Philip Tosteson

email: ptoste@math.uchicago.edu

I am an L.E. Dickson Instructor and an NSF postdoctoral fellow at the University of Chicago. My postdoctoral mentor is 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, hyperplane arrangements, representations of categories, Koszul duality, and posets.

Research

Extremal stability for configuration spaces, preprint.

We study the behavior of the homology of unordered configuration spaces along extremal rays.

Categorifications of rational Hilbert series and characters of FSop modules, to appear in Algebra and Number Theory.

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 Dan Petersen, to appear in Geometry and Topology

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 Jeremy Miller, to appear in Transactions of the American Mathematical Society

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, to appear in Compositio Mathematica.

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 Amanda Bower, Rachel Insoft, Shiyu Li, and Steven Miller, Journal of Combinatorial Theory, Series A. (135 (2015), 130--160).

The Average Gap Distribution for Generalized Zeckendorf Decompositions, with Olivia Beckwith, Amanda Bower, Louis Gaudet, Rachel Insoft, Shiyu Li and Steven Miller, the Fibonacci Quarterly (51 (2013), 13--27).

Slides

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?"

Inclusion Exclusion and Rep Stability for Non-Manifolds