Andrew Putman
University of Notre Dame, IN·Mathematics & Statistics
Symmetry turns out to govern a vast range of mathematical phenomena. Even objects that do not look geometric can be studied through their symmetries, which give them a hidden geometry that can answer questions no other method can.
- Fields
Mathematics & Statistics
- Topics
geometric group theory
geometric topology
mapping class groups
- Grants
- 1 known grant, through Jul 2027
What they work on
- Extend results on stable homology, the algebraic hole-counting invariants that settle down as groups grow, from mapping class groups and related groups to their finite-index subgroups.
- Determine when mapping class groups and automorphism groups of free groups are commensurable with fundamental groups of compact Kähler manifolds, linking geometric topology to algebraic geometry.
- Prove a version of representation stability for the homology of the Torelli group, a key subgroup of the mapping class group of a surface.
- Clarify the relationship between the unstable homology of arithmetic groups, which are groups of integer-like matrices, and the Steinberg representation.
More about the research
This project studies the topological side of that symmetry. Topology is the study of large-scale properties of spaces that survive bending and stretching, such as the presence of high-dimensional holes. The goal is to connect objects from geometric topology and geometric group theory with ideas from algebraic geometry and representation theory.
Why it matters
This is pure mathematics with no stated application. It deepens understanding of the topology of important infinite groups and builds bridges between geometric topology, group theory, algebraic geometry and representation theory. These connections could let methods from one field answer open questions in another.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Michelle Chu
University of Minnesota, MN·Mathematics & Statistics·Early-career
Study virtual properties of the fundamental groups of hyperbolic 3-manifolds — properties held by finite-index subgroups, which correspond to finite-sheeted covering spaces of the manifold.
Nicholas Salter
University of Notre Dame, IN·Mathematics & Statistics·Early-career
Establish asphericality, the absence of higher-dimensional holes, for special strata of Abelian differentials using the PI's Abel-Jacobi flow method, as progress toward the Kontsevich–Zorich conjecture.
Mark Brittenham
University of Nebraska at Omaha, NE·Mathematics & Statistics
Investigate the L-space conjecture for toroidal 3-manifolds, which links left-orderable fundamental groups, foliations and Floer homology, using the notion of slope detection.
The school
University of Notre Dame: 19 labs on PI Pages, $346M a year on research, 277 research doctorates a year.
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