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…
In this talk we will first briefly review some geometric inverse problems we have studied for the one-dimensional heat, wave and Burgers equations. Then, we will consider a one-dimensional fluid-solid interaction model governed by the Burgers equation with a time varying interface. This is a preliminary simplified version of other more complicate and more realistic…
In this talk, we will investigate the acoustic transmission eigenvalue problem associated with an inhomogeneous media with a conductive boundary. These are a new class of eigenvalue problems that are not elliptic, not self-adjoint, and nonlinear, which gives the possibility of complex eigenvalues. The talk will consider the case of an Isotropic and Anisotropic scatterer.…
Quantum computers are physical machines that process information using the principles of quantum mechanics, which in turn is underpinned by linear algebra. The talk will start with a review of Lie algebras (consisting of matrices under the operation of commutator) and their role in quantum mechanics. The dynamical Lie algebra (DLA) of a quantum system…
A number of continuous interacting particle systems can be described as collections of Brownian particles on the real line whose collision dynamics are mediated by the local times associated with the gaps between adjacent particles. Examples of systems in this class include the ordered particle dynamics of rank-based diffusions and certain eigenvalue processes arising in…
This talk will begin by constructing a new algebra based on a generalization of the set of relations for Ozsváth-Szabó's bordered Heegaard Floer algebra $B(n,k)$. Then, I will strengthen an existing result by establishing an isomorphism between our new algebra and the algebra $\widetilde{B}(\mathcal{V})$ associated with an arrangement of real affine hyperplanes. If time permits, I’ll…
Matrix computations are ubiquitous in scientific computing and data science. However, many existing methods consume enormous computational resources for solving increasingly large and complex problems. In this talk, I will describe my work on exploiting mathematical structures and hardware capabilities to accelerate matrix computations. I will focus on a randomized algorithm, namely RChol, for computing…