Georgia Institute of Technology College of Sciences

Given a basis for a linear subspace U of nxn matrices, we study the problem of either producing a rank-one matrix in U, or certifying that none exist. While this problem is NP-Hard in the worst case, we present a polynomial time algorithm to solve this problem in the generic setting under mild conditions on the dimension of U. Our algorithm is based on Hilbert’s Nullstellensatz and a “lifted” adaptation of the simultaneous diagonalization algorithm for tensor decompositions. We extend our results to the more general setting in which the set of rank-one matrices is replaced by an algebraic set. Time permitting, we will discuss applications to quantum separability testing and tensor decompositions. This talk is based on joint work with Harm Derksen, Nathaniel Johnston, and Aravindan Vijayaraghavan.

There are lots of ways to measure the complexity of a knot. Some come from knot diagrams, and others come from topological or geometric quantities extracted from some auxiliary space. In this talk, I’ll describe a geometry property, which we call “twist positivity”, that often puts strong restrictions on how the braid and bridge index are related. I’ll describe some old and new results about twist positivity, as well as some new applications towards knot concordance. In particular, I’ll describe how using a suite of numerical knot invariants (including the braid index) in tandem allows one to prove that there are infinitely many positive braid knots which all represent distinct smooth concordance classes. This confirms a prediction of the slice-ribbon conjecture. Everything I’ll discuss is joint work with Hugh Morton. I will assume very little background about knot invariants for this talk – all are welcome!

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. 

I will give essentially matching upper and lower bounds for the expected max-sliced 1-Wasserstein distance between a probability measure on a separable Hilbert space and its empirical distribution from n samples. A version of this result for Banach spaces will also be presented. From this, we will derive an upper bound for the expected max-sliced 2-Wasserstein distance between a symmetric probability measure on a Euclidean space and its symmetrized empirical distribution.

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.

In this talk I will present a computer-assisted method to study solutions vanishing at infinity in differential equations on R^n. Such solutions arise naturally in various models, in the form of traveling waves or localized patterns for instance, and involve multiple challenges to address both on the numerical and on the analytical side. Using spectral techniques, I will explain how Fourier series can serve as an approximation of the solution as well as an efficient mean for the construction of a fixed-point operator for the proof. To illustrate the method, I will present applications to the constructive proof of localized patterns in the 2D Swift-Hohenberg equation and in the Gray-Scott model. The method extends to non-local equations and proofs of solitary travelling waves in the (capillary-gravity) Whitham equation will be exposed.