We provide a generalization of Jouanolou duality that is applicable to a plethora of situations. The environment where this generalized duality takes place is a new class of rings, that we introduce and call weakly Gorenstein. As a main consequence, we obtain a new general framework to investigate blowup algebras. We use our results to…
Despite the success of deep learning-based algorithms, it is widely known that neural networks may fail to be robust to adversarial perturbations of data. In response to this, a popular paradigm that has been developed to enforce robustness of learning models is adversarial training (AT), but this paradigm introduces many computational and theoretical difficulties. Recent…
Abstract: Social networks play a crucial role in how we share and discuss ideas. This talk delves into the mathematics behind the spread of information (such as opinions or beliefs) on these networks. Imagine a network as a complex web, where each node represents an individual and the connections represent interactions. Opinions, beliefs, arguments, and other…
Virtual (Zoom link: https://ncsu.zoom.us/j/93542716306?pwd=c1Iyc0s2bUJhWUR0R2ZNcmlLSHJqQT09)
Models for networks that evolve and change over time are ubiquitous in a host of domains including modeling social networks, understanding the evolution of systems in proteomics, the study of the growth and spread of epidemics etc. This talk will give a brief summary of three recent findings in this area where stochastic approximation techniques…
Models for networks that evolve and change over time are ubiquitous in a host of domains including modeling social networks, understanding the evolution of systems in proteomics, the study of the growth and spread of epidemics etc. This talk will give a brief summary of three recent findings in this area where stochastic approximation techniques…
If you are interested in learning more about applied math research from your fellow students, or you want a friendly and constructive environment to practice presenting your own research, AMGSS is for you! This is an informational and sign-up meeting, so come to learn more about AMGSS and/or to sign up to present your research.
A fundamental question in Differential Geometry which has spurred much recent work is whether every orientable compact surface with boundary can be minimally embedded in the Euclidean 3-ball with free boundary. I will discuss work (joint with M. Karpukhin, R. Kusner, and D. Stern) which resolves this problem.
Fun activity: Prepare a 5-10 minutes presentation on a math object that you want others to know about. That could include their definition, a couple of examples, some fun fact or property, an open problem, etc. This is meant to be an informal and fun discussion .
Computational and applied mathematics are growing in popularity and relevance. As students prepare for either the job market or graduate school, it is increasingly more important that their mathematics courses reflect the changing landscape of the field. In this talk, I will detail my development and initial implementation of two new courses: Applied Calculus I…
Central-upwind schemes are Riemann-problem-solver-free Godunov-type finite-volume schemes, which are, in fact, non-oscillatory central schemes with a certain upwind flavor: derivation of the central-upwind numerical fluxes is based on the one-sided local speeds of propagation, which can be estimated using the largest and smallest eigenvalues of the Jacobian. I will introduce two new classes of central-upwind…
The goal of this talk is to price American-type financial contracts in the presence of Knightian uncertainty. In particular, instead of choosing a particular probabilistic model to represent the price process of some underlying asset (on which an American option is written), we first restrict our attention to the whole class of models that are…
Dense linear systems in large-scale kernel approximation in machine learning, discretization of boundary integral equations in mathematical physics, and low-rank approximation of Schur complements in large sparse matrix factorization often employ a multilevel structure of low-rank off-diagonal blocks. To solve such systems efficiently, we present a GPU-based parallel recursive skeletonization solver that utilizes batched dense…
As with weather, there is a limit to how far into the future we can accurately forecast the future course of epidemics. Yet, we often need to make plans further into the future than we can reasonably forecast, and in many cases our actions themselves will change the course of an epidemic. One solution to this conundrum is the construction of "planning scenarios" where we project the likely course of an epidemic well beyond the…
In this talk, I will introduce a novel Riemannian geometric model for wave propagation in gas giant planets. Terrestrial planets and gas giants have one key difference: in gas, density goes to zero at the surface, and seismic waves come to a full stop. We model the sound speed in a planet by a Riemannian…
In Stanley’s seminal work “Cohen-Macaulay Complexes”, Stanley conjectured that all h vectors of matroid complexes are pure O-sequences. We constructed coparking functions on matroids with extra restrictions and showed that the degree sequences of coparking functions are the same as h vectors of matroid complexes. By this construction, we proved that Stanley’s conjecture is true…
In this talk, I shall present a relaxation formula and duality theory for the multi-marginal Coulomb cost that appears in optimal transport problems arising in Density Functional Theory. The related optimization problems involve probabilities on the entire space and, as minimizing sequences may lose mass at infinity, it is natural to expect relaxed solutions which…
We consider weakly compressible flows coupled with a cloud system that models the dynamics of warm clouds. Our goal is to explicitly describe the evolution of uncertainties that arise due to unknown input data, such as model parameters and initial or boundary conditions. The developed stochastic Galerkin method combines the space-time approximation obtained by a…
We start by giving a short introduction to geometric inverse problems. Then we present our recent results regarding inverse problems for the minimal surface equation. Minimal surfaces are solutions to a quasilinear elliptic equation. We determine a minimal surface up to an isometry from the corresponding Dirichlet-to-Neumann map in dimension 2. Applications of the results…
Quantum hardware has advanced to the point where it is now possible to perform simulations of small physical systems. Although the current capabilities are limited, given the rapid advancement it is an opportune time to develop novel algorithms for the simulation of quantum matter, and to develop those that make it possible to make connections…