## Seminars and Colloquia by Series

### Dual representation of polynomial modules with applications to partial differential equations

Series
Dissertation Defense
Time
Friday, April 15, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 or ONLINE
Speaker
Marc HärkönenGeorgia Tech

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.

### Exponential decay of intersection volume and applications

Series
Combinatorics Seminar
Time
Friday, April 15, 2022 - 09:00 for 1 hour (actually 50 minutes)
Location
Zoom
Speaker
Hong LiuECOPRO, IBS

Please Note: Note the unusual time!

When do two balls in a metric space have small intersection? We give some natural conditions to guarantee an exponential decay on the volume of such intersections. Our proof is conceptually simple, making use of concentration of measure on a "slice." We will discuss a couple of applications of this volume estimate in coding theory. This is joint work with Jaehoon Kim and Tuan Tran.

### Formal grammar modeling three-stranded DNA:RNA braids

Series
Mathematical Biology Seminar
Time
Wednesday, April 13, 2022 - 10:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Margherita Maria FerrariUniversity of South Florida

Meeting Link: (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.

### Fast algorithms for $(\Delta+1)$-edge-coloring

Series
Graph Theory Seminar
Time
Tuesday, April 12, 2022 - 15:45 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Abhishek DhawanGeorgia Institute of Technology

Vizing's Theorem states that simple graphs can be edge-colored using $\Delta+1$ colors. The problem of developing efficient $(\Delta+1)$-edge-coloring algorithms has been a major challenge. The algorithms involve iteratively finding small subgraphs $H$ such that one can extend a partial coloring by modifying the colors of the edges in $H$. In a recent paper, Bernshteyn showed one can find $H$ such that $e(H) = \mathrm{poly}(\Delta)(\log n)^2$.  With this result, he defines a $(\Delta+1)$-edge-coloring algorithm which runs in $\mathrm{poly}(\Delta, \log n)$ rounds. We improve on this by showing we can find $H$ such that $e(H) = \mathrm{poly}(\Delta)\log n$. As a result, we define a distributed algorithm that improves on Bernshteyn's by a factor of $\mathrm{poly}(\log n)$. We further apply the idea to define a randomized sequential algorithm which computes a proper $(\Delta+1)$-edge-coloring in $\mathrm{poly}(\Delta)n$ time. Under the assumption that $\Delta$ is a constant, the previous best bound is $O(n\log n)$ due to Sinnamon.

### Baker-Lorscheid (Hyperfield) Multiplicities in Two Variables

Series
Algebra Seminar
Time
Tuesday, April 12, 2022 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Trevor GunnGeorgia Tech

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.

### Upsilon invariant for graphs and homology cobordism group of homology cylinders

Series
Geometry Topology Seminar
Time
Monday, April 11, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
skies 006
Speaker
Akram AlishahiUGA

Upsilon is an invariant of knots defined using knot Floer homology by Ozsváth, Szabó and Stipsicz. In this talk, we discuss a generalization of their invariant for embedded graphs in rational homology spheres satisfying specific properties. Our construction will use a generalization of Heegaard Floer homology for “generalized tangles” called tangle Floer homology. As a result, we get a family of homomorphisms from the homology cobordism group of homology cylinders (over a surface of genus 0), which is an enlargement of the mapping class group defined by Graoufaldis and Levine.

### Learning Operators with Coupled Attention

Series
Applied and Computational Mathematics Seminar
Time
Monday, April 11, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
https://gatech.zoom.us/j/96551543941
Speaker
Paris PerdikarisUniversity of Pennsylvania

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.

### Incest and infanticide: a branching process with deletions and mergers that matches the threshold for hypercube percolation

Series
Combinatorics Seminar
Time
Friday, April 8, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Fiona SkermanUppsala University / Simon&#039;s Institute

We define a graph process based on a discrete branching process with deletions and mergers, which is inspired by the 4-cycle structure of the hypercube $\mathcal{Q}_d$ for large $d$. We prove survival and extinction under certain conditions on $p$ and $q$ that heuristically match the known expansions of the critical probabilities for bond percolation on the hypercube. Joint work with Laura Eslava and Sarah Penington. Based on https://arxiv.org/abs/2104.04407.

### Stiffness and rigidity in random dynamics

Series
CDSNS Colloquium
Time
Friday, April 8, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Online via Zoom
Speaker
Aaron BrownNorthwestern University

Consider two volume-preserving, smooth diffeomorphisms f and g of a compact manifold M.  Define the random walk on M by selecting either f or g (i.i.d.) at each iterate.  A number of questions arise in this setting:

1. What are the closed subsets of M invariant under both f and g?
2. What are the stationary measures on M for the random walk.  In particular, are the stationary measures invariant under f and g?

Conjecturally, for a generic pair of f and g we should be able to answer the above.  I will describe one sufficient criteria on f and g underwhich we can give some partial answers to the above questions.  Such a criteria is expected to be generic amoung pairs of (volume-preserving) diffeomorphisms and should be able to be verified in a number of naturally occurring geometric settings where the above questions are not fully answered.

### Parallel server systems under an extended Heavy traffic condition

Series
ACO Student Seminar
Time
Friday, April 8, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Eyal CastielIsrael Institute of Technology