Nicholas Salter
University of Notre Dame, IN·Mathematics & Statistics·Early-career
A moduli space is a mathematical "world map" that lists and organizes every possible version of some object, such as every polynomial of a given form. These spaces sit where algebraic geometry, the study of polynomial equations and their solutions, meets topology, the study of shape.
- Fields
Mathematics & Statistics
- Topics
moduli spaces
asphericality
fundamental groups
- Grants
- 1 known grant, through Jun 2029
What they work on
- 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.
- Develop the geometric theory of fundamental groups, the groups of loops in a space, for linear systems on algebraic surfaces and for strata of Abelian differentials, drawing on geometric group theory.
- Study asphericality conjectures for two families of single-variable polynomials, the equicritical stratification and spaces of fewnomials (polynomials with few terms), using topological methods.
More about the research
A striking pattern is that many of them are aspherical: once a space's loops (its fundamental group) are understood, it has no hidden higher-dimensional "holes." When a space is aspherical, its loops capture its entire shape. Long-standing conjectures of Arnol'd, Thom, and Kontsevich–Zorich predict asphericality for important families, but these remain largely unsolved. The project proposes new ways in, using topology and geometric group theory, which studies groups through the geometry of the spaces they act on.
Why it matters
This is pure mathematics. It aims to resolve long-standing asphericality conjectures and to give geometric group theory new families of groups to study. Related areas of geometry and topology have a history of applications in physics, data science, computer vision and robotics, which this work could eventually inform. The Math Circles Institute aims to give K-12 teachers new enrichment materials for their classrooms.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Andrew Putman
University of Notre Dame, IN·Mathematics & Statistics
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.
Denis Osin
Vanderbilt University, TN·Mathematics & Statistics
Study rigidity properties of acylindrically hyperbolic groups by addressing auxiliary questions and conjectures connecting analytic and geometric properties, motivated by open problems on quasi-isometric and measure equivalence rigidity.
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
University of Notre Dame: 19 labs on PI Pages, $346M a year on research, 277 research doctorates a year.
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