arXiv subjects

Philip Tosteson

Publications and source records attributed to Philip Tosteson.

15 recordsLinked to original sources

arXiv2026

Stable homology of strata of abelian differentials

We show that the homology of strata of abelian differentials stabilizes in a range where the number of simple zeros is large relative to the homological degree. In this range, we show that the rational cohomology agrees with the restriction of the tautological classes to the stratum, and that the rational Picard group is trivial for unprojectivized strata. Our proof method is to develop an $h$-principle for these strata, valid in a range of homological degrees that increases with the number of simple zeros. The same approach also applies to higher order differentials.

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arXiv2025

Representation stability for moduli spaces of admissible covers

We prove a representation stability result for the sequence of spaces $\overline M_{g, n}^A$ of pointed admissible $A$-covers of stable $n$-pointed genus-$g$ curves, for an abelian group $A$. For fixed genus $g$ and homology degree $i$, we give the sequence of rational homology groups $H_i(\overline{M}_{g, n}^A;\mathbb Q)$ the structure of a module over a combinatorial category, a la Sam--Snowden, and prove that this module is generated in degree at most $g + 5 i$. This implies that the generating function for the ranks of the homology groups is rational, with poles in the set $\left\{-1, -\frac{1}{2}, \ldots, -\frac{1}{|A|^2\cdot(g + 5i)}\right\}$. In the case where $A$ is the trivial group, our work significantly improves on previous representation stability results on the Deligne--Mumford compactification $\overline M_{g, n}$.

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arXiv2025

Homological stability and Manin's conjecture for rational curves on quartic del Pezzo surfaces

We prove a version of Manin's conjecture (over $\mathbb{F}_{q}$ for $q$ large) and the Cohen--Jones--Segal conjecture (over $\mathbb{C}$) for maps from rational curves to split quartic del Pezzo surfaces. The proofs share a common method which builds upon prior work of the first and fourth authors. The main ingredients of this method are (i) the construction of bar complexes formalizing the inclusion-exclusion principle and its point counting estimates, (ii) dimension estimates for spaces of rational curves using conic bundle structures, (iii) estimates of error terms using arguments of Sawin--Shusterman based on Katz's results, and (iv) a certain virtual height zeta function revealing the compatibility of bar complexes and Peyre's constant. Our argument substantiates the heuristic approach to Manin's conjecture over global function fields given by Batyrev and Ellenberg--Venkatesh in this case.

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arXiv2024

Homology of spaces of curves on blowups

We consider the space of holomorphic maps from a compact Riemann surface to a projective space blown up at finitely many points. We show that the homology of this mapping space equals that of the space of continuous maps that intersect the exceptional divisors positively, once the degree of the maps is sufficiently positive compared to the degree of homology. The proof uses a version of Vassiliev's method of simplicial resolution. As a consequence, we obtain a homological stability result for rational curves on the degree $5$ del Pezzo surface, which is analogous to a case of the Batyrev--Manin conjectures on rational point counts.

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arXiv2024

Hilbert series of representations of categories of $G$-sets

Let $G$ be a finite group. A contravariant functor from the category of finite free $G$-sets to vector spaces has an associated Hilbert series, which records the underlying sequence of $G^n$ representations, $n \in \mathbb N$. We prove that this Hilbert series is rational with denominator given by linear polynomials with coefficients in the field generated by the character table of $G$.

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arXiv2022

Polynomial representations of the Witt Lie algebra

The Witt algebra W_n is the Lie algebra of all derivations of the n-variable polynomial ring V_n=C[x_1, ..., x_n] (or of algebraic vector fields on A^n). A representation of W_n is polynomial if it arises as a subquotient of a sum of tensor powers of V_n. Our main theorems assert that finitely generated polynomial representations of W_n are noetherian and have rational Hilbert series. A key intermediate result states polynomial representations of the infinite Witt algebra are equivalent to representations of Fin^op, where Fin is the category of finite sets. We also show that polynomial representations of W_n are equivalent to polynomial representations of the endomorphism monoid of A^n. These equivalences are a special case of an operadic version of Schur--Weyl duality, which we establish.

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arXiv2022

Extremal stability for configuration spaces

We study stability patterns in the high dimensional rational homology of unordered configuration spaces of manifolds. Our results follow from a general approach to stability phenomena in the homology of Lie algebras, which may be of independent interest.

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arXiv2022

Representation stability for the Kontsevich space of stable maps

For a fixed algebraic variety $X$, curve class $α\in N_1(X)$, and genus $g \in \mathbb N$, we consider the sequence of $S_n$ representations obtained from the homology of the Kontsevich space of stable maps to $X$, $\bar M_{g,n}(X,α)$. Using the category of finite sets and surjections, we prove a representation stability theorem that governs the behavior of this sequence of representations for $n$ sufficiently large.

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arXiv2021

Stability in the homology of Deligne-Mumford compactifications

Using the the theory of FS^op modules, we study the asymptotic behavior of the homology of $\overline M_{g,n}$, the Deligne--Mumford compactification of the moduli space of curves, for $n >> 0$. An FS^op module is a contravariant functor from the category of finite sets and surjections to vector spaces. Via maps that glue on marked P^1's, we give the homology of $\overline M_{g,n}$ the structure of an FS^op module and bound its degree of generation. As a consequence, we prove that the generating function $\sum_{n} \dim(H_i(\overline M_{g,n})) t^n$ is rational, and its denominator has roots in the set $\{1, 1/2, \dots, 1/p(g,i)\}$ where $p(g,i)$ is a polynomial of order $O(g^2 i^2)$. We also obtain restrictions on the decomposition of the homology of $\overline M_{g,n}$ into irreducible $S_n$ representations.

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arXiv2021

Categorifications of rational Hilbert series and characters of $FS^{op}$ modules

We introduce a method for associating a chain complex to a module over a combinatorial category, such that if the complex is exact then the module has a rational Hilbert series. We prove homology--vanishing theorems for these complexes for several combinatorial categories including: the category of finite sets and injections, the opposite of the category of finite sets and surjections, and the category of finite dimensional vector spaces over a finite field and injections. Our main applications are to modules over the opposite of the category of finite sets and surjections, known as $FS^{op}$ modules. We obtain many constraints on the sequence of symmetric group representations underlying a finitely generated $FS^{op}$ module. In particular, we describe its character in terms of functions that we call character exponentials. Our results have new consequences for the character of the homology of the moduli space of stable marked curves, and for the equivariant Kazhdan-Luzstig polynomial of the braid matroid.

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arXiv2021

Factorization statistics and bug-eyed configuration spaces

A recent theorem of Hyde proves that the factorizations statistics of a random polynomial over a finite field are governed by the action of the symmetric group on the configuration space of $n$ distinct ordered points in $\mathbb R^3$. Hyde asked whether this result could be explained geometrically. We give a geometric proof of Hyde's theorem as an instance of the Grothendieck--Lefschetz trace formula applied to an interesting, highly nonseparated algebraic space. An advantage of our method is that it generalizes uniformly to an arbitrary Weyl group. In the process we study certain non-Hausdorff models for complements of hyperplane arrangements, first introduced by Proudfoot.

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arXiv2019

Representation stability for pure braid group Milnor fibers

We prove a representation stability result for the Milnor fiber associated to the pure braid group. Our result connects previous work of Simona Settepenella to representation stability in the sense of Church--Ellenberg--Farb, answering a question of Graham Denham. We also use our result to compute the stable integral homology of the Milnor fiber.

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arXiv2017

Lattice Spectral Sequences and Cohomology of Configuration Spaces

For a topological space $X$, we introduce a criterion for the $\rm FI$ module $H^i({\rm Conf}_n(X))$ to be finitely generated and give several applications. For instance, if $C$ is a finite connected $CW$ complex, then $X = C \times \mathbb{R}^2$ satisfies the criterion. Our main tool is a spectral sequence that we derive from the homological algebra of the partition lattice.

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arXiv2014

The Distribution of Gaps between Summands in Generalized Zeckendorf Decompositions

Zeckendorf proved that any integer can be decomposed uniquely as a sum of non-adjacent Fibonacci numbers, $F_n$. Using continued fractions, Lekkerkerker proved the average number of summands of an $m \in [F_n, F_{n+1})$ is essentially $n/(φ^2 +1)$, with $φ$ the golden ratio. Miller-Wang generalized this by adopting a combinatorial perspective, proving that for any positive linear recurrence the number of summands in decompositions for integers in $[G_n, G_{n+1})$ converges to a Gaussian distribution. We prove the probability of a gap larger than the recurrence length converges to decaying geometrically, and that the distribution of the smaller gaps depends in a computable way on the coefficients of the recurrence. These results hold both for the average over all $m \in [G_n, G_{n+1})$, as well as holding almost surely for the gap measure associated to individual $m$. The techniques can also be used to determine the distribution of the longest gap between summands, which we prove is similar to the distribution of the longest gap between heads in tosses of a biased coin. It is a double exponential strongly concentrated about the mean, and is on the order of $\log n$ with computable constants depending on the recurrence.

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arXiv2012

The Average Gap Distribution for Generalized Zeckendorf Decompositions

An interesting characterization of the Fibonacci numbers is that, if we write them as $F_1 = 1$, $F_2 = 2$, $F_3 = 3$, $F_4 = 5, ...$, then every positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers. This is now known as Zeckendorf's theorem [21], and similar decompositions exist for many other sequences ${G_{n+1} = c_1 G_{n} + ... + c_L G_{n+1-L}}$ arising from recurrence relations. Much more is known. Using continued fraction approaches, Lekkerkerker [15] proved the average number of summands needed for integers in $[G_n, G_{n+1})$ is on the order of $C_{\rm Lek} n$ for a non-zero constant; this was improved by others to show the number of summands has Gaussian fluctuations about this mean. Kolo$\breve{\rm g}$lu, Kopp, Miller and Wang [17, 18] recently recast the problem combinatorially, reproving and generalizing these results. We use this new perspective to investigate the distribution of gaps between summands. We explore the average behavior over all $m \in [G_n, G_{n+1})$ for special choices of the $c_i$'s. Specifically, we study the case where each $c_i \in {0,1}$ and there is a $g$ such that there are always exactly $g-1$ zeros between two non-zero $c_i$'s; note this includes the Fibonacci, Tribonacci and many other important special cases. We prove there are no gaps of length less than $g$, and the probability of a gap of length $j > g$ decays geometrically, with the decay ratio equal to the largest root of the recurrence relation. These methods are combinatorial and apply to related problems; we end with a discussion of similar results for far-difference (i.e., signed) decompositions.

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