Let x_1,...,x_n be an overdetermined spanning set for the Euclidean space R^k, where n > k. Let L be the integer span of these vectors. Then L is an additive subgroup of R^n. When is it discrete in R^n? Naturally, this depends on the choice of the spanning set, but in which way? We will […]
We introduce the notion of linear multifractional stable sheets in the broad sense (LMSS) to include both linear multifractional Brownian sheets and linear multifractional stable sheets. The purpose of the framework is to study the existence and joint continuity of the local times of LMSS, and also the local Holder condition of the local times […]
Graph products of groups were introduced in E. Green’s thesis in the 90’s as generalizations of Right-Angled Artin Groups. These have become objects of intense study due to their key roles in topology and group theory. Recently, Caspers and Fima introduced graph products of von Neumann algebras. Since their inception, several structural aspects such as […]
Given a unital AF (approximately finite-dimensional) algebra A equipped with a faithful tracial state, we equip each (norm-closed two-sided) ideal of A with a metrized quantum vector bundle structure, when […]
We comment on the main steps to take when studying some variational problems. This includes optimization problems arising in geometry, machine learning, non linear elasticity, fluid mechanics, etc... For the sake of illustration, in this talk, we keep our focus on a minimization problem obtained after a time-discretization of the incompressible Navier-Stokes equations. Elementary geometric […]
In this talk we discuss the Hilbert space approach, or the variational approach, in the study of questions of existence and multiplicity for some two-point boundary-value problems for nonlinear, second order, ordinary differential equations (ODEs). We illustrate the use of the Hilbert space approach in obtaining some old existence results for periodic solutions of a […]
Using elementary methods from differential equations and analysis we will consider the existence and multiplicity of solutions to semilinear partial differential equations with boundary conditions.
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 […]
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 […]
