MASTER THESIS PROJECT
Lie groupoids and effective equivalences in the theory of differentiable dynamical systems
Generally (and loosely) speaking, the theory of differentiable dynamical systems is concerned with the study of the spaces of orbits arising from actions of Lie groups on manifolds [6]. These orbit spaces sit in a much larger class of spaces, the spaces of orbits of Lie groupoids. The notion of Lie groupoid elegantly unifies and generalizes several independent notions, including that of manifold, Lie group, Lie group action, principal bundle, and foliation. While ordinary Lie groups only describe global symmetries of geometric structures, Lie groupoids offer the additional possibility of talking about local symmetries of such structures, within a unified framework. In the classical theory of dynamical systems, too, the use of the language of Lie groupoids would seem to be natural, if not necessary. For example, the flow of an arbitrary vector field on a manifold does not correspond, in general, to a one-parameter group of diffeomorphisms of the manifold (that is, an action of the additive group of the real numbers on the manifold), but it does always correspond to a Lie groupoid, known as the flow groupoid of the vector field. Moreover, certain apparently cumbersome definitions admit a straightforward reformulation using the language of Lie groupoids: for instance, topological equivalence of vector fields simply amounts to isomorphism of the corresponding flow groupoids. The notion of Lie groupoid, however, seems to be absent from most of the literature on dynamical systems.
When one seeks to unify and generalize the classical theory of differentiable dynamical systems by introducing Lie groupoids into its foundations, there are a couple of questions that need to be answered: What is to be understood rigorously by the orbit space of a Lie groupoid? What are the relevant geometric properties of these orbit spaces that one is supposed to study? As an example, consider a constant vector field in euclidean space. It is intuitively obvious that the space of orbits (that is to say integral curves) in this case should just be a euclidean plane, pictured as cutting the orbits transversely. The same conclusion can be drawn for an arbitrary Lie group action whose orbits are the same as in this example. If on the other hand a Lie group acts transitively on a manifold, then the orbit space should consist of just a single point. In each one of these examples, the orbit space happens to be a manifold. The appropriate geometric structure to consider on the orbit space of the group of rotations acting on the euclidean plane, however, is no longer that of an ordinary smooth or topological manifold. It should also be noted that if one follows the idea of studying orbit spaces by choosing suitable submanifolds cutting the orbits transversely, one is usually forced to leave the realm of ordinary Lie group actions, a circumstance which further motivates the introduction of the more flexible language of Lie groupoids.
As a possible simultaneous answer to the two questions raised above, in a 2015 paper [7] I introduced a (mathematically rigorous) notion of effective equivalence for Lie groupoids, and I proposed that two Lie groupoids should be viewed as giving rise to the same orbit space (up to isomorphism) when they are related by one such effective equivalence. The geometric properties of orbit spaces that one should study, then, are precisely those properties of Lie groupoids which are invariant under effective equivalence.
If you choose to work on this master thesis project, you will help me to develop the point of view outlined above. You will learn about Lie groupoids, an extensive topic which includes the ordinary theory of Lie groups, but which at the same time goes far beyond, with important applications in differential geometry and mathematical physics [2,5]. Your tasks will include:
a) searching through the existing literature on dynamical systems, control theory etc. for possible applications of the notion of effective equivalence and, more generally, of the Lie groupoid viewpoint;
b) improving the current understanding of the notion of effective equivalence by working out new examples and studying certain questions that remain open;
c) helping me improve, organize in readable form, and type some of the research that I have carried out so far on this topic, but not yet published.
The required background is differential topology at the basic level [3,4] and perhaps some familiarity with transformation groups [1]. Familiarity with aspects of the theory of dynamical systems would of course be very helpful, but not indispensable.
References
[1] Bredon, Glen E.; Introduction to compact transformation groups; Academic Press, New York, 1972.
[2] Cannas da Silva, Ana, and Weinstein, Alan; Geometric models for noncommutative algebras; 1998.
[3] Guillemin, Victor, and Pollack, Alan; Differential topology; Prentice-Hall, 1974.
[4] Lang, Serge; Fundamentals of differential geometry; Springer-Verlag, 2001.
[5] Moerdijk, I., and Mrcun, J.; Introduction to foliations and Lie groupoids; Cambridge University press, 2003.
[6] Smale, S.; Differentiable dynamical systems; Bull. Am. Math. Soc. 73 (1967) pp. 747–817.
[7] Trentinaglia, G.; Reduced smooth stacks? Theory Appl. Categ. 30 (2015) no. 31 pp. 1032–1066.