In this project I will study topological dynamical systems: configurations of points that evolve over time according to a fixed rule. The key feature is continuity: if two states start out very similar, they remain similar after the rule is applied, there are no sudden jumps. This may sound abstract, but many familiar situations fit this picture. Examples include how a fluid mixes when you stir it, or how a population changes from one season to the next.
Because such systems appear in many different forms, it is useful to have a single mathematical language that can describe them all. In this project, that unifying language is provided by groupoids. A groupoid is a way of organizing symmetries and "allowed moves" within a system. Unlike a group, where every pair of moves can be combined, a groupoid allows partial combination: you can only compose moves when they actually fit together. This flexibility makes groupoids well suited for dynamical systems, where symmetries often depend on where you are in the system.
My focus will be on identifying and computing invariants, that is quantities that stay the same even when the system is transformed in certain ways. Invariants matter because they let us recognize deep similarities between systems that may look very different on the surface.
I will study invariants from two complementary viewpoints. In the first, dynamical systems are treated using ideas from quantum theory, where key quantities can fail to commute, reflecting the kind of behavior associated to Heisenberg's uncertainty principle. In the second, I use tools from topology, which studies what remains unchanged when objects are smoothly deformed, like bending or stretching without tearing.
Computing these invariants is valuable because it can reveal patterns and constraints shared by whole families of dynamical systems, allowing us to understand and predict their behavior at a broad structural level rather than case by case.
Background
- Crainic, M.; Moerdijk, I. A homology theory for étale groupoids. J. Reine Angew. Math. 521, 25-46 (2000).
- Davis, J. F., Lück, W. Spaces over a category and assembly maps in isomorphism conjectures in K- and L-theory. K-Theory 15, No. 3, 201-252 (1998).
- Matui, H. Homology and topological full groups of étale groupoids on totally disconnected spaces. Proc. Lond. Math. Soc. (3) 104, No. 1, 27-56 (2012).
- Meyer, R., Nest, R. The Baum-Connes conjecture via localisation of categories. Topology 45, No. 2, 209-259 (2006).
