Traces in higher categories, stable homotopy theory and dynamics

Fixed points of functions are important in many areas of mathematics and there are many different approaches to identifying and counting fixed points. I’m interested in a family of invariants that are defined using homological and homotopical techniques. In previous work I have developed a categorical structure that describes these invariants and applied it to define relative, equivariant and fiberwise fixed point invariants. Working with coauthors, I have shown that the categorical structure also captures many of the properties of classical invariants that have been essential for computations. As a natural extension of this work the I intend to explore generalizations to two types of dynamical systems: • the iterates of an endomorphism f of a topological space X and • continuous maps g from the product of X and the real numbers (or the product of X and I) to X. In the first case an initial goal is to understand fixed points of fn for fixed values of n. The next step is to understand how the invariants for different choices of n fit together. In the second case the goal is to understand in- variants for sets that are invariant under g, i.e. sets A so that g(A, t) is a subset of A for all t, or points of X that are fixed for some particular t. Some homological and homotopical invariants have been developed for these types of maps and their fixed points, but many of these invariants are much less sophisticated than the corresponding fixed point invariants. My goal is to extend my unique approach to fixed point invariants to these generalizations. In previous work I has found that the categorical structure I defined to capture classical invariants provides guidance that is very helpful when extending the underlying ideas.