## Grants and Contracts Details

### Description

There are many generalizations of the Lefschetz fixed point theorem and its converse,
including equivariant and fiberwise generalizations. However, many closely related
questions, such as corresponding invariants for coincidences and periodic points,
are currently unresolved and seem inaccessible using classical approaches.
In previous work, the PI has defined a trace in bicategories and shown that it gives
a new approach to the converse to the Lefschetz fixed point theorem. Additionally,
her approach generalizes more easily than the classical techniques, and proves relative,
equivariant and fiberwise generalizations of the Lefschetz fixed point theorem and its
converse.
In this project she proposes further extensions to coincidences and periodic points.
Little is known about these generalizations, but the PI has shown that her approach
gives new generalizations of the Lefschetz coincidence theorem and she expects that it
will also apply to the converse of the Lefschetz coincidence theorem. She also hopes
to generalize several classical properties of fixed point invariants to the trace in bicategories.
These properties have been extremely useful for computations of classical
invariants and generalizations to bicategories would imply corresponding results, and
possibly computations, for the refinements and generalizations.
Intellectual merit: The standard approaches to the Lefschetz fixed point theorem
and its converse have focused on integer valued invariants: the Lefschetz number and
Nielsen number. These invariants are easy to define, but they are very difficult to
generalize. The PI’s approach produces invariants that are elements of stable homotopy
groups of spheres and twisted loop spaces, as well as more algebraic targets. These
invariants readily generalize and agree with the integer valued invariants in the classical
cases.
Broader impact: As a graduate student and especially as a post-doc, the PI has
had many opportunities to mentor graduate students and graduate school bound undergraduates.
She intends to continue mentoring focusing on graduate students whose
interests are most compatible with her experience.
As a graduate student, the PI founded a student chapter of the Association for
Women in Mathematics (AWM). Recently a similar chapter has been founded at the
University of Kentucky. The PI intends to use her experience to support the leadership
of the University of Kentucky chapter as they identify future directions for their
organization. She is particuarly interested in helping the students develop a range of
activities that both serve the broader community and support local graduate students.
The PI also hopes to give University of Kentucky graduate students interested in
topology a broader experience of the field. Her current focus is on expanding the participation
of graduate students in the topology seminar and facilitating greater interaction
between graduate students and external visitors.

Status | Finished |
---|---|

Effective start/end date | 6/1/12 → 11/30/15 |

### Funding

- National Science Foundation: $106,705.00

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