A virtual property of a 3-manifold is a property satisfied by a finite cover of the 3-manifold. The study of such properties has been at the heart of several major developments in 3-manifold topology in the past decade. In this talk I will provide motivation and background on these virtual properties and discuss some recent results. Zoom…
Optimization is a broad area with ramifications in many disciplines, including machine learning, control theory, signal processing, robotics, computer vision, power systems, and quantum information. I will talk about some novel algorithmic and theoretical results in two broad classes of optimization problems. The first class of problems are semidefinite programs (SDP). I will present the…
Schubert calculus has its origins in enumerative questions asked by the geometers of the 19th century, such as "how many lines meet four fixed lines in three-space?" These problems can be recast as questions about the structure of cohomology rings of geometric spaces such as flag varieties. Borel's isomorphism identifies the cohomology of the complete…
Carleson embedding theorem is a building block for many singular integral operators and the main instrument in proving ``Leibniz rule" for fractional derivatives (Kato--Ponce, Kenig). It is also an essential step in all known ``corona theorems’’. Multi-parameter embedding is a tool to prove more complicated Leibniz rules that are also widely used in well-posedness questions…
In 2011, Lutwak, Yang and Zhang extended the definition of the Lp-Minkowski convex combination (p ≥ 1) from convex bodies containing the origin in their interiors to all measurable subsets in R n , and as a consequence, extended the Lp-Brunn-Minkowski inequality to the setting of all measurable sets. In this talk, I will present…
A Rademacher sum X is a random variable characterized by real numbers a_1, ..., a_n, and is equal to X = a_1 x_1 + ... + a_n x_n, where x_1, ..., x_n are independent signs (uniformly selected from {-1, 1}). A conjecture by Bogusław Tomaszewski, 1986: all Rademacher sums X satisfy Pr >= 1/2. We…
Consider a metric space (S,d) with an upper curvature bound in the sense of Alexandrov (i.e.~via triangle comparison). We show that if (S,d) is homeomorphically equivalent to the 2-sphere, then it is conformally equivalent to the 2-sphere. The method of proof is through harmonic maps, and we show that the conformal equivalence is achieved by…
I shall prove a neat inequality concerning convex bodies and semi-norms, and show that cylinders (i.e. direct products of intervals with convex bodies of lower dimension) give equality cases for it. I shall explain its connections to the Log-Brunn-Minkowski conjecture. If time permits, I will discuss related results and surrounding questions. Based on a joint…
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…