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Organizing Committee
University of Salzburg, Austria
Arizona State University
Chemnitz University of Technology
NC State University
University of Pittsburgh
Abstract
In spite of their omnipresence and importance, a number of questions about knots remain elusive. Addressing them solicits techniques from a range of mathematical disciplines at the interface of algebra, analysis, geometry, modeling, and low-dimensional topology. Some of the most exciting recent avenues of research include optimizing geometry, quantum knot invariants, and applications in material sciences, physics, and molecular biology.
This workshop emphasizes bridging the gap between theoretical, computational, and experimental approaches in knot theory and its applications, including artificial intelligence.
This workshop emphasizes bridging the gap between theoretical, computational, and experimental approaches in knot theory and its applications, including artificial intelligence.
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Workshop Schedule
Monday, May 22, 2023
8:50 - 9:00 AM EDT
Welcome
11th Floor Lecture Hall
9:00 - 9:45 AM EDT
Elastic knots
11th Floor Lecture Hall
Speaker
Heiko von der Mosel, RWTH Aachen University
Session Chair
Elizabeth Denne, Washington & Lee University
Abstract
To model knotted springy wires we minimize in prescribed knot classes a total energy consisting of the classic Euler-Bernoulli bending energy and an additive repulsive potential.
The ultimate goal is to characterize the shape of such minimizing knots
for various knot classes. For that we send a prefactor of the repulsive potential
to zero and analyze the limiting configurations -- so-called elastic knots. For all torus knot classes T(2,b) we established the doubly-covered circle as the unique elastic knot, which confirms mechanical and numerical experiments. There are, however, instances when numerical gradient flows seem to get stuck in different configurations exhibiting some symmetry. To provide analytic support for these rare observations we use the symmetric criticality principle to find symmetric elastic knots exhibiting these symmetries.
In this talk we give a survey on the analytic results, show some of the numerical
simulations obtained by Bartels, Riege and Reiter, and address many open questions.
10:00 - 10:20 AM EDT
Coffee Break
11th Floor Collaborative Space
10:20 - 11:05 AM EDT
The unknotting number and reinforcement learning
11th Floor Lecture Hall
Speaker
András Juhász, University of Oxford
Session Chair
Elizabeth Denne, Washington & Lee University
Abstract
The unknotting number of a knot K is the minimal number of crossing changes required to turn K into the unknot, taken over all diagrams of K. There is no algorithm known to compute it, and even finding the unknotting number of a fixed diagram becomes computationally unfeasible when the crossing number is large. Reinforcement learning is a machine learning paradigm where an agent learns to perform various actions by interacting with an environment in order to maximise a reward. I will discuss how reinforcement learning can help with the study of the unknotting number. This is joint work in progress with Sam Blackwell, Alex Davies, Thomas Edlich, and Marc Lackenby.
11:20 AM - 12:05 PM EDT
The topology, geometry, and combinatorics of feedforward neural networks
11th Floor Lecture Hall
Speaker
J. Elisenda Grigsby, Boston College
Session Chair
Elizabeth Denne, Washington & Lee University
Abstract
Neural networks are a class of parameterized functions that have proven remarkably successful at making predictions about unseen data from finite labeled data sets. They do so even in settings when classical results suggest that they ought to be overfitting (aka memorizing) the data.
I will begin by describing the structure of neural networks and how they learn. I will then advertise one of the theoretical questions animating the field: how does the relationship between the number of parameters and the size of the data set impact the dynamics of how they learn? Along the way I will emphasize the many ways in which topology, geometry, and combinatorics play a role in the field.
12:15 - 1:45 PM EDT
Lunch/Free Time
1:45 - 2:30 PM EDT
DNA self assembly and DNA knots
11th Floor Lecture Hall
Speaker
Nataša Jonoska, University of South Florida
Session Chair
Adam Lowrance, Vassar College
Abstract
We present a range of topological questions associated with DNA self-assembly and three dimensional structures. The questions vary from topological graph theory related to DNA strand routing a three-dimensional mesh, to questions in knot theory related to structural embeddings in 3D, to algebraic descriptions related to Jones monoids associated with DNA origami.
2:45 - 3:30 PM EDT
Universal construction and its applications
11th Floor Lecture Hall
Speaker
Mikhail Khovanov, Columbia University
Session Chair
Adam Lowrance, Vassar College
Abstract
We'll review the universal construction of topological theories in low dimensions and describe examples and their applications in topology, representation theory and number theory.
3:45 - 4:05 PM EDT
Coffee Break
11th Floor Collaborative Space
4:05 - 4:50 PM EDT
Machine learning and hard problems in topology
11th Floor Lecture Hall
Speaker
Sergei Gukov, Caltech
Session Chair
Adam Lowrance, Vassar College
Abstract
I will describe several ways in which tools from reinforcement learning and natural language processing can help with problems in knot theory and low-dimensional topology, including hard challenges such as the smooth Poincare conjecture in dimension 4 and its variants. I will try to make the talk self-contained; in particular, no prior knowledge of machine learning is required, and a large part of the talk will give a gentle introduction to various algorithms and architectures.
5:00 - 6:30 PM EDT
Reception
11th Floor Collaborative Space
Tuesday, May 23, 2023
9:00 - 9:45 AM EDT
Random graph embeddings
11th Floor Lecture Hall
Speaker
Jason Cantarella, University of Georgia
Session Chair
Allison Moore, Virginia Commonwealth University
Abstract
Knots occur in all kinds of systems, including graphs. We're used to finding knots in loops, but what about knotted theta curves? Or knotted graphs? In this talk, we discuss a general framework for constructing random embeddings of arbitrary graphs. This is a conditional probability problem, but setting up the conditions correctly requires some ideas from algebraic topology. We'll view the problem as embedding a simplicial 1-complex and see that the embedding data is (in a precise sense) dual to the simplicial homology of the 1-complex. This will enable us to make some new exact calculations for random knots.