Georgia Institute of Technology College of Sciences

In 1939, Wolfgang Gröbner proposed using differential operators to represent ideals in a polynomial ring. Using Macaulay inverse systems, he showed a one-to-one correspondence between primary ideals whose variety is a rational point, and finite dimensional vector spaces of differential operators with constant coefficients. The question for general ideals was left open. Significant progress was made in the 1960's by analysts, culminating in a deep result known as the Ehrenpreis-Palamodov fundamental principle, connecting polynomial ideals and modules to solution sets of linear, homogeneous partial differential equations with constant coefficients. 

This talk aims to survey classical results, and provide new constructions, applications, and insights, merging concepts from analysis and nonlinear algebra. We offer a new formulation generalizing Gröbner's duality for arbitrary polynomial ideals and modules and connect it to the analysis of PDEs. This framework is amenable to the development of symbolic and numerical algorithms. We also study some applications of algebraic methods in problems from analysis.

Link: https://gatech.zoom.us/j/95997197594?pwd=RDN2T01oR2JlaEcyQXJCN1c4dnZaUT09

Meeting Link: https://gatech.zoom.us/j/94882290086 (Meeting ID: 948 8229 0086, Passcode: 264830)

Abstract: R-loops are three-stranded structures formed by a DNA:RNA hybrid and a single strand of DNA, often appearing during transcription. Although R-loops can threaten genome integrity, recent studies have shown that they also play regulatory roles in physiological processes. However, little is known about their structure and formation. In this talk, we introduce a model for R-loops based on formal grammars, that are systems to generate words widely applied in molecular biology. In this framework, R-loops are described as strings of symbols representing the braiding of the strands in the structure, where each symbol corresponds to a different state of the braided structure. We discuss approaches to develop a stochastic grammar for R-loop prediction using experimental data, as well as refinements of the model by incorporating the effect of DNA topology on R-loop formation.

For polynomials in 1 variable, Matt Baker and Oliver Lorschied were able to connect results about roots of polynomials over valued
fields (Newton polygons) and over real fields (Descartes's rule) by looking at factorization of polynomials over the tropical and signed
hyperfields respectively. In this talk, I will describe some ongoing work with Andreas Gross about extending these ideas to two or more
variables. Our main tool is the use of resultants to transform questions about 0-dimensional systems of equations to factoring a single
homogeneous polynomial.

Supervised operator learning is an emerging machine learning paradigm with applications to modeling the evolution of spatio-temporal dynamical systems and approximating general black-box relationships between functional data. We propose a novel operator learning method, LOCA (Learning Operators with Coupled Attention), motivated from the recent success of the attention mechanism. In our architecture, the input functions are mapped to a finite set of features which are then averaged with attention weights that depend on the output query locations. By coupling these attention weights together with an integral transform, LOCA is able to explicitly learn correlations in the target output functions, enabling us to approximate nonlinear operators even when the number of output function measurementsin the training set is very small. Our formulation is accompanied by rigorous approximation theoretic guarantees on the universal expressiveness of the proposed model. Empirically, we evaluate the performance of LOCA on several operator learning scenarios involving systems governed by ordinary and partial differential equations, as well as a black-box climate prediction problem. Through these scenarios we demonstrate state of the art accuracy, robustness with respect to noisy input data, and a consistently small spread of errors over testing data sets, even for out-of-distribution prediction tasks.