In this talk, I will discuss how incorporating geometric information into classical learning algorithms can improve their performance. The main focus will be on optimal mass transport (OMT), which has evolved as a major method to analyze distributional data.  In particular, I will show how embeddings can be used to build OMT-based classifiers, both in supervised and unsupervised learning settings. The proposed framework significantly…

Qualitatively, a tipping point in a dynamical system is when a small change in system inputs causes the system to move to a drastically different state. The discussion of tipping points in climate and related fields has become increasingly urgent as scientists are concerned that different aspects of Earth’s climate could tip to a qualitatively different state without…

In many recent works, analysis and number theory go beyond working side by side and team up in an interconnected back and forth interplay to become a powerful force. Here I describe two distinct meetings of the pair, which result in sharp counts for equilateral triangles in Euclidean space and statistics for how often a random polynomial has Galois group not isomorphic to the full symmetric group. https://ncsu.zoom.us/j/91896366693?pwd=YnFuZURGc1NNenRTQ3YrbjVTK0dQZz09 Meeting ID: 918 9636 6693 Passcode: 875811

Speaker: Lisa Fauci, Professor of Mathematics, Tulane University Abstract: In the past decade, the study of the fluid dynamics of swimming organisms has flourished. My research has been centered on answering questions about biophysics using computational models of fluid flow: How do mammalian sperm penetrate the outer layer of an egg for successful fertilization?  How…

The stochastic processes of evolution have generated DNA, RNA, and protein sequences. These sequences determine how these entities chemically interact with themselves and each other, form physical structures, and functionally behave as signals and/or machines within cells. My research involves reconstructing the history of the stochastic processes that led to the sequences we observe today…

We define vertex-colourings for edge-coloured digraphs, which unify the theory of P-partitions and proper vertex-colourings of graphs. Furthermore, we use our vertex-colourings to define generalized chromatic functions, which merge the chromatic symmetric and quasisymmetric functions and generating functions of P-partitions. We also discuss the relations between generalized chromatic functions, Schur functions in noncommuting variables, and the well-known Stanley-Stembridge (3+1)-free conjecture.

Stochastic fluctuations drive biological processes from particle diffusion to neuronal spike times. The goal of this talk is to use a variety of mathematical frameworks to understand such fluctuations and derive insight into the corresponding applications. We start by considering a novel stochastic process motivated by astrocytes, glial cells that ensheath neuronal synapses and can…

Machine learning (ML) has achieved unprecedented empirical success in diverse applications. It now has been applied to solve scientific problems, which has become an emerging field, Scientific Machine Learning (SciML). Many ML techniques, however, are very complex and sophisticated, commonly requiring many trial-and-error and tricks. These result in a lack of robustness and interpretability, which…

In this talk, we study an exploration version of continuous time expected utility maximization problem with reinforcement learning. It is shown that the optimal feedback policy is Gaussian. We then prove a policy improvement theorem. An implementable reinforcement learning algorithm is designed. Numerical examples are provided for illustrations. https://ncsu.zoom.us/j/95758380569?pwd=OFZKWnVQTkJVTTNPU1R2TkhXQzdPZz09 Meeting ID: 957 5838 0569 Passcode: 832132

The concept of multidegrees provides the right generalization of the degree of a projective variety to a multiprojective setting. The study of multidegrees goes back to seminal work by van der Waerden in 1929. We will slowly introduce the notion of multidegrees of a multiprojective variety. A complete characterization of the positivity of multidegrees will…

Recent years have witnessed tremendous progress in developing and analyzing quantum computing algorithms for quantum dynamics simulation of bounded operators (Hamiltonian simulation). However, many scientific and engineering problems require the efficient treatment of unbounded operators, which frequently arise due to the discretization of differential operators. Such applications include molecular dynamics, electronic structure theory, quantum control…

Many problems in scientific computing require minimizing nonsmooth optimization problems. In many applications, it is common to minimize the sum of a smooth nonconvex function and a nonsmooth convex function. For example, imaging  and data science applications require minimizing a measure of data misfit plus a sparsifying L1- or total-variation regularizer.  We develop a novel…

A hyperplane arrangement is a union of codimension one linear spaces.  These simple objects provide fertile ground for interactions between combinatorics, algebra, algebraic geometry, topology, and group actions.  The combinatorics of an arrangement is encoded by the pattern of intersections among the hyperplanes, called its intersection lattice.  On the other hand, a key algebraic object…

What is the cheapest way to superhedge a path-dependent derivative security? If liquid European calls and the underlying risky stock can be used for hedging, then the lowest superhedging price corresponds to the highest expected cost of the exotic claim. Each expected cost is associated to a probabilistic model which makes the risky stock a martingale…

The solution of large sparse linear systems is an essential building block in many science and engineering applications. It is also often the main computational bottleneck. For large problems, direct solvers (based on, e.g., LU or Cholesky factorizations) can require a significant amount of computing resources. By contrast, iterative solvers (e.g., CG and GMRES) can…

 Hyperbolic polynomials are coordinate free generalization to the notion of real rooted polynomials. A special class of hyperbolic polynomials are determinantal polynomials and they bound spectrahedra, feasible sets of semidefinite programming. I shall discuss some techniques of real algebraic geometry to deal with convex semialgebraic sets such as spectrahedra, and  demonstrate the applications of hyperbolic…

We study the problem of prediction of binary sequences with expert advice in the online setting, which is a classic example of online machine learning. We interpret the binary sequence as the price history of a stock, and view the predictor as an investor, which converts the problem into a stock prediction problem. In this framework, an investor, who predicts the daily…

Schubert calculus has its origins in enumerative questions asked by the geometers of the 19th century, such as “how many lines meet four fixed lines in three-space?”  These problems can be recast as questions about the structure of cohomology rings of geometric spaces such as flag varieties.  Borel’s isomorphism identifies the cohomology of the complete…

Many polynomials in combinatorics (and in other areas of mathematics) have nice properties such as having all of their roots being real numbers, or having all of their coefficients being nonnegative.  By surveying recent advances in the Hodge theory of matroids (namely, the nonnegativity of Kazhdan-Lusztig polynomials of matroids and Dowling and Wilson's top-heavy conjecture…

In this brief talk, I introduce Archimedean Riesz spaces, also known as vector lattices, and their Archimedean Riesz space tensor product as defined by Fremlin. I define what it means to be an ideal in a Riesz space, then provide a counterexample to the statement ``the Fremlin tensor product of ideals is an ideal." Specifically,…