to follow

Building on the concept of diagonal dimension introduced by Li, Liao, and Winter in 2023, we propose a topological dimension for an inclusion pair of C*-algebras. This new framework allows for finite values in cases of Cartan inclusions that are not diagonal. In this talk, we present calculations for both upper and lower bounds concerning the inclusion of the unitization of c_0(\mathbb{N}) into the Toeplitz algebra. This work is a collaboration with W. Winter.

The landmark completion of the Elliott classification program for unital separable simple nuclear C*-algebras saw three regularity properties rise to prominence: Z-stability, a C*-algebraic analogue of von Neumann algebras' McDuffness; finite nuclear dimension, an operator algebraic version of having finite Lebesgue dimension; and strict comparison, a generalization of tracial comparison in II_1 factors. Given their relevance to classification, most of the investigations into their interplay have focused on the simple nuclear case.

 The purpose of this talk is to advertise the general study of these properties and discuss their applications both within and outside operator algebras. Concretely, I will explain how characterizing when certain twisted group C*-algebras are Z-stable can provide new partial solutions to a well-known problem in generalized time-frequency analysis; this is joint work with U. Enstad. If time allows, I will also briefly discuss how a different incarnation of tracial comparison (finite radius of comparison) for non-commutative tori relates to the existence of smooth Gabor frames; this last part is joint work with U. Enstad and also H. Thiel.

Guentner, Willet and Yu defined a notion of dynamic asymptotic dimension for an étale groupoid that can be used to bound the nuclear dimension of its groupoid C*-algebra.  To have finite dynamic asymptotic dimension, the isotropy subgroups of the groupoid must be locally finite.  I will discuss 1) how to use similar ideas to bound the nuclear dimension of the C*-algebra of a groupoid with `large' isotropy subgroups and 2) the limitations of that approach. In an application to the C*-algebra of a directed graph,  if the C*-algebra is stably finite, then its nuclear dimension is at most 1.  This is joint work with Dana Williams. 

Structural mechanics plays a crucial role in soft matter physics, mechanobiology, metamaterials, pattern formation, active matter, and soft robotics. What unites these seemingly disparate topics is the natural balance that emerges between elasticity, geometry, and stability. This seminar will serve as a high-level overview of our work on several problems concerning the stability of structures. I will cover three topics: (1) shapeshifting shells; (2) mechanical metamaterials; and (3) elastogranular mechanics.

I will begin by discussing our development of a generalized, stimuli-responsive shell theory. (1) Non-mechanical stimuli including heat, swelling, and growth further complicate the nonlinear mechanics of shells, as simultaneously solving multiple field equations to capture multiphysics phenomena requires significant computational expense. We present a general shell theory to account for non-mechanical stimuli, in which the effects of the stimuli are
generalized into three forms: those that add mass to the shell, those that increase the area of the shell through the natural stretch, and those that change the curvature of the shell through the natural curvature. I will show how this model can capture the morphogenesis of the optic cup, the snapping of the Venus flytrap, leaf growth, and the buckling of electrically active polymer plates. (2) I will then discuss how cutting thin sheets and shells, a process
inspired by the art of kirigami, enables the design of functional mechanical metamaterials. We create linear actuators, artificial muscles, soft robotic grippers, and mechanical logic units by systematically cutting and stretching thin sheets. (3) Finally, if time permits, I will introduce our work on the interactions between elastic and granular matter, which we refer to as elastogranular mechanics. Such interactions occur across all lengths, from morphogenesis, to root growth, to stabilizing soil against erosion. We show how combining rocks and string in the absence of any adhesive we can create large, load bearing structures like columns, beams, and arches. I will finish with a general phase diagram for elastogranular behavior.