Georgia Institute of Technology College of Sciences

We introduce the notion of TDFOE studying the thermodynamic formalism on topological Markov shifts where the potential is random, depending on a Gibbs random walk on a compact metric space.

We introduce the associated Pressure Out of Equilibrium, and discuss its properties (e.g. variational principle), and applications (Azuma inequality for potentials expanding on average w.r.t. to a Gibbs measure).

We discuss the associated Semi-Ruelle operator which generates the POE, and its properties (e.g. existence of conformal measures, spectral radius). We provide an application by studying Effectively Expanding on Average diffeomorphisms on closed manifolds (this is a $C^1$-open condition, extends previously studied conditions).

We prove quasi-compactness (and later a spectral gap) for the Averaged Semi-Ruelle operator on a Sobolev space.

Notably, the diffeos are allowed to be highly dissipative. Finally, we mention a few applications such as: Dimension bounds on the stationary measure, and CLT and LLT for the (not necessarily invariant) volume measure.

On September 8, OpenAI announced a Lean-certified resolution of the Navier-Stokes Millennium Problem by furnishing a classical solution to the forced 3d Navier-Stokes equations which blows up at the origin in finite time. The purpose of this talk will be to provide (i) historical context and relevance for the Millennium Problem; and (ii) give an overall view of the techniques developed towards its resolution from the last ten years or so. I will avoid technical details, and endeavor to make the talk accessible to those outside PDE. While I will briefly address the ongoing priority dispute and allegations of misconduct by OpenAI, this will not be the focus of the talk. 

Given a link in a closed 3-manifold, which Dehn surgery multislopes give rise to 3-manifolds with taut foliations? In this talk, I will discuss the ziggurat phenomenon: restricting to those foliations transverse to a fixed flow on the link complement, the set of multislopes typically has a fractal staircase shape with rational corners. In joint work with Thomas Massoni, we explain the ziggurat phenomenon in some contexts using tools from contact geometry. For those who have already seen a talk about this work, I'll try to emphasize some different aspects.

Abstract: In the early 2000s, substantial work was devoted to understanding the generally distributed processor-sharing queue, which can be viewed as an idealization of round-robin or time-sharing protocols arising in computer-system applications. The fluid limit for a processor-sharing queue with reneging was established, but the diffusion limit for a processor-sharing queue with impatience was obtained only under a “soft deadlines” formulation, in which the patience times of jobs are tracked but jobs do not actually leave the queue when their patience times expire. This limitation was largely a consequence of the methodology underlying that research program: much of the analysis relied on heavy-traffic limits and the state-space-collapse framework introduced by Bramson and Williams, which is not naturally suited to systems with reneging. More recent work has developed methods for obtaining diffusion approximations of measure-valued queueing systems using classical central limit theorem analysis, martingale methods, and SPDE-valued limits (see work by Ramanan and by the author of this talk). Using these methods, the diffusion approximation for a processor-sharing queue with true reneging can be obtained.

Bio: Eva Loeser is a postdoctoral research associate in the Applied Probability Group in the Department of Statistics and Operations Research at the University of North Carolina at Chapel Hill. Her research interests include stochastic processes, particularly fluid and diffusion approximations of stochastic systems, high-dimensional stochastic processes, and universal limiting objects such as stochastic partial differential equations (SPDEs) and semimartingale reflecting Brownian motions (SRBMs). The models she studies arise in applications including systems biology, computer systems, queueing theory, operations research, and mathematical physics.