Related papers: On the construction of permutation complexes for p…
Farahat and Higman constructed an algebra $\mathrm{FH}$ interpolating the centres of symmetric group algebras $Z(\mathbb{Z}S_n)$ by proving that the structure constants in these rings are "polynomial in $n$". Inspired by a construction of…
We show that the homology of modules for Hurwitz spaces stabilizes and compute its stable value. As one consequence, we compute the moments of Selmer groups in quadratic twist families of abelian varieties over suitably large function…
We develop cohomological and homological theories for a profinite group $G$ with coefficients in the Pontryagin dual categories of pro-discrete and ind-profinite $G$-modules, respectively. The standard results of group (co)homology hold for…
We determine the Gross-Hopkins duals of certain higher real $K$-theory spectra. More specifically, let $p$ be an odd prime, and consider the Morava $E$-theory spectrum of height $n=p-1$. It is known, in the expert circles, that for certain…
In the present article, we combine some techniques in the harmonic analysis together with the geometric approach given by modules over sheaves of rings of twisted differential operators ($\mathcal{D}$-modules), and reformulate the…
This is the second of two papers treating faithful actions of simple algebraic groups on irreducible modules and on the associated Grassmannian varieties; in the first paper we considered the module itself and its projective space, while…
Greenlees and Sadofsky showed that the classifying spaces of finite groups are self-dual with respect to Morava K-theory K(n). Their duality map was constructed using a transfer map. We generalize their duality map and prove a K(n)-version…
Let $\mathcal{M}_{0,n+1}$ be the moduli space of genus zero Riemann surfaces with $n+1$ marked points. In this paper we compute $H_*^{\Sigma_n}(\mathcal{M}_{0,n+1};\mathbb{F}_p)$ and $H_*^{\Sigma_n}(\mathcal{M}_{0,n+1};\mathbb{F}_p(\pm 1))$…
We show for g > 6 that the second homology group of the Torelli group of a surface of genus g and 1 boundary component is generated as an Sp(2g,Z)-module by the image under the stabilization map of the second homology group of the Torelli…
For an arrangement with complement X and fundamental group G, we relate the truncated cohomology ring, H^{<=2}(X), to the second nilpotent quotient, G/G_3. We define invariants of G/G_3 by counting normal subgroups of a fixed prime index p,…
We consider bases for the cohomology space of regular semisimple Hessenberg varieties, consisting of the classes that naturally arise from the Bialynicki-Birula decomposition of the Hessenberg varieties. We give an explicit combinatorial…
Recent classification of $\frac{3}{2}$-transitive permutation groups leaves us with six families of groups which are $2$-transitive, or Frobenius, or one-dimensional affine, or the affine solvable subgroups of $ \mathrm{AGL}(2, q)$, or…
For a profinite group $G$, let $(\text{-})^{hG}$, $(\text{-})^{h_dG}$, and $(\text{-})^{h'G}$ denote continuous homotopy fixed points for profinite $G$-spectra, discrete $G$-spectra, and continuous $G$-spectra (coming from towers of…
Using a previous classification result on symmetric additive 2-cocycles, we collect a variety of facts about the Lubin-Tate cohomology of formal groups to compute the 2-primary component of the scheme of symmetric multiplicative 2-cocycles.…
Solid abelian groups, as introduced by Dustin Clausen and Peter Scholze, form a subcategory of all condensed abelian groups satisfying some ''completeness'' conditions and having favourable categorical properties. Given a profinite ring…
In this paper, we give a Gr\"obner-Shirshov basis for the finitely presented semigroup algebra $\mathbf{k}[S_n(Sym_n)]$ defined by permutation relations of symmetric type. As an application, by the Composition-Diamond Lemma, we obtain…
Multipersistence homology modules were introduced by G.Carlsson and A.Zomorodian which gave, together with G.Singh, an algorithm to compute their Groebner bases. Although their algorithm has polynomial complexity when the chain modules are…
We discuss a homological method for transferring algebra structures on complexes along suitably nice homotopy equivalences, including those obtained after an application of the Perturbation Lemma. We study the implications for the Homotopy…
We have shown that the n-th Morava K-theory K^*(X) for a CW-spectrum X with action of Morava stabilizer group G_n can be recovered from the system of some height-(n+1) cohomology groups E^*(Z) with G_{n+1}-action indexed by finite…
We describe and compute the homotopy of spectra of topological modular forms of level 3. We give some computations related to the "building complex" associated to level 3 structures at the prime 2. Finally, we note the existence of a number…