We discuss extensions of several inequalities in harmonic analysis to the setting of families of orthonormal functions. While the case of Sobolev-type inequalities is classical, newer results concern the Strichartz inequality, the Stein-Tomas inequality and Sogge’s spectral cluster estimates, among others. Of particular interest is the dependence of the constants in the resulting bounds on…
This work explores the recovery stochastic volatility models (SVMs) from market models for the VIX futures term structure. Market models have more flexibility for fitting of curves than do SVMs, and therefore they are better-suited for pricing VIX futures and derivatives. But the VIX itself is a derivative of the S&P500 (SPX) and it is…
The study of extremals for Steklov eigenvalues has revitalised the theory of free boundary minimal surfaces. One of the most basic open questions can be phrased as follows: Can a surface of any given topology be realised as an embedded free boundary minimal surface in the 3-dimensional Euclidean unit ball? We will answer this question…
Factor systems are a tool for working on the extension problem for algebraic structures such as groups, Lie algebras, and associative algebras. We construct the Leibniz algebra analogue to a series of group-theoretic results from W. R. Scott’s Group Theory. Fixing a pair of Leibniz algebras A and B, we develop a correspondence between factor systems…
We consider the restriction to the unit circle of random degree-n polynomials with iid coefficients (Kac polynomials). Recent work of Yakir and Zeitouni shows that for Gaussian coefficients, the minimum modulus (suitably rescaled) follows a limiting exponential distribution. We show this is a universal phenomenon, extending their result to arbitrary sub-Gaussian coefficients, such as Rademacher…
In this talk, we will explain a (mysterious?) connection between the combinatorics of affine buildings and representation theory in type A. If a group acts simply and transitively on the vertices of an affine building in type A_n, it gives rise to a certain combinatorial structure called a triangle presentation. We will describe how triangle presentations…
In my talk, I will explain the approach of finding solutions to nonlinear PDEs via tug-of-war games. I will focus on the context of p-Laplacian and the non-local geometric p-Laplacian. Zoom meeting: Link
A curve of constant width has the property that any two parallel supporting lines are the same distance apart in all directions. A fundamental problem involving these curves is to find one which encloses the smallest amount of area for a given width. This problem was resolved long ago and has a few relatively simple solutions.…
Chair: Alen Alexanderian (alexanderian@ncsu.edu, contact for Zoom access)
Zoom ID: 939 3643 7199 (opens 15 min prior to the meeting) Passcode: 9917
In a Hilbert space, for convex optimization, we report on recent advances regarding the acceleration of first-order algorithms. We rely on inertial dynamics with damping driven by the Hessian, and the link between continuous dynamic systems and algorithms obtained by temporal discretization. We first review the classical results, from Polyak's heavy ball with friction method…
This is a report on the joint work with Matthew Levy. We use surjection operations on integral cochains tof a topological space X (described by McClure-Smith and Berger-Fresse) to describe a complex computing (co)homology of the cartesian power of X with some diagonals removed. Host: Radmila Sazdanovic ZOOM link: https://ncsu.zoom.us/j/97278681300
An important problem in computer vision is to understand the space of images that can be captured by an arrangement of cameras. A description of this space allows for statistical estimation methods to reconstruct three-dimensional models of the scene that was imaged. The set of images captured by an arrangement of pinhole cameras is usually…
Let X and Y be random variables. Quasi-independence models are log-linear models that describe a situation in which some states of X and Y cannot occur together, but X and Y are otherwise independent. We characterize which quasi-independence models have rational maximum likelihood estimator, or MLE, based on combinatorial features of the bipartite graph associated…
Suppose we model n votes in an election between two candidates as n i.i.d. uniform random variables in {-1,1}, so that 1 represents a vote for the first candidate, and -1 represents a vote for the other candidate. Then, for each vote, we flip a biased coin (with fixed probability larger than 1/2 of landing…
We consider a finite group scheme G, and its associated representation category rep G. Here one can think of a finite discrete group, or an infinitesimal group scheme, such as the kernel of the r-th Frobenius map for GL_n over F_p. Via a standard tensor categorical construction one has Drinfeld's center Z(rep G) of the…
We survey the known results concerning the minimal Riesz energy on sufficiently smooth sets, and present some new results on fractal sets, which are the key examples of non-rectifiable sets. In particular, we will talk about the connection of these problems to ergodic theory and probability theory: the key step in our proofs is applying…
The Ricci curvature Ric(g) is a symmetric 2-tensor on a Riemannian manifold (M,g) that encodes curvature information. It features in several interesting geometric PDEs such as the Ricci flow and the Einstein equation. The nature of Ric(g) as a differential operator -- nonlinear and degenerate elliptic -- make these equations particularly challenging. Host: Peter McGrath Instructions to join: Zoom…