Stability Patterns in the Homology of Moduli Spaces

Purdue University, West Lafayette IN

Investigators

Abstract

Homology is a mathematical tool that was introduced over a hundred years ago to measure features of shape that remain invariant under deformation. This helped make rigorous many calculations in calculus and physics involving integrals of functions defined on spaces with “holes.” Homology gives an algebraic measure of holes in geometric objects, allowing algebraic techniques to give geometric information, for example, showing that a given geometric object cannot be deformed into another one. The aim of this project is to study patterns in the homology of families of spaces coming from number theory and geometry. These patterns simplify homology calculations by reducing infinite calculations to finite calculations. In this project, new methods involving the use of computers in theoretical algebraic topology and number theory will be developed. The project will enhance graduate and postdoctoral training in algebraic topology through mentoring, seminars, and conferences. The PI will promote diversity and inclusion through participation in a summer bridge program designed for students from underrepresented racial and ethnic groups to better prepare them for graduate school. This project aims to improve understanding of the homology of arithmetic groups, a central concept in number theory, algebraic K-theory, and even the theory of manifolds. In high dimensions, the homology is known to vanish, and, in low dimensions, the homology is known to stabilize. These stable homology groups have been completely calculated in many cases. The project focuses on two ranges of dimensions: just below where the homology is known to vanish and just above the stable range. Conjecturally, the highest degree homology groups should exhibit a pattern called “extremal stability” and the homology near the stable range should exhibit a different pattern called “secondary stability.” Highly connected simplicial complexes and operadic cells will be used to try to establish these conjectures. The connection with algebraic K-theory will be a key point of emphasis when studying the homology of arithmetic groups. Similar patterns will be investigated in spaces coming from geometric topology such as various moduli spaces and configuration spaces. 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.