Georgia Institute of Technology College of Sciences

The following is a well-known and difficult problem in rare event simulation: given a set and a Gaussian distribution, estimate the probability that a sample from the Gaussian distribution falls outside the set. Previous approaches to this question are generally inefficient in high dimensions. One key challenge with this problem is that the probability of interest is normally extremely small. I'll discuss a new, provably efficient method to solve this problem for a general polytope and general Gaussian distribution. Moreover, in practice, the algorithm seems to substantially outperform our theoretical guarantees and we conjecture that our analysis is not tight. Proving the desired efficiency relies on a careful analysis of (highly) correlated functions of a Gaussian random vector.Joint work with Ton Dieker.

Thermodynamics provides a robust conceptual framework and set of laws that govern the exchange of energy and matter. Although these laws were originally articulated for macroscopic objects, it is hard to deny that nanoscale systems, as well, often exhibit “thermodynamic-like” behavior. To what extent can the venerable laws of thermodynamics be scaled down to apply to individual microscopic systems, and what new features emerge at the nanoscale? I will review recent progress toward answering these questions, with a focus on the second law of thermodynamics. I will argue that the inequalities ordinarily used to express the second law can be replaced by stronger equalities, known as fluctuation relations, which relate equilibrium properties to far-from-equilibrium fluctuations. The discovery and experimental validation of these relations has stimulated interest in the feedback control of small systems, the closely related Maxwell demon paradox, and the interpretation of the thermodynamic arrow of time. These developments have led to new tools for the analysis of non-equilibrium experiments and simulations, and they have refined our understanding of irreversibility and the second law. Bio Chris Jarzynski received an AB degree in physics from Princeton University in 1987, and a PhD in physics from the University of California, Berkeley in 1994. After postdoctoral positions at the University of Washington in Seattle and at Los Alamos National Laboratory in New Mexico, he became a staff member in the Theoretical Division at Los Alamos. In 2006, he moved to the University of Maryland, College Park, where he is now a Distinguished University Professor in the Department of Chemistry and Biochemistry, with joint appointments in the Institute for Physical Science and Technology and the Department of Physics. His research is primarily in the area of nonequilibrium statistical physics, where he has contributed to an understanding of how the laws of thermodynamics apply to nanoscale systems. He has been the recipient of a Fulbright Fellowship, the 2005 Sackler Prize in the Physical Sciences, and the 2019 Lars Onsager Prize in Theoretical Statistical Physics. He is a Fellow of the American Physical Society and the American Academy of Arts and Sciences. Contact: Professor Jennifer Curtis Email: jennifer.curtis@physics.gatech.edu

It is a classical theorem in algebraic topology that the loop space of a suitable Lie group can be modeled by an infinite dimensional variety, called the loop Grassmannian. It is also well known that there is an algebraic analog of loop Grassmannians, known as the affine Grassmannian of an algebraic groop (this is an ind-variety). I will explain how in motivic homotopy theory, the topological result has the "expected" analog: the Gm-loop space of a suitable algebraic group is A^1-equivalent to the affine Grassmannian.

I will review various ways of modeling the homotopy theory of spaces: several model categories of simplicial sheaves and simplicial presheaves, and related infinity categorical constructions.