Petersen and Tenner defined the depth statistic for Coxeter group elements which, in the symmetric group, can be described in terms of a cost-minimization problem over the factorizations of a permutation into transpositions. We generalize that cost function to the other classical (finite and affine) Weyl groups, letting the cost of an individual reflection t…
I will describe how an invariant measure with Cauchy-distributed marginals arises from a supercritical stochastic heat equation with an additional, independent additive noise. Joint work with Chiranjib Mukherjee. Zoom meeting: Link
Once per semester, the Teaching and Learning Seminar will host 10-minute "lightning talks" in which graduate students and/or faculty members will talk about some element of their teaching that they'd like to share. This may be a cool example you came up with for a class you are teaching, an innovative teaching technique or application…
Deep neural networks (DNNs) have been a successful model across diverse machine learning tasks, increasingly capturing the interest for their potential in engineering problems where PDEs have long been the dominant model. This talk delves into efficient training for PDE operator learning in both the forward and the inverse problems setting. Firstly, we address the curse…
Schoen-Yau proved the spacetime positive energy theorem by reducing it to the time-symmetric (Riemannian) case using the Jang equation. To acquire solutions to the Jang equation, they introduced a family of regularized equations and took the limit of regularized solutions, whereas a sequence of regularized solutions could blow up in some bounded regions enclosed by apparent horizons. They analyzed the blowup behavior near and outside the apparent horizons, but what happens inside…
Reduced-order models (ROMs) are a critical tool for sensitivity analysis, parameter inference, and uncertainty quantification where high-fidelity models would be computationally intractable. Galerkin POD-ROMs are one particular class of ROMs which project high-fidelity model equations onto a set of model solutions to construct ROMs retaining original model parameters and physics, enabling accurate sensitivity analysis, parameter inference,…
Zoom Meeting: https://ncsu.zoom.us/j/93046132033?pwd=dkZiTjlKazgzK2Q3aXJra1g2R1Q0dz09 Meeting ID: 930 4613 2033 Passcode: 075251
In 1995, Stanley introduced the chromatic symmetric function of a graph, a symmetric function analog of the classical chromatic polynomial of a graph. The Stanley-Stembridge e-positivity conjecture is a long-standing conjecture that states that the chromatic symmetric function of a certain class of graphs, called incomparability graphs of (3+1)-free posets, has nonnegative coefficients when expanded…
In this talk I will first review recent results which characterize adversarial training (AT) of binary classifiers as nonlocal perimeter regularization. Then I will speak about a probabilistic generalization of AT which also admits such a geometric interpretation, albeit with a different nonlocal perimeter. Using suitable relaxations one can prove the existence of solutions for…
As part of a professional development project aimed at engaging in the Discipline of Noticing as conceptualised by John Mason, some colleagues and I wrote and shared brief-but-vivid accounts of our practice. Evident in these accounts is the catalyst for the subsequent change in my practice of introducing short, pre-recorded videos to complement, or substitute…
It is well-known that nearly any function can be approximated arbitrarily-well by a neural network with non-linear activations. However, one cannot guarantee that the weights in the neural network remain bounded in norm as the approximation error goes to zero, which is an important consideration when practically training neural networks. This raises the question: What…
All BMA seminars have a virtual option with the following Zoom Link: https://ncsu.zoom.us/j/93046132033?pwd=dkZiTjlKazgzK2Q3aXJra1g2R1Q0dz09 Meeting ID: 930 4613 2033 Passcode: 075251
We examine the asymptotic behaviors of solutions to Hamilton-Jacobi equations while varying the underlying domains. We establish a connection between the convergence of these solutions and the regularity of the additive eigenvalues in relation to the domains. To accomplish this, we introduce a framework based on Mather measures that enables us to compute the one-sided derivative…
Extreme weather events are of growing concern for societies because under climate change their frequency and intensity are expected to increase significantly. Unfortunately, general circulation models (GCMs)--currently the primary tool for climate projections--cannot characterize weather extremes accurately. Here, we report on advances in the application of a multi-scale deep learning framework, trained on reanalysis data,…
In this talk we consider an Electro-Magnetic Wave operator on a complete Riemannian manifold, which generalizes the standard wave equation to include first order and zeroth order terms. The Cauchy Problem for the Electro-Magnetic Wave operator with zero initial values and a smooth compactly supported forcing function has a unique smooth solution. We study the…
In this talk, we consider an inverse problem for the nonlinear Boltzmann equation with a time-dependent kernel in dimensions 2 and higher. We establish a logarithm-type stability result for the collision kernel from measurements under certain additional conditions. A uniqueness result is derived as an immediate consequence of the stability result. Our approach relies on…
The purpose of this project is to develop, test, and document performance of dual-domain clustering algorithms for spatiotemporal datasets, tailored to pandemic preparedness and endgame challenges. Dual-domain clustering refers to the unsupervised learning clustering method performed on data with both application-specific attributes (e.g., number of infectious) and geographic information (e.g., latitude and longitude of data…