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. […]
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, […]
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 […]
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 […]
Abstract: An isometry between two normed vector spaces is a linear map that preserves the norm (i.e., the length of each output agrees with the length of its input). For […]
Abstract: Consider n independent, biased coins, each with a known probability of heads. Presented with an ordering of these coins, flip (i.e., toss) each coin once, in that order, until we have observed both a head and a tail, or flipped all coins. The Unanimous Vote problem asks us to find the ordering that minimizes […]
