Award 1501007, Variational Problems and Dynamics, was active from May 31, 2015 through October 31, 2020 on a no-cost extension of the original 3 year grant. The project focused on proving and applying functional inequalities in a range of problems coming from physics and quantum information theory. In addition to the PI, the grant supported 3 Rutgers graduate students: Francis Seuffert, David Herrera and Erik Amorim. Francis Seuffert, who completed his PhD under my direction in Spring 2017,  is now a postdoctoral fellow at University of Pennsylvania. David Herrera is a current graduate student working on his PhD under my direction.  Erik Amorim, who was a  thesis student of a colleague, defended his thesis in Fall 2020.  Although Amorim was not my thesis student, he and the PI worked closely, resulting in a significant publication in Linear Algebra and its Applications. The final no-cost extension permitted planned work with Herrera and Amorim to be completed, and provided the means to continue collaborating after the pandemic precluded meeting in my office. 

The analysis of the equations that govern physical processes often requires a precise quantitative understanding of the relative sizes of various quantities involved in these processes, and this is provided by mathematical inequalities, often of a geometric nature. The quest for a better understanding of these processes is in part the quest for new and more precise mathematical inequalities, and that was the focus of the research supported by this ward. The supported research has resulted in 21 publications, all acknowledging support.  Two of these were by supported graduate students, and one was a joint work of the PI and a graduate student. The balance were papers by the PI and his collaborators. Seven of the papers deal with problems in kinetic theory; that is, the study of the Boltzmann equation which describes the behavior of rarefied gasses. Of particular interest in this group is the paper published by the PI and Carvalho and Loss in Annals of Probability, to appear in 2021. This paper culminates a long line of research by these authors going back almost 20 years, and it solves an old conjecture of Mark Kac for the physically significant case of hard sphere collisions in three dimensional space. Also of note in this area in the work by the same authors on a quantum version of the Kac model. Another focus was on problems coming from quantum information theory. Here, a particularly significant result is the paper by Carlen and Mass in Journal of Functional Analysis, 2017, which provides a framework for writing certain Quantum Markov Semigroups, objects used to describe the evolution of quantum system out of equilibrium, as "steepest descent" with respect to the quantum relative entropy. Moreover, in the same paper this framework was then used to prove some new sharp entropy inequalities, one of which had recently been conjectured by people working in quantum information. 

The results in these 21 papers relate to a wide range of problems, and that is reflected in the fact that the papers are published not only in mathematics journals, but also physics journals, and one is published in IEEE transactions on information theory. The broader impact of the award lies partly in this direct connection to fields outside of mathematics and its further potential for application in other fields. It also lies  in the involvement of graduate students, contributing to training of the next generation of researchers.  Finally, the grant has permitted the PI to travel widely not only to collaborate, but to disseminate the results at many meeting in the U.S. and abroad. In particular, the grant covered travel making possible visits by the PI to the Mittag-Leffler Institute in Sweden, the Oberwolfach Mathematics Research Institute in Germany and the Shroedinger Institute in Austria, among others,  that were particularly important for dissemination of the results. 

The intellectual merit of the proposed research lies in the fact that mathematical issues that were investigated are challenging and interesting as mathematics per se, but also have been motivated by a wide range of applied problems, so that the results can be expected to be broadly useful to other researchers.

Last Modified: 01/23/2021
Modified by: Eric A Carlen