email: ptoste@unc.edu
I am an assistant professor in the mathematics department at UNC Chapel Hill. Previously, I was an L.E. Dickson Instructor and an NSF postdoctoral fellow at the University of Chicago, working with Benson Farb. Before moving to Chicago, I did my PhD at the University of Michigan, where my advisor was Andrew Snowden.
I am interested in questions involving topology, combinatorics, and representation theory, especially ones from the area of representation stability. I like to think about configuration spaces, moduli spaces of curves and algebraic maps, hyperplane arrangements, representations of categories, operads, Koszul duality, and poset homology.
Hilbert series of representations of categories of G-sets preprint.
Polynomial representations of the Witt Lie algebra, with S. Sam and A. Snowden preprint.
This paper considers representations of the monoid of polynomial endomorphisms of C^n, and relates them to representations of the category of finite sets.
Representation stability for the Kontsevich space of stable maps, preprint.
This paper uses the category of finite sets and surjections to prove a representation stability theorem for the Kontsevich moduli space of stable maps.
Extremal stability for configuration spaces, with B. Knusdsen and J. Miller, Mathematische Annalen (to appear).
This paper studies the behavior of the homology of unordered configuration spaces along extremal rays.
Categorifications of rational Hilbert series and characters of FSop modules, Algebra and Number Theory (to appear).
This paper is about the Hilbert series of modules over combinatorial categories, especially the opposite of the category of finite sets and surjections (known as FSop). It uses methods from poset topology to unify several results by categorifying them. In the case of FSop, this categorification has new consequences for equivariant Hilbert series.
Factorization Statistics and Bug-Eyed Configuration Spaces, with D. Petersen, Geometry and Topology 25 (2021) 3691-3723.
We gave a geometric explanation for a surpising connection, discovered by Trevor Hyde, between factorization statistics of polynomials and configurations in R^3. We also generalized this connection to arbitrary Coxeter groups.
Homological Stability for Pure Braid Group Milnor Fibers, with J. Miller, Trans. Amer. Math. Soc. 374 (2021).
We proved a representation stability theorem for the Milnor fiber associated to the type A braid arrangement.
Stability in the Homology of Deligne-Mumford Compactifications, Compositio Mathematica (2021) Volume 157, Issue 12.
This paper uses the category of finite sets and surjections to study the moduli space of stable marked curves from the point of view of representation stability.
Lattice Spectral Sequences and Cohomology of Configuration Spaces, preprint.
This paper proves a global version of the Goresky-MacPherson formula, and applies it to prove representation stability for configuration spaces of non-manifolds.
The Distribution of Gaps between Summands in Generalized Zeckendorf Decompositions, with A. Bower, R. Insoft, S. Li, and S. Miller, Journal of Combinatorial Theory, Series A. (135 (2015), 130--160).
The Average Gap Distribution for Generalized Zeckendorf Decompositions, with O. Beckwith, A. Bower, L. Gaudet, R. Insoft, S. Li and S. Miller, the Fibonacci Quarterly (51 (2013), 13--27).
Representation Stability, Configuration Spaces, and Deligne–Mumford Compactifications, slides from my Thesis defense
Cutting and Pasting: Rethinking how we measure, a Michigan math club talk based on Schanuel's article "What is the length of a Potato?"