Homotopical Methods in Fixed Point Theory

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This award supports participation in the summer school “Homotopical Methods in Fixed Point Theory,” taking place July 11 to 15, 2022 at the University of Colorado, Boulder. This workshop will be an opportunity for participants from a wide variety of institutions and backgrounds to learn advanced topics in algebraic topology, an area of mathematics that uses tools from algebra to study certain invariant properties of geometric objects. The advantage of the workshop is that the questions and problems in this field are easy to state and motivate, yet their study naturally leads one to consider modern developments in the field of algebraic topology. The active-learning format, and emphasis on co-creating mathematics in community with one another, will allow for significant collaboration between participants and for interactions with the scientific committee. The approachability of the topic and the format of the workshop will attract participants coming from institutions with smaller topology research groups, as well as those with marginalized or underrepresented aspects of their identity. The scientific committee itself is made up of early-career researchers, and more than half of the organizing and scientific committee are women. This representation and the diverse perspectives it offers for the design of the summer school will have a positive impact on attendees.The scientific goal of the summer school is to introduce participants to tools and ideas from algebraic topology and homotopy theory through the lens of fixed point theory. The workshop will be structured around mini-courses taught in an active-learning style and course topics will range from classical fixed point theory to modern tools such as duality, spectra, and trace methods in algebraic K-theory. This range reflects recent and superficially unrelated advances in homotopy theory and higher category theory that have provided new approaches to topological fixed point theory. These approaches have refined classical invariants defined using simplicial structures that were often difficult to generalize. This shift in perspective also illuminated the centrality of additivity for useful fixed point invariants. Since algebraic K-theory is the universal home for additive invariants, it is natural to look for connections between fixed point theory and algebraic K-theory. This connection is further supported by the recent important developments in computations of K-groups. Work of some members of the scientific committee has established that these computational approaches can be effectively modeled using the new forms of fixed point invariants. This provides a very different approach to these important developments and a new source of intuition. https://sites.google.com/colorado.edu/fixedpointtheory2020/This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.