Deep learning method has emerged as a competitive mesh-free method for solving partial differential equations (PDEs). The idea is to represent solutions of PDEs by neural networks to take advantage of the rich expressiveness of neural networks representation. In this talk, we will explore the applicability of this powerful framework to the kinetic equation, which…
Gaussian processes (GPs) are widely employed as versatile modeling and predictive tools in spatial statistics, functional data analysis, computer modeling and diverse applications of machine learning. They have been widely studied over Euclidean spaces, where they are specified using covariance functions or covariograms for modelling complex dependencies. There is a growing literature on GPs over…
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,…
Effective webpages are an increasingly important medium for disseminating information related to classes and research. For TAs, they provide a critical way to share material with your class whereas for those entering the job market, they are often checked by potential employers interested in obtaining more information about candidates. Hence it is important to construct…
One common problem in machine learning is to train a computer to correctly classify some data set when the potential set of features is known. However, the problem of classification becomes even more tricky if an adversary gains access to the input data set and can modify the features. In this talk, I will first…
Advisor is Mette Olufsen.
In this talk, I will present 3 problems on fluid structure interaction: 1) Flight stability of wedges: Recent experiments have shown that cones of intermediate apex angles display orientational stability with apex leading in flight. Here we show in experiments and simulations that analogous results hold in the two-dimensional context of solid wedges or triangular prisms in planar…
In this talk, I will present new results on the symmetry reduction of gas dynamic systems of PDEs following the general framework presented by Lev Ovsyannikov in his article "The “podmodeli” program. Gas dynamics" https://www.sciencedirect.com/science/article/pii/0021892894901376 The gas dynamics systems of equations, with an arbitrary state equation, has an 11-dimensional Lie algebra of symmetries which generates a group…
Speaker’s webpage: https://math.cas.lehigh.edu/andrew-harder In particle physics, many quantities of interest are expressed in terms of Feynman integrals. These integrals are attached to combinatorial objects called Feynman graphs, and can be expressed as integrals over (infinite) domains inside the real plane. In examples, one often finds that Feynman integrals are equal to special values of functions that…
This article is devoted to a general class of one dimensional NLS problems with a cubic nonlinearity. The question of obtaining scattering, global in time solutions for such problems has attracted a lot of attention in recent years, and many global well-posedness results have been proved for a number of models under the assumption that…
Classically, the primary objects one was concerned with in algebraic geometry were the zero sets of systems of polynomial functions, which we call varieties. Since the work of Grothendieck, the main objects of study in modern algebraic geometry are schemes. In this talk, I will introduce the concept of an affine scheme, a key part…
Of primary importance in computational science and applications is quantification and improvement of predictive capabilities of large-scale parameterized models, which often require the use of multi-query techniques that are intractable for computationally expensive models. This research focuses on three primary thrusts: sensitivity analysis, reduced-order modeling, and uncertainty quantification and algorithm and implementation choices that scale favorably…
The Morrey Conjecture concerns quasi-convexity and rank-one convexity of functions. While the former implies the latter, it's unclear if the converse is true. Sverak proved the conjecture in 3D, but it remains unresolved in the planar case. Analyzing these properties analytically is difficult, especially for vector-valued functions, hence we perform numerical simulations using example functions…
Disorder is a fact of life, and controlling it on the nanoscale is complex, expensive, and of limited use. On the other hand, disordered materials do offer a range of possible applications if we know how to identify their useful features. To this end, we propose an approach through Topological Learning to quantify disorder and…
Fusion categories are algebraic gadgets that have seen many applications in topology and mathematical physics. In particular, they can be used to encode topological quantum field theories in the sense of Atiyah. Classical examples of fusion categories include C-Rep(G), the category of finite dimensional complex representations of a finite group G. Because of their connections…
Motivated by motion of myxobacteria, we review several kinetic approaches for modeling motion of self-propelled, interacting rods. We will focus on collisional models of Boltzmann type and discuss the derivation of the governing equations, the range of their validity, and present some analytical and numerical results. We will show that collisional models have a natural…
SAS 4201 (Zoom link: https://ncsu.zoom.us/j/93542716306?pwd=c1Iyc0s2bUJhWUR0R2ZNcmlLSHJqQT09)
Hypertoric varieties are quaternionic analogs of toric varieties, important for their interaction with the combinatorics of matroids as well as for their prominent place in the rapidly expanding field of algebraic symplectic and hyperkahler geometry. In the last decade, hypertoric varieties have appeared prominently in investigations of symplectic duality, a mathematical incarnation of 3d mirror…
The classical Descartes’ rule of signs provides an easily computable upper bound for the number of positive real roots of a univariate polynomial with real coefficients. Descartes' rule of signs is of special importance in applications where positive solutions to polynomial systems are the object of study. This is the case in reaction network theory…
Deep neural networks (DNNs) have gained undeniable success as high-dimensional function approximators in countless applications. However, there is a significant hidden cost behind triumphs - the cost of training. Typically, DNN training is posed as a stochastic optimization problem with respect to the learnable DNN weights. With millions of weights, a non-convex and non-smooth objective…