# Workshop on Lie Groups, Singular Spaces, and Higher Structures

## Description

Geometric approaches to classical mechanics and quantum mechanics have evolved over several decades.

The phase space of a classical mechanical system is modeled by a symplectic manifold; geometric quantization consists of mathematical procedures that start with the phase space of a classical mechanical system and build a corresponding quantum mechanical system. On the classical level, time evolution is modeled by a Hamiltonian flow on phase space, and symmetries of phase space are modeled by Hamiltonian group actions.

These ideas have lead to extensive mathematical activity in the last half a century, connecting symplectic geometry with Lie theory and representation theory. This activity focuses on smooth — infinitely differentiable — objects: smooth functions, maps, group actions. Approaching these objects geometrically means that one seeks to work with their intrinsic structures that are independent of choices of coordinates. (Smooth) manifolds provide a convenient playfield for exactly this: differential calculus on objects beyond the Cartesian spaces $R^n$ .

On many occasions, starting with smooth manifolds, one ends up with new objects that are no longer smooth manifolds. In classical mechanics, this can occur as a result of introducing constraints into the system. More generally, this can occur as a result of the Marsden-Weinstein reduction procedure, in which the presence of symmetries allows us to reduce the number of degrees of freedom by passing to a sub-quotient of our original system. Another context is that of hybrid dynamical systems, such as the “hopping hoop”. These situations create the need for differential calculus on objects that are more general than manifolds. Differential calculus beyond manifolds can be done through several different approaches. Naive but powerful such approaches include diffeology in the sense of Souriau, and differential structures in the sense of Sikorski. Modern approaches, often inspired by algebraic geometry and adapted to the $C∞$ context, include — among others — Lie groupoids and differentiable stacks, and $C∞$ schemes.

The purpose of this conference is to create a discussion between experts that work on a number of such approaches to singular spaces as well as on the broader themes that lead to these structures: Hamiltonian group actions, completely integrable systems, and geometric quantization.

The “Cartan package” on smooth manifolds consists of the standard relationships between vector fields and flows and between these and differential forms. There is no satisfactory “Cartan package” for singular spaces; even within individual approaches, there are often several inequivalent definitions of differential forms and of vector fields. The value of such definitions can be evaluated according to whether they provide a framework in which one can adapt standard results about manifolds to more general contexts of singular spaces. Our proposed speakers have developed such adaptations in various contexts: de Rham theories of symplectic reduced spaces and on leaf spaces of foliations; applications of category-theoretic structures to hybrid dynamical systems; applications of Lie groupoids and Lie algebroids to Cartan realization problems; classification of symplectic toric stacks.

The discussion that we will foster in this conference will shed light on relationships between different approaches to singular spaces. Any achievement of any one of these approaches becomes a testing ground for the other approaches. For example, Sjamaar’s de Rham theory on reduced spaces incorporates integration and Stokes’s theorem, but “remembers” the original ambient manifold. On reduced spaces, can one incorporate integration and Stokes’s theorem into the intrinsically defined diffeological differential forms? Or, when viewing such reduced spaces as $C∞$ schemes, is there a notion of differential forms that includes the Marsden-Weinstein symplectic form on the reduced spaces and that allows integration and Stokes’s theorem?

Informally, this conference is also connected with the sixty-first birthdays of Eugene Lerman and Reyer Sjamaar, whose 1991 Annals paper “Stratified symplectic spaces and reduction” has had tremendous impact on this field and continues to provide motivation, as well as testing-ground, for various approaches to differential calculus beyond manifolds.