## Proper Groupoids

There are a couple of definitions of a groupoid mathematically, but intuitively one may think of a groupoid as a space equipped with a set of rules which say which points of the space one can move between. A great example is a knight on a chessboard: if the knight starts on a particular square, every possible combination of moves that the knight can do is encoded in the groupoid. Since a knight can actually move to any square on the chessboard, perhaps this is not so interesting. A bishop, which is constrained to all squares of the same colour, is more interesting: the groupoid encodes all the moves that the bishop can make regardless of where it starts on the chessboard. The groupoid even encodes the different ways a piece can start at one square and end on another; for example, if a rook starts in the upper left corner and wants to get to the lower right corner, it could go all the way across horizontally, and then all the way down, or go all the way down first and then across. These are considered two different ways, and the are distinguished as such in the groupoid.

A little more formally, a Lie groupoid is a space (the objects) equipped with a space of invertible arrows (the "ways" to get from one point in the space to another), satisfying some techinical conditions. The spaces are smooth manifolds, and properness simply ensures that the space of arrows does not get too wild. It turns out that proper Lie groupoids locally (in a very loose sense of the word "local") look like representations of compact groups, and so they have very nice orbit spaces. I am interested in examining local and global invariants of proper Lie groupoids from this perspective.

Much of my research falls under the category of proper Lie groupoids. Lie group actions are examples of Lie groupoids, and so the results on differential forms on orbit spaces of compact Lie group actions, proper Lie group actions, and proper Lie groupoids, respectively discussed in [W:PhD], [KW], and [W:gpds] all fall under this umbrella. One can also talk about actions of proper groupoids, and compute their Bredon cohomology in certain cases (see [FSW]). I am also interested in invariants obtained via the orbit space, and I have focused so far on orbifolds [W:orb] and circle actions [CDGMW] in this regard. In particular, the orbit spaces of these Lie groupoids are uniquely determined (up to a mild form of equivalence) by their topology, their stratifications, and certain integer labels associated to the strata. Most of these can be obtained from the differential space structure (the smooth functions) on the space.

- [CDGMW] Suzanne Craig, Naiche Downey, Lucas Goad, Michael J. Mahoney, Jordan Watts, Orbit spaces of linear circle actions, Involve, Vol. 12 (2019), 941-959.
- [FSW] Carla Farsi, Laura Scull, and Jordan Watts, Classifying spaces and Bredon (co)homology for transitive groupoids, Proc. Amer. Math. Soc., Vol. 148 (2020), 2717-2737.
- [KW] Yael Karshon and Jordan Watts, Basic forms and orbit spaces: a diffeological approach, SIGMA Symmetry Integrability Geom. Methods Appl., Vol. 12 (2016), 026, 19 pages.
- [W:orb] Jordan Watts, The differential structure of an orbifold, Rocky Mountain J. Math., Vol. 47 (2017), 289-327.
- [W:gpds] Jordan Watts, The orbit space and basic forms of a proper Lie groupoid, In: Current Trends in Analysis, its Applications and Computation: Proceedings of the 12th ISAAC Congress, Aveiro, Portugal, 2019; Trends in Mathematics: Research Perspectives; Birkhäuser, 2022.
- [W:PhD] Jordan Watts, Diffeologies, Differential Spaces, and Symplectic Geometry, PhD Thesis, Department of Mathematics, University of Toronto, 108 pages, 2012.