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…
On a Riemannian manifold with boundary, the X-ray transform integrates a function or a tensor field along all geodesics through the manifold. The reconstruction of the integrand of interest from its X-ray transform is the basis of important inverse problems in seismology and medical imaging. For this transform, statements of injectivity and microlocal stability in the interior of the domain were established some…
Noncrossing partitions and nonnesting partitions are both counted by Catalan numbers. Noncrossing partitions on admit a natural cyclic action of order 2n, induced by the Kreweras complement. Nonnesting partitions admit a natural toggle-based action; in fact, they admit one such action for each choice of Coxeter element of the symmetric group. We prove that the latter actions all…
How to use mathematics to see inside something that is too large or too precious to be opened up? Would you like to learn about the mathematical theory behind medical and seismic imaging? Attend Triangle Area Inverse Problems Weekend at NC State University from Friday October 4 thru Sunday October 6 to learn more. The event is…
The trace reconstruction problem asks to reconstruct an unknown n-bit string x given independent random "traces" of x, where a random trace is obtained by first deleting each bit of x independently with some probability (say 0.5), and then outputting the concatenation of the remaining bits of x. A basic question is to determine the…
Applications of machine learning (ML) in mixed-integer program (MIP) optimization is an active area of research. The human-designed heuristics used by MIP solvers rely on domain expertise and years of experience with the expectation of applicability to only a specific problem class. The underlying question is whether training can simulate expertise and experience. We consider…
The travel time map of a compact length space with a closed measurement set takes any point of the space to the function which measures the distance from this point to every point in the measurement set. The range of this map is called the travel time data, and it appears in the proofs of…
The Schur functions are an important basis of the ring of symmetric functions, and Schur’s Q-functions enjoy many analogous properties as a basis of the subring Gamma. We will begin by discussing various properties and bases of symmetric functions, before moving on to the comparisons between Schur functions and Schur’s Q-functions. In particular, plethysm is…