I will talk about various notions of equivalence for manifolds and morphisms and the relationships between them. Questions, interruptions, and detours are strongly encouraged!
Quasi-stationary distributions (QSDs) are those almost invariant to a diffusion process over exponentially long time. Representing important transient stochastic dynamics, they arise frequently in applications especially in chemical reactions and population systems admitting extinction states. This talk will present some rigorous results on the existence, uniqueness, concentration, and convergence of QSDs along with their connections to the spectra of the Fokker-Planck operators.
Empirical Bayes provides a powerful approach to learning and adapting to latent structure in data. Theory and algorithms for empirical Bayes have a rich literature for sequence models, but are less understood in settings where latent variables and data interact through more complex designs.
In this work, we study empirical Bayes estimation of an i.i.d. prior in Bayesian linear models, via the nonparametric maximum likelihood estimator (NPMLE). We introduce and study a system of gradient flow equations for optimizing the marginal log-likelihood, jointly over the prior and posterior measures in its Gibbs variational representation using a smoothed reparametrization of the regression coefficients. A diffusion-based implementation yields a Langevin dynamics MCEM algorithm, where the prior law evolves continuously over time to optimize a sequence-model log-likelihood defined by the coordinates of the current Langevin iterate.
We show consistency of the NPMLE under mild conditions, including settings of random sub-Gaussian designs under high-dimensional asymptotics. In high noise, we prove a uniform log-Sobolev inequality for the mixing of Langevin dynamics, for possibly misspecified priors and non-log-concave posteriors. We then establish polynomial-time convergence of the joint gradient flow to a near-NPMLE if the marginal negative log-likelihood is convex in a sub-level set of the initialization.
This is joint work with Leying Guan, Yandi Shen, and Yihong Wu.
Motivated by problems in data science, we study the following questions:
(1) Given a Hilbert space V and a group G of linear isometries, does there exist a bilipschitz embedding of the quotient metric space V/G into a Hilbert space?
(2) What are necessary and sufficient conditions for such embeddings?
(3) Which embeddings minimally distort the metric?
We answer these questions in a variety of settings, and we conclude with several open problems.
Unicritical polynomials, typically written in the form $z^d+c$, have been widely studied in arithmetic and complex dynamics and are characterized by their one finite critical point. The behavior of a map's critical points under iteration often determines the dynamics of the entire map. Rational maps where the critical points have a finite forward orbit are called post-critically finite (PCF), and these are of great interest in arithmetic dynamics. They are viewed as a dynamical analogue of abelian varieties with complex multiplication and often display interesting dynamical behavior. The family of (single-cycle normalized) dynamical Belyi polynomials have two fixed critical points, so they are PCF by construction, and these maps provide a new testing ground for conjectures and ideas related to post-critically finite polynomials. Using this family, we can begin to explore properties of polynomial maps with two critical points. In this talk we will discuss applications of this family in arithmetic dynamics; in particular, how this family can be used to determine more general reduction properties of PCF polynomials.
The fine curve graph of a surface is a graph whose vertices are essential simple closed curves in the surface and whose edges connect disjoint curves. Following a rich history of hyperbolicity in various graphs based on surfaces, the fine curve was shown to be hyperbolic by Bowden–Hensel–Webb. Given how well-studied the curve graph and the case of “up to isotopy” is, we ask: what about the mysterious part of the fine curve graph not captured by isotopy classes? In this talk, we introduce the result that the subgraph of the fine curve graph spanned by curves in a single isotopy class is not hyperbolic; indeed, it contains a flat of EVERY dimension. Along the way, we will discuss how to not prove this theorem as we explore proofs of hyperbolicity of related complexes. This work is joint with Ryan Dickmann.
Please Note: Available on zoom at: https://gatech.zoom.us/j/98258240051
We shall discuss the quantum dynamics associated with ergodic Schroedinger operators. Anderson localization (pure point spectrum with exponentially decaying eigenfunctions) has been obtained for a variety of ergodic operator families, but it is well known that Anderson localization is highly unstable and can also be destroyed by generic rank one perturbations. For quasiperiodic operators, it also sensitively depends on the arithmetic properties of the phase (a seemingly irrelevant parameter from the point of view of the physics of the problem) and doesn’t hold generically. These instabilities are also present for the physically relevant notion of dynamical localization.
In this talk we will introduce the notion of the transport exponent, explain its stability, and explain how logarithmic upper bounds may be obtained in the quasi-periodic setting for all relevant parameters. This is based on joint work with S. Jitomirskaya.
Sidorenko's conjecture can be formulated as "Let $H$ be a bipartite graph, and $\rho\in [0,1]$. Of all the graphs with edge density $\rho$, the graph(-limit) obtained by picking edges uniformly at random minimizes the homomorphism density of $H$." This conjecture, first formulated in 1991 by Sidorenko, has received considerable attention over the last decades, and yet remains open in the general case.
It was shown recently [Blekherman, Raymond, Singh, Thomas, 2020] that sums-of-squares in Razborov's flag algebra are not strong enough to prove even small, known cases of the conjecture. To circumvent this, we introduce a novel kind of derivation of flags. Due to their combinatoric nature, we can use them to systematically gain knowledge on global minimizers of problems in extremal graph theory. We combine them with the flag algebra method to find new proofs for various cases of Sidorenko's conjecture.
I will discuss a nonlinear elliptic system of partial differential equations arising in Riemannian geometry and General Relativity. Specifically, I will present recent advances on the analysis of asymptotically Euclidean, initial data sets for Einstein’s field equations. In collaboration with Bruno Le Floch (Sorbonne University) I proved that solutions to the Einstein constraints can be glued together along possibly nested conical domains. The constructed solutions may have arbitrarily low decay at infinity, while enjoying (super-)harmonic estimates within possibly narrow cones at infinity. Importantly, our localized seed-to-solution method, as we call it, leads to a proof of a conjecture by Alessandro Carlotto and Richard Schoen on the localization problem at infinity, and generalize P. LeFloch and Nguyen’s theorem on the asymptotic localization problem. This lecture will be based on https://arxiv.org/abs/2312.17706
For the past 25 years, a key player in contact topology has been the Floer-theoretic invariant called Legendrian contact homology. I'll discuss a package of new invariants for Legendrian knots and links that builds on Legendrian contact homology and is derived from rational symplectic field theory. This includes a Poisson bracket on Legendrian contact homology and a symplectic structure on augmentation varieties. Time permitting, I'll also describe an unexpected connection to cluster theory for a family of Legendrian links associated to positive braids. Parts of this are joint work in progress with Roger Casals, Honghao Gao, Linhui Shen, and Daping Weng.
