Integrable cross-ratio maps are solutions to one of the discrete integrable equations on quad-graphs. These maps may be of interest to many mathematicians; just to name a few uses, discrete holomorphic functions, orthogonal circle packings, and polygon recutting are all special cases of integrable cross-ratio maps. The goal of my research is to find an…
In recent years much attention has turned to rigidity, and almost-rigidity, problems involving lower scalar curvature bounds. In this talk, I'll discuss some contributions to this area, including some new stability theorems for spheres. Some of this is joint work with Davi Maximo, and some is joint with Paul Sweeney Jr.
A representation of the category of finite sets is a linear algebraic object, which roughly consists of a sequence of representations V_n of the symmetric group S_n related by transition maps. These representations occur naturally in several places including in the study of Kazhdan-Lusztig polynomials of braid matroids, the homology of moduli spaces of curves,…
In this talk, we will examine the time evolution of viscoelastic solids within a framework that allows for collisions and self-contact. In the static and quasi-static regimes, corresponding existence results have been shown through variational descriptions of the problem. For the fully dynamical case, however, collisions have so far either been ignored or handled using…
Symmetry is prevalent in a variety of machine learning and scientific computing tasks, including computer vision and computational modeling of physical and engineering systems. Empirical studies have demonstrated that machine learning models designed to integrate the intrinsic symmetry of their tasks often exhibit substantially improved performance. Despite extensive theoretical and engineering advancements in symmetry-preserving machine…
By slicing a Heegaard diagram for a knot K in $S^3$, it is possible to retrieve the knot Floer homology of K as a tensor product of bimodules over an $\A_{\infty}$ algebra corresponding to the slice. The first step in this process is to assign an $\mathcal{A}_{\infty}$ algebra to this slice, which can also be…
Dimension four is the lowest dimension where smooth and topological manifolds can differ; any difference between these categories is known as exotica. In particular, a smooth 4-manifold is \emph{exotic} if there is another smooth 4-manifold which is homeomorphic but not diffeomorphic to it. There is a wealth of literature, mostly written between 1983 and 2008,…
Multiple financial assets’ time-series data is stored in a matrix upon which we perform principal component analysis to find predominant factors in the market. Random matrix theory helps us to identify the number of factors present in the data, with the top eigenvalue-eigenvector pair bearing a strong resemblance to the market’s capitalization-weighted portfolio. This resemblance…
A fundamental problem in knot theory is determining when two distinct knot diagrams represent the same knot. This is traditionally addressed through the use of invariants such as the knot group, Alexander polynomial, and Jones polynomial, among others. While the knot group distinguishes prime knots, it is known to be incomplete as a knot invariant.…
Symplectic manifolds show up in many areas of mathematics, including topology, geometry, representation theory, and dynamics. In the talk I will give an introduction to symplectic manifolds and then discuss embeddings of 3-manifolds into symplectic 4-manifolds. This is joint work with Ali Daemi and Mike Miller Eismeier.
Motivated by the problem of classifying quantum symmetries of non-semisimple, finite-dimensional associative algebras, we define a notion of connection between bounded quivers and build a bicategory of bounded quivers and quiver connections. We prove this bicategory is equivalent to a bicategory of basic algebras, bimodules, and intertwiners with some additional structure. Speaker’s website: https://math.sciences.ncsu.edu/people/sthomps6/
Machine learning has become a widely popular and successful paradigm, including for data-driven science. A major application is forecasting complex dynamical systems. Artificial neural networks (ANN) have evolved as a clear leading approach, and recurrent neural networks (RNN) are considered to be especially well suited. Reservoir computers (RC) have emerged for simplicity and computational advantages.…
We consider the random Cayley graphs of a sequence of finite nilpotent groups of diverging sizes G = G(n), whose ranks and nilpotency classes are uniformly bounded. For some k = k(n) such that 1 << log k << log |G|, we pick a random set of generators S = S(n) by sampling k elements…
We analyze a randomized preconditioned Cholesky-QR algorithm for computing the thin QR factorization of real matrices with full rank. Using a perturbation analysis that is transparent and identifies clearly all factors that contribute to error amplification, we identify steps of the algorithm that can be performed in lower precision while maintaining accuracy. The numerical experiments…
Generalizing unknotting number, n-adjacent knots have n crossings such that changing any non-empty subset of them results in the unknot. We determine the 2-adjacent knots through 12-crossings, with one exception. Using Heegaard Floer d-invariants and the Alexander polynomial, we develop a new technique to obstruct 2-adjacency, and we prove conjectures of Ito and Kato regarding…
One of the main approaches to topological data analysis is to study discrete invariants of (multiparameter) persistence modules. In this talk, we discuss recent research which utilizes techniques from (relative) homological algebra to interpret classical examples of invariants and define new invariants. The Hilbert function and (generalizations of) the rank invariant serve as our main…
Finding combinatorial interpretations for the structure constants of Schubert polynomials is a long-standing open problem in algebraic combinatorics. In the case where one of the Schubert polynomials is a Schur polynomial, the structure constants are encoded in a poset called the “k-Bruhat order”. In studying the k-Bruhat order, Bergeron and Sottile were led to introduce…