Related papers: Representation stability for ordered Hurwitz space…

We consider for two based graphs $G$ and $H$ the sequence of graphs $G_k$ given by the wedge sum of $G$ and $k$ copies of $H$. These graphs have an action of the symmetric group $\Sigma_k$ by permuting the $H$-summands. We show that the…

Hurwitz numbers are a weighted count of degree d ramified covers of curves with specified ramification profiles at marked points on the codomain curve. Isomorphism classes of these covers can be included as a dense open set in a moduli…

In this paper we study homological stability for spaces ${\rm Hom}(\mathbb{Z}^n,G)$ of pairwise commuting $n$-tuples in a Lie group $G$. We prove that for each $n\geqslant 1$, these spaces satisfy rational homological stability as $G$…

We prove a general representation stability result for polynomial coefficient systems which lets us prove representation stability and secondary homological stability for many families of groups with polynomial coefficients. This gives two…

We prove geometric and cohomological stabilization results for the universal smooth degree $d$ hypersurface section of a fixed smooth projective variety as $d$ goes to infinity. We show that relative configuration spaces of the universal…

For fixed j and w, we study the j-th homology of the configuration space of n labeled disks of width 1 in an infinite strip of width w. As n grows, the homology groups grow exponentially in rank, suggesting a generalized representation…

We introduce the idea of *representation stability* (and several variations) for a sequence of representations V_n of groups G_n. A central application of the new viewpoint we introduce here is the importation of representation theory into…

Secondary homological stability is a recently discovered stability pattern for the homology of a sequence of spaces exhibiting homological stability in a range where homological stability does not hold. We prove secondary homological…

The ordered configuration space of $n$ open unit squares in the $w$ by $h$ rectangle exhibits homological stability in the space direction. That is, for fixed $n$ and fixed homological degree $k$, once the underlying rectangle is large…

We prove homological stability for two different flavours of asymptotic monopole moduli spaces, namely moduli spaces of framed Dirac monopoles and moduli spaces of ideal monopoles. The former are Gibbons-Manton torus bundles over…

We prove a homological stability theorem for the moduli spaces of manifolds diffeomorphic to g(S^n x S^n), provided n > 2. This generalises Harer's stability theorem for the homology of mapping class groups. Combined with previous work of…

In [Chu12], Church used representation stability to prove that the space of configurations of distinct unordered points in a closed manifold exhibit rational homological stability. A second proof was also given by Randal-Williams in [RW11]…

We prove a motivic stabilization result for the cohomology of the local systems on configuration spaces of varieties over $\mathbb{C}$ attached to character polynomials. Our approach interprets the stabilization as a probabilistic…

We consider the Temperley-Lieb algebras $\textrm{TL}_n(\delta)$ at $\delta = 1$. Since $\delta = 1$, we can consider the multiplicative monoid structure and ask how this monoid acts on topological spaces. Given a monoid action on a…

The problem of characterizing normed ordered spaces which admit a representation in the algebraic, order and norm sense as a subspace of $C(X)$, the space of all continuous functions on a compact Hausdorff space is a classical problem that…

Wahl recently proved that the homology of the non-orientable mapping class group stabilizes as the genus increases. In this short note we analyse the situation where the underlying non-orientable surfaces have marked points.

The study of the moduli of covers of the projective line leads to the theory of Hurwitz varieties covering configuration varieties. Certain one-dimensional slices of these coverings are particularly interesting Belyi maps. We present…

In this paper we will study the representations of isomorphisms between bases of topological spaces. It turns out that the perfect setting for this study is that of regular open subsets of complete metric spaces, but we have achieved some…

Given a smooth, projective curve $Y$, a point $y_0 \in Y$, a positive integer $n$, and a transitive subgroup $G$ of the symmetric group $S_{d}$ we study smooth, proper families, parameterized by algebraic varieties, of pointed degree $d$…

In [Pal13] (arXiv:1106.4540) the second author proved that the sequence of "oriented" configuration spaces on an open connected manifold exhibits homological stability as the number of particles goes to infinity. To complement that result…