Related papers: The complexity of a flat groupoid

The complexity of a homogeneous space $G/H$ under a reductive group $G$ is by definition the codimension of generic orbits in $G/H$ of a Borel subgroup $B\subseteq G$. We give a representation-theoretic interpretation of this number as the…

Recently it was shown that, for every fixed k>1, given a finite simply connected simplicial complex X, the kth homotopy group \pi_k(X) can be computed in time polynomial in the number n of simplices of X. We prove that this problem is…

Let S be a Noetherian scheme, f:X->Y a surjective S-morphism of S-schemes, with X of finite type over S. We discuss what makes Y of finite type. First, we prove that if S is excellent, Y is reduced, and f is universally open, then Y is of…

Let k be a field, and let {\pi}:\tilde{X} -> X be a proper birational morphism of irreducible k-varieties, where \tilde{X} is smooth and X has at worst quotient singularities. When the characteristic of k is zero, a theorem of Koll\'ar in…

Let G be a connected simply-connected reductive algebraic group. In this article, we consider the normal algebraic varieties equipped with a horospherical G-action such that the quotient of a G-stable open subset is a curve. Let X be such a…

Let $X$ and $S$ be complex spaces with $X$ countable at infinity and $S$ reduced locally pure dimensional. Let $\pi:X\to S$ be an universally-$n$-equidimensional morphism (i.e open with constant pure $n$-dimensional fibers). If there is a…

By developing a theory of deformations over nilpotent Lie algebras, based on Schlessinger's deformation theory over Artinian rings, this paper investigates the pro-l-unipotent fundamental group of a variety X. If X is smooth and proper,…

Let a complex algebraic reductive group $\bf G$ act on a complex algebraic manifold $\bf X$. For a $\bf G$-invariant subvariety $\Xi$ of the nilpotent cone $\mathcal{N}(\mathfrak{g}^*)\subset \mathfrak{g}^*$ we define a notion of…

We study the palindrome complexity of infinite sequences on finite alphabets, i.e., the number of palindromic factors (blocks) of given length occurring in a given sequence. We survey the known results and obtain new results for some…

Let $f\colon Y \to X$ be a proper flat morphism of locally noetherian schemes. Then, the locus in $X$ over which $f$ is smooth is stable under generization. We prove that under suitable assumptions on the formal fibers of $X$, the same…

We study three graph complexes related to the higher genus Grothendieck-Teichm\"uller Lie algebra and diffeomorphism groups of manifolds. We show how the cohomology of these graph complexes is related, and we compute the cohomology as the…

We show that any homomorphism between Noetherian $F$-finite rings can be factored into a regular morphism between Noetherian $F$-finite rings followed by a surjection. This result establishes an analog of the 'smooth-by-surjective'…

A smooth cuboid can be identified with a $3\times 3$ matrix of linear forms, with coefficients in a field $K$, whose determinant describes a smooth cubic in the projective plane. To each such matrix one can associate a group scheme over…

Let $K$ be a number field, and let $\mathcal{X}$ be a proper regular flat scheme over $\mathcal{O}_{K}$ with a generic fiber $X$ geometrically connected over $K$. We prove that there is an exact sequence up to finite groups $0\rightarrow…

This work concerns representations of a finite flat group scheme $G$, defined over a noetherian commutative ring $R$. The focus is on lattices, namely, finitely generated $G$-modules that are projective as $R$-modules, and on the full…

We show that the parametrised topological complexity of Cohen, Farber and Weinberger gives an invariant of group epimorphisms. We extend various bounds for the topological complexity of groups to obtain bounds for the parametrised…

A class of topological spaces is topologically rigid if any two spaces with the same fundamental group are also homeomorphic. Topological rigidity, in addition to its intrinsic interest, has been useful for solving abstract commensurability…

We study the complexity of the isomorphism relation for various classes of closed subgroups of the group of permutations of the natural numbers. We use the setting of Borel reducibility between equivalence relations on Polish spaces. For…

What are simplest ways to construct a finite group from its atomic constituents? To understand part-whole relations between finite simple groups and the global structure of finite groups, we axiomatize complexity measures on finite groups.…

We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group G, we show that a finite subset X with |X X \^{-1} X |/ |X| bounded is…