Georgia Institute of Technology College of Sciences

Given a collection of multivariate polynomials $F, F_1, …F_n$, to determine whether $F$ can be written as $F= \sum G_iF_i$ is an important problem, both in theory and practice. In this talk, I will explain how some problems of this nature, even in 2 or 3 variables, can become interesting and difficult. An unexpected connection to a conjecture on sumsets will be explained. As the time of writing, the conjecture is unsolved by both humans and AIs.  

In cosmology, a basic explanation of the observed concentration of mass in singular structures is provided by the Zeldovich approximation, which takes the form of free-streaming flow for perturbations of a uniform Einstein-de Sitter universe in co-moving coordinates. The adhesion model suppresses multi-streaming by introducing viscosity. We study mass flow in this model by analysis of Lagrangian advection in the zero-viscosity limit. Under mild conditions, we show that a unique limiting Lagrangian semi-flow exists. Limiting particle paths stick together after collision and are characterized uniquely by a differential inclusion. The absolutely continuous part of the mass measure satisfies a Monge-Ampère equation related to convexification of the free-streaming velocity potential.

The use of Monge-Ampère equations and optimal transport theory for the reconstruction of inverse Lagrangian maps in cosmology was introduced in work of Brenier and Frisch et al (2003). We show that the singular part of the mass measure can differ from the Alexandrov solution to the Monge-Ampère equation, however, when flows along singular structures merge, as shown by analysis of a 2D Riemann problem. In a neighborhood of merging singular structures in our examples, we show that reconstruction yielding a monotone Lagrangian map cannot be exact a.e., even off of the singularities themselves.

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.

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. 

Unconditional Schauder frames in Hilbert spaces provide unconditionally convergent reconstruction formulas for every element of the space. Such expansions have been a major theme across various branches of theoretical mathematics and have also proved useful in applied fields such as signal processing. Consequently, the characterization and analysis of unconditional Schauder frames have been important research topics over the past decades. In this talk, we will first present a complete characterization of unconditional Schauder frames. Second, we will talk about several applications of this characterization. 

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.