Jeremy Miller
Purdue University, IN·Mathematics & Statistics
Homology is a mathematical tool for counting and describing the "holes" in a shape, or topological space. It is central to algebraic topology, the field that studies shapes using algebra.
- Fields
Mathematics & Statistics
- Topics
algebraic topology
homology
homological stability
- Grants
- 1 known grant, through Aug 2028
What they work on
- Prove homological stability and its variants, meaning that the hole-counting homology settles down as spaces grow, for classifying spaces of arithmetic groups, using tools such as locally symmetric spaces and homological algebra.
- Establish stability patterns for moduli spaces of graphs and Hurwitz spaces using graph complexes, operads and combinatorial simplicial complexes.
- Apply stability results to analytic number theory over function fields, algebraic K-theory and multiple polylogarithms, drawing on étale cohomology and Hopf algebras.
More about the research
Homology also appears in physics and economics. For example, the Aharonov-Bohm effect in electromagnetism is homological in nature, and homology can be used to prove that Nash equilibria exist. The project studies patterns that appear when one looks at the homology of a whole sequence of related spaces. It then uses those patterns to learn about other areas of mathematics, especially number theory.
Why it matters
This is pure mathematics. Stability results for these spaces would add to the understanding of how the shape of mathematical spaces relates to counting problems in number theory over function fields, to algebraic K-theory, and to multiple polylogarithms. Homology itself already has applications in physics, such as the Aharonov-Bohm effect, and in economics.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Peter Patzt
University of Oklahoma, OK·Mathematics & Statistics
Study cohomological and representation stability for general linear groups and their principal congruence subgroups, which are matrix groups defined by a divisibility condition. Describe how the cohomology behaves as matrix size grows.
Aaron Landesman
Harvard University, MA·Mathematics & Statistics
Make progress on the Cohen-Lenstra conjectures about class groups of quadratic fields over function fields by computing relevant homology groups with new topological tools.
Hannah Larson
University of California-Berkeley, CA·Mathematics & Statistics·Early-career
Develop new intersection-theory methods for moduli spaces of low-genus algebraic curves, spaces whose points each represent a curve, and connect them to closely related moduli spaces.
The school
Purdue University: 43 labs on PI Pages, $932M a year on research, 833 research doctorates a year.
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