Please join us for pre-seminar Tea

Please join us for pre-seminar Tea

In this talk, I will discuss our recent theoretical advancements in generative modeling. The first part of the presentation will focus on learning distributions with symmetry. I will introduce results on the sample complexity of empirical estimations of probability divergences for group-invariant distributions, and present performance guarantees for GANs and score-based generative models that incorporate symmetry. Notably, I will offer the first quantitative comparison between data augmentation and directly embedding symmetry into models, highlighting the latter as a more fundamental approach for efficient learning. These findings underscore how incorporating symmetry into generative models can significantly enhance learning efficiency, particularly in data-limited scenarios. The second part will cover $\alpha$-divergences with Wasserstein-1 regularization. These divergences can be interpreted as $\alpha$-divergences constrained to Lipschitz test functions in their variational form. I will demonstrate how generative learning can be made agnostic to assumptions about target distributions, including those with heavy tails or low-dimensional and fractal supports, through the use of these divergences as objective functionals. I will outline the conditions for the finiteness of these divergences under minimal assumptions on the target distribution along with the gradient flow formulation associated with them. This framework provides guarantees for various machine learning algorithms that optimize over this class of divergences. 

In this talk, we will start by describing how classical tools from probability

offer a robust framework to understand the dynamics of waves via appropriate ensembles

on phase space rather than particular microscopic dynamical trajectories. We will continue

by explaining the fundamental shift in paradigm that arises from the “correct” scaling in this

context and how it opened the door to unveil the random structures of nonlinear waves that

live on high frequencies and fine scales as they propagate. We will then discuss how these 

ideas broke the logjam in the study of the Gibbs measures associated to nonlinear 

Schrödinger equations in the context of equilibrium statistical mechanics and of the 

hyperbolic $\Phi^4_3$ model in the context of constructive quantum field theory. 

We will end with some open challenges about the long-time propagation of randomness 

and out-of-equilibrium dynamics.

Please join us for pre-seminar Tea

I will describe a joint project with Tobias Ekholm, Pietro Longhi, and Vivek Shende constructing a map from the HOMFLYPT skein module of a 3-manifold M to that of its branched cover arising from the projection of a Lagrangian 3-manifold L in the cotangent bundle of M. The map is defined by counting holomorphic curves and is a vast generalization of the quantum UV-IR map of Neitzke and Yan, which is a close cousin of the quantum trace map of Bonahon and Wong. The existence of this map has some interesting consequences in the theory of skein-valued curve counts, and I will discuss some of them if time permits. 

Please join us for pre-seminar Tea

The Jones polynomial of a link can be computed diagrammatically by using skein relations which encode the representation theory of SL(2). By considering the vector space spanned by links drawn on a surface and imposing these skein relations, we obtain an algebra known as the Kauffman bracket skein algebra of the surface. Replacing SL(2) by SL(3) or any other higher rank Lie group gives rise to a new skein algebra involving not only links but also certain graphs called webs. In this talk, we will discuss some of the complications involved with studying skein algebras built from webs on surfaces and then discuss how the use of stated skein algebras helps us get around these.

Please join us for pre-seminar Tea

In this talk, we explore adaptations of semidefinite programming relaxations for solving many-body physics problems. Our approach transforms a high-dimensional PDE problem into a convex optimization problem, setting it apart from traditional non-convex methods that rely on nonlinear re-parameterizations of the solution. In the context of statistical mechanics, we demonstrate how a mean-field type solution for an interacting particle Fokker-Planck equation can be provably recovered without resorting to non-convex optimization. For quantum mechanical systems, we present a similar technique to obtain the ground state of a quantum system and introduce a near-linear time algorithm for solving the convex program using hierarchical matrices.

Please join us for pre-seminar Tea

A relaxed-pace seminar on impromptu subjects related to the interests of the audience. Everyone is welcome. The subjects are geometry, probability, combinatorics, dynamics, and more!

Please join us for pre-seminar Tea

Please join us for pre-seminar Tea

Single particle cryo-EM is an increasingly popular technique for determining 3-D molecular structures at high resolution. The 2017 Nobel Prize in Chemistry was awarded to three of the pioneers of cryo-EM, and already in the early stages of the global pandemic, cryo-EM was successfully applied to image the SARS-CoV-2 spike protein. We will discuss the mathematical principles and computational methods for reconstruction using cryo-EM, focusing on its sample complexity (i.e., the number of images required for reconstruction as a function of the signal-to-noise ratio) and reconstruction of flexible molecules.

The classical uniformization theorem for Riemann surfaces is well-established for both compact and non-compact connected surfaces. However,  in the discrete setting, the discrete uniformization theorem is only known for closed polyhedral surfaces.  This talk will propose a discrete uniformization problem for all surfaces and relate it to the classical Weyl problem in the hyperbolic 3-space and the Cauchy-Alexandrov rigidity theorem on convex polytopes.  Some of our progress will be discussed,  including a discrete Schwarz-Pick-Ahlfors lemma and its application.  This is joint work with Yanwen Luo.

Please join us for pre-seminar Tea