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Description
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.
Status  Finished 

Effective start/end date  9/1/15 → 8/31/18 
Funding
 Simons Foundation
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Projects
 1 Finished

Traces in higher categories, stable homotopy theory and dynamics
9/1/15 → 8/31/18
Project: Research project