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, 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 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 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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