In the past few years, topological deep learning (TDL), a term coined by us in 2017, has become an emerging paradigm in artificial intelligence (AI) and data science. TDL is built on persistent homology (PH), an algebraic topology technique that bridges the gap between complex geometry and abstract topology through multiscale analysis. While TDL has made huge strides in a wide variety of scientific and engineering disciplines, it has many limitations. I will discuss our recent effort in extending TDL from graphs to manifolds and curves, using algebraic topology, geometric topology, and differential geometry. I will also discuss how TDL led to victories in worldwide annual competitions in computer-aided drug design, the discoveries of SARS-CoV-2 evolutionary mechanism, and the accurate forecasting of emerging dominant viral variants.

The classical theory of error-correcting codes primarily uses the Hamming metric to measure distances. Within this framework, Maximum Distance Separable (MDS) codes are highly valued due to their optimal parameters, enabling the correction of the maximum number of errors for a given code rate. In 1997, Rosenbloom and Tsfasman introduced the m-metric codes (now known as NRT metric codes), identifying analogs of Reed-Solomon codes that still possess the MDS property. Later, Skriganov provided an explicit construction of these codes using Hasse derivatives.

In this talk, we will introduce and discuss our analogs of Reed-Solomon codes in a related context, also employing Hasse derivatives for our construction.

This is joint work with Mahir Bilen Can.

The Gromov–Hausdorff distance between two abstract metric spaces provides a (dis)-similarity measure quantifying how far the two metric spaces are from being isometric. Although the inception of the distance was due to M. Gromov in the context of hyperbolic groups, it has recently been shown to provide a robust theoretical framework for shape and dataset comparison. Consequently, the computational aspects and various bounds on the Gromov–Hausdorff distance are receiving a lot of attention from both applied and theoretical communities. In this talk, I give an overview of the Gromov–Hausdorff distance, delineating its relation to the well-known Hausdorff distance. The main focus of the talk is to present interesting lower bounds on the former by a constant multiple of the latter on interesting spaces like the circle, closed Riemannian manifolds, and metric graphs.

Rees algebras represent an essential algebraic tool in the study of singularities of algebraic varieties, as they arise, for instance, as homogeneous coordinate rings of blowups or graphs of rational maps.

In this talk, I will discuss the problem of finding the defining equations of Rees algebras. Although this is wide open in general, the problem becomes treatable in the case of height-two perfect ideals with a linear presentation, where one can use a combination of homological methods and linear algebra, inspired by classical elimination theory.

This is part of joint work with E. Price and M. Weaver (arxiv:2308.16010 and arxiv:2409.14238

This will be an informal working group, learning about the long time asymptotic behavior of the defocusing NLS equation, based on a combination of Riemann-Hilbert and d-bar techniques.

Today:

1. A summary of what the past 4 weeks have yielded: a description of the behavior of “all” solutions of the nonlinear Schroedinger equation in the long-time regime.

2. A summary of the fundamental steps in the analysis.

3. If there is any time at the end, some “sick tricks”.

The Boltzmann equation describes the time evolution of the density function in a phase (position-velocity) space for a classical particle (molecule) under the influence of other particles in a diluted (or rarified) gas, evolving in vacuum for a given initial distribution. The Enskog equation introduces a function in the collision operator for the Boltzmann equation, allowing one to take into account the interactions between molecules at a small distance away, rather than solely at the point of collision. In this talk, we discuss modern results on the stochastic treatment of the spatially homogeneous Boltzmann equation, the Enskog equation, and the connection between methods used in each system. Regularity results and the motivation behind this probabilistic treatment will be given.

Curve registration is a set of techniques to align functional data in the presence of time warping—phase variation in the functional observations. In this talk, we will present a Bayesian framework to estimate the parameters of an ODE model when the observations contain stochastic fluctuations in both amplitude and phase with a Gaussian process defining the time warping model. To facilitate such a framework, a new method for curve registration using Hamiltonian Monte Carlo will be presented along with a hierarchical model that links a basis fit of the data to solutions of an ODE model, allowing for parameter estimation.

Many physical phenomena involve the nonlinear, conservative transfer of energy from weakly damped degrees of freedom driven by an external force to other modes that are more strongly damped. For example, in hydrodynamic turbulence, energy enters the system primarily at large spatial scales, but at high Reynolds number, dissipative effects are only significant at very high frequencies. In this talk, I will discuss nonlinear energy transfer and the existence of invariant measures for a class of degeneratly forced SDEs on R^d with a bilinear nonlinearity B(x,x) constrained to possess various properties common to finite-dimensional fluid models and a linear damping term -Ax that acts only on a proper subset of the phase space. Existence of an invariant measure is straightforward if kerA = {0}, but when the kerA is nontrivial, an invariant measure can exist only if the nonlinearity transfers enough energy from the undamped modes to the damped modes. We develop a set of sufficient dynamical conditions on B that guarantees the existence of an invariant measure and prove that they hold “generically” within our constraint class of nonlinearities provided that dim(kerA) < 2d/3 and the stochastic forcing acts directly on at least two degrees of freedom.

In many phylogenetic models, the number of parameters to estimate grows with the number of taxa under study. However, parsimonious models of evolution demand local similarity in parameters on subtrees. To achieve scalable inference in such a setting, we employ auto-correlated, shrinkage-based models. We compare inference under these models to previous state-of-the art in a variety of applied settings. In one example, we investigate the heritable clock structure of various surface glycoproteins of influenza A virus in the absence of prior knowledge about molecular clock placement. In another example, we estimate the phylogenetic location of environmental shifts in the ancestry of Anolis lizards.