The concept of entropy was introduced into ergodic theory by Kolmogorov in the late 1950s and has become a pervasive tool in dynamics with applications to areas such as Riemannian geometry, analytic number theory, and Diophantine approximation. Kolmogorov's approach is based on Shannon's theory of information from the 1940s and is most generally applicable to actions of groups satisfying a kind of internal finite approximation property called amenability.
A new approach to entropy in dynamics was initiated by Bowen some ten years ago and then further developed by Li and myself. In this case one externalizes the finite approximation of the dynamics so that it occurs outside the acting group, and then counts these models in the spirit of Boltzmann's work in statistical mechanics. This notion of entropy applies to the much larger class of acting groups satisfying the property of soficity, which includes free groups. In fact it is not known whether nonsofic groups exist.
I will discuss some of the basic ideas at play in both amenable and sofic entropy, and describe how the passage from single transformations to actions of general amenable and sofic groups marks a shift in applications away from geometry and smooth dynamics and more towards noncommutative harmonic analysis and operator algebras.

September 14 3:30 PM SDRICH 207

Marcelo Disconzi (Vanderbilt University) Viscous Fluids in General Relativity

We consider the problem of describing relativistic viscous fluids. More specifically, we study Einstein's equations coupled to a relativistic version of the Navier-Stokes equations. After motivating the problem and reviewing its history, we present recent results about well-posedness and causality of the equations of motion. If time allows, applications to physics will be briefly discussed.

October 11 3:30 PM SDRICH 207

Frank Morgan (Williams College) The Isoperimetric Problem with Density

The fact that the round sphere provides the least-perimeter way to enclose given volume in Euclidean space has inspired millennia of theory and applications. What happens if you give space a density, weighting both perimeter and volume? Interest in such densities has exploded since their appearance in Perelman's proof of the Poincaré Conjecture. We'll discuss some conjectures and recent results, some by undergraduates.

November 9 3:30 PM SDRICH 207

Lotfi Hermi (Florida International University) Isoperimetric Inequalities for Wedge-Like Domains

By introducing geometric factors and physical parameters that lend themselves to the Payne interpretation in Weinstein fractional space, we prove new isoperimetric inequalities for the fundamental eigenvalue of the Dirichlet problem for wedge-like membranes in two dimensions.
These inequalities generalize and improve sharp inequalities that date back to the works of Payne-Weinberger (1960), Payne-Rayner (1973), Crooke-Sperb (1978), and Chiti (1982). We also motivate, conjecture, and prove an isoperimetric inequality relating the fundamental eigenvalue of a wedge-like membrane to its ``relative torsional rigidity'', which we introduce. This new sharp universal inequality beats both Faber-Krahn and Payne-Weinberger at the isoperimetric game, which we will show numerically for certain triangles. The central tool of this recent development is the use of a new weighted form of the Kohler-Jobin symmetrization method, which we develop. Similar considerations can be undertaken for convex cones in higher dimensions.