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Computational and Applied Mathematics: Wei Zhu, Georgia Institute of Technology, Symmetry-Preserving Machine Learning: Theory and Applications

SAS 4201

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…

Triangle Topology Seminar: Lisa Piccirillo UT Austin, New constructions and invariants of exotic 4-manifolds.

SAS 4201

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,…

Stochastics and Discrete Analysis Seminar: Andrew Papanicolaou, NC State, Principal Eigenportfolios and Primary Factors

SAS 4201

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…

Pure Math Graduate Student Seminar: Reeshad Arian, NC State, Quandles and Knots

SAS 2106

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.…

Geometry Topology Seminar: Tye Lidman, NC State, 3-manifolds in symplectic 4-manifolds

SAS 4201

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.

Algebra and Combinatorics Seminar: Sean Thompson, NC State, Quiver connections and bimodules of basic algebras

SAS 4201

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/

Computational and Applied Mathematics: Erik Bollt, Clarkson University, Next-Generation Reservoir Computing, and On Explaining the Surprising Success of a Random Neural Network for Forecasting Chaos

SAS 4201

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.…

Applied Mathematics Graduate Student Seminar: James Garrison, NC State, Randomized Preconditioned Cholesky-QR in Mixed Precision

SAS 4201

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…

Pure Math Graduate Student Seminar: Everett Meike, NC State, Cataloguing 2-adjacent knots

SAS 2106

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…

Geometry and Topology Seminar: Eric Hanson, NC State, Topological data analysis via relative homological algebra

SAS 4201

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…

Algebra and Combinatorics Seminar: Nick Mayers, NC State, The quantum k-Bruhat order

SAS 4201

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…

Differential Equations and Nonlinear Analysis Seminar: Jone Apraiz, Universidad del País Vasco, Inverse problem for a one-dimensional fluid-solid interaction model

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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…

Computational and Applied Mathematics: Isaac Harris, Purdue University, Transmission Eigenvalue Problems for a Scatterer with a Conductive Boundary

SAS 4201

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.…

Colloquium: Bojko Bakalov, NC State, Dynamical Lie algebras

SAS 4201

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…

Stochastics/Discrete Analysis Seminar: Peter Rudzis, UNC Chapel Hill, Well-posedness and stationarity of infinite systems of competing Brownian particles

SAS 4201

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…