Georgia Institute of Technology College of Sciences

A fundamental conjecture in number theory is the Riemann hypothesis, which implies the prime number theorem with an optimally strong error term. While a proof remains elusive, many results in number theory can nonetheless be proved using weaker inputs. I will discuss how one such weaker input, subconvexity, can be used to prove strong results on the equidistribution of geometric objects such as lattice points on the sphere. If time permits, I will also discuss how various proofs of subconvexity reduce to understanding period integrals of automorphic forms.

In this talk, we discuss generative modeling algorithms motivated by the time reversal and reflection properties of diffusion processes. Score-based diffusion models (SBDM) have recently emerged as state-of-the-art approaches for image generation. We develop SBDMs in the infinite-dimensional setting, that is, we model the training data as functions supported on a rectangular domain. Besides the quest for generating images at ever higher resolution, our primary motivation is to create a well-posed infinite-dimensional learning problem so that we can discretize it consistently at multiple resolution levels. We demonstrate how to overcome two shortcomings of current SBDM approaches in the infinite-dimensional setting by ensuring the well-posedness of forward and reverse processes, and derive the convergence of the approximation of multilevel training. We illustrate that approximating the score function with an operator network is beneficial for multilevel training.

In the second part of this talk, we propose the Reflected Schrodinger Bridge algorithm: an entropy-regularized optimal transport approach tailored for generating data within diverse bounded domains. We derive reflected forward-backward stochastic differential equations with Neumann and Robin boundary conditions, extend divergence-based likelihood training to bounded domains, and demonstrate its scalability in constrained generative modeling.

Flow-based models are widely used in generative tasks, including normalizing flow, where a neural network transports from a data distribution P to a normal distribution. This work develops a flow-based model that transports from P to an arbitrary Q (which can be pre-determined or induced as the solution to an optimization problem), where both distributions are only accessible via finite samples. We propose to learn the dynamic optimal transport between P and Q by training a flow neural network. The model is trained to optimally find an invertible transport map between P and Q by minimizing the transport cost. The trained optimal transport flow subsequently allows for performing many downstream tasks, including infinitesimal density ratio estimation (DRE) and distribution interpolation in the latent space for generative models. The effectiveness of the proposed model on high-dimensional data is demonstrated by strong empirical performance on high-dimensional DRE, OT baselines, and image-to-image translation.

The “hard direction” of the Giroux correspondence states that any two open books representing the same contact structure is related by a sequence of positive stabilisations and destabilisations. We give a proof of this statement using convex surface theory. This is a joint work with Joan Licata.