Title: The Chronicles of Fractal Geometry: Fractal Strings, and Functorial Harps Abstract: In this talk, we explore the colorful analytical world of fractal geometry. We introduce fractal strings in the sense of Lapidus, both intuitively and by way of rigorous constructions. We examine rich illustrations of higher dimensional fractals and p-adic fractal strings. Then, we […]
Title: Review of differential geometry Abstract: 1. Given the embedding of a sphere of radius rho centered at the origin of \R^3 from spherical coordinates, what is the pullback of the flat […]
Title: A crash course in Bornologies Abstract: By a bornology on a nonempty set X, we mean a family of subsets that contains the singletons, that is stable under finite unions, and that is stable under taking subsets. The prototype for a bornology is the so-called metric bornology: the family of metrically bounded subsets of […]
Title: What can chicken McNuggets tell us about symmetric functions, positive polynomials, random norms, and AF algebras? Abstract: Numerical semigroups are combinatorial objects that lead to deep and subtle questions. With tools from complex, harmonic, and functional analysis, probability theory, algebraic combinatorics, and computer-aided design, we answer virtually all asymptotic questions about factorization lengths in […]
Title: Domains of Quantum Metrics on AF algebras Abstract: Given a compact quantum metric space (A, L), we prove that the domain of L coincides with A if and only if A is finite-dimensional. Intuitively, this should allow for different quantum metrics with distinct domains when A is infinite-dimensional, and we show how to explicitly […]
Title: Exceptional Sets for Divergent Fourier Series Abstract: A survey of some old and newer results on divergent Fourier series with some comments on how they relate to undergraduate analysis […]
Title: Geometric classification problems with the Bergman metric Abstract: One of the common problems in mathematics is the classification problem: When are two mathematical structures really the same? The classification problem appears throughout undergraduate mathematics courses in different forms. For example, in an abstract algebra course, one asks when are two groups isomorphic? In a […]
Title: Transfinite Apollonian metric Abstract: The concept of transfinite diameter of compact sets in the complex plane was introduced by Fekete in 1923. It is a generalization of the standard diameter of sets and has found many applications in the study of conformal mappings. The Apollonian metric was introduced by A. Beardon in 1995 and […]
Abstract: The three-dimensional incompressible Euler equations describe the motion of an ideal fluid, yet the mechanisms that govern the possible loss of regularity of smooth solutions remain only partially understood. A classical result of Beale, Kato, and Majda shows that if a smooth solution breaks down in finite time, then the time integral of the […]
Abstract: In general, the objective of algebraic topology is to classify spaces using some algebraic invariants or up to some notion of equivalence. In the area of equivariant homotopy theory, the goal is the same but now spaces equipped with a group action are considered and algebraic invariants of choice are homotopy groups. It turns […]
Abstract: Let C be a compact convex set (in a locally convex topological vector space). By Choquet’s theorem, every point in C is the barycenter of a probability measure supported on the extreme points. When this representing measure is unique, C is called a simplex. Simplices arise naturally in various fields of mathematics: the space […]
Abstract: We study metrics on completely positive maps, and in particular on quantum channels, induced by seminorms from noncommutative geometry. Using an infinite-dimensional analogue of the Choi–Jamiołkowski correspondence, we construct such metrics and show that, under suitable assumptions, they satisfy stability and chaining. I will present the main ideas and explain how spectral triples and […]
