time: 12:00 PM
Location: Hill 705
Abstract: We consider a non-local shape optimization problem, which is motivated by a simple model for swarming and other self-assembly/aggregation models, and prove the existence of different phases. A technical key ingredient, which we establish, is that a strictly subharmonic function cannot be constant on a set of positive measure.

(With Rupert Frank)

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