The Effros-Shen algebra corresponding to an irrational number $\theta$ can be described by an inductive sequence of direct sums of matrix algebras, where the continued fraction expansion of $\theta$ encodes the dimensions of the summands, and how the matrix algebras at the nth level fit into the summands at the (n+1)th level. In recent work, […]
Title: Developments in Noncommutative Fractal Geometry Abstract: As a noncommutative fractal geometer, I look for new expressions of the geometry of a fractal through the lens of noncommutative geometry. At the quantum scale, the wave function of a particle, but not its path in space, can be studied. Riemannian methods often rely on smooth paths to encode […]
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 metric in \R^3? i.e., what is the "round metric" on the 2-sphere of radius rho? 2. If we impose a complex structure on S^2 via […]
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. […]
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 […]
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 […]
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 […]
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. […]
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, […]
