Ensemble control deals with the problem of using a finite number of control inputs to simultaneously steer a large population (in the limit, a continuum) of individual control systems. As a dual, ensemble estimation deals with the problem of using a finite number of measurement outputs to estimate the initial state of every individual system in the (continuum) ensemble. We introduce in the talk a novel class of ensembles of nonlinear control systems, termed distinguished ensemble systems. Every such system has two key components, namely a set of finely structured control vector fields and a set of co-structured observation functions. In the first half of the talk, we demonstrate that the structure of a distinguished ensemble system can significantly simplify the analysis of ensemble controllability and observability. Moreover, such a structure can be used as a principle for ensemble system design. In the second half of the talk, we address the issue about existence of a distinguished ensemble system for a given manifold. We will focus on the case where the state space of every individual control system is an arbitrary semi-simple Lie group or its homogeneous space.

This is joint work with Robert Yuncken. We developed a coordinate-free geometric definition and a unified approach to various pseudodifferential calculi. The starting point of the construction is the identification of an appropriate groupoid, which is a purely geometric object. These groupoids are suitable adaptations of Connes' tangent groupoid. We show how, once the appropriate groupoid has been constructed, the details of the calculus are developed on its foundation. Our construction recovers the classical pseudodifferential calculus as well as, for example, the Heisenberg calculus (on contact manifolds) and its generalizations (to filtered manifolds).

The simplex category, the category of finite totally ordered sets, is a strict test category, meaning that there exists a product preserving Quillen equivalence between simplicial sets, presheaves on the simplex category, and "nice topological spaces". It is not clear however how the combinatorics of the simplex category relate to spectra, the objects which represent cohomology theories for spaces. All the well known models of spectra, Adams spectra, symmetric spectra, L-spectra, etc. may or do use simplicial sets as their presentation for spaces, but only one almost forgotten model, Kan's semisimplicial spectra of 1963 leverage the combinatorics of the simplex category for elegance and insight.
    We will begin by asking why the simplex category is a strict test category and why the the category of globes is not. We will then then develop a strict test category, the theta category of Joyal, and construct an elegant suspension functor for cellular sets, presheaves on the category theta. We will then present a conjectured new model of spectra based on this category which recovers the best aspects of Kan's semisimplicial spectra. We will  also comment on how this model of spectra may provide a universal algebraic lens to the distinction between stable and unstable homotopy theory.