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…
The Z-hat invariant, proposed by Gukov, Pei, Putrov, and Vafa, is a fascinating q-series invariant of three-manifolds related to various areas of mathematics and physics. We want to find a topological quantum field theory (TQFT) that computes this invariant to understand it better. This invariant cannot come from TQFT in the sense of Atiyah, as…
We are concerned with the rigorous justification of the so-called hard congestion limit from a compressible system with singular pressure towards a mixed compressible-incompressible system modeling partially congested dynamics, in the framework of BV solutions. We will consider small BV perturbations of reference solutions constituted by (possibly interacting) large interfaces, and we will analyze the dynamics of…
Please SAVE THE DATE for the Department's Awards Ceremony and End of Semester Meeting. ceremony and tea to follow.
Deep learning is one of the most universal techniques in the modern big data era, achieving remarkable success across imaging, healthcare, natural language processing, and more. As applications begin to rely more heavily on deep learning, it is crucial that we understand how these algorithms make predictions and how we can make them better (e.g.,…
Once per semester, the Teaching and Learning Seminar will host 10-minute "lightning talks" in which 4 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…
Please save the date for the College of Sciences’ annual Faculty and Staff meeting at 4 p.m. on Wednesday, April 20, in 301 Riddick Hall. Interim Dean John Blondin will join us as we discuss topics that are important to you. There will be time for you to ask questions too.A reception in the Riddick Hearth…
Deep learning is one of the most universal techniques in the modern big data era, achieving remarkable success across imaging, healthcare, natural language processing, and more. As applications begin to rely more heavily on deep learning, it is crucial that we understand how these algorithms make predictions and how we can make them better (e.g.,…
Symmetric functions are very interesting combinatorial objects. After decomposing certain symmetric functions in terms of different bases, we can often combinatorially interpret the resulting coefficients. This is made particularly easy if those coefficients are positive (or controllably negative). The chromatic symmetric function is one such symmetric function, and its decomposition has been of interest for…
Congenital heart disease affects 1 in 100 infants and is the leading cause of infant mortality in the US. Among the most severe forms of congenital heart disease is single ventricle physiology, in which the heart develops with only one functional pumping chamber. These patients typically undergo three open chest surgeries, culminating in the Fontan…
Many areas of interest within the Earth are anisotropic, meaning that the speed of sound is different in different directions. It turns out that pressure waves are far better behaved than shear waves, but fortunately the different polarizations are coupled together through algebraic geometry. I will explain the surprising power of algebraic geometry in the…