Parallel Transport with Pole Ladder: a Third Order Scheme in Affine Connection Spaces which is Exact in Affine Symmetric Spaces

Parallel transport is an important step in many discrete algorithms for statistical computing on manifolds. Numerical methods based on Jacobi fields or geodesics parallelograms are currently used in geometric data processing. In this last class, pole ladder is a simplification of Schild's ladder for the parallel transport along geodesics that was shown to be particularly simple and numerically stable in Lie groups. So far, these methods were shown to be first order approximations of the Riemannian parallel transport, but higher order error terms are difficult to establish. In this paper, we build on a BCH-type formula on affine connection spaces to establish the behavior of one pole ladder step up to order 5. It is remarkable that the scheme is of order three in general affine connection spaces with a symmetric connection, much higher than expected. Moreover, the fourth-order term involves the covariant derivative of the curvature only, which is vanishing in locally symmetric space. We show that pole ladder is actually locally exact in these spaces, and even almost surely globally exact in Riemannian symmetric manifolds. These properties make pole ladder a very attractive alternative to other methods in general affine manifolds with a symmetric connection

Hecke operators and Q-groups associated to self-adjoint homogeneous cones

Let G be a reductive algebraic group associated to a self-adjoint homogeneous cone defined over Q, and let G' be an appropriate neat arithmetic subgroup of G. We present two algorithms to compute the action of the Hecke operators on the integral cohomology of G'. This simultaneously generalizes the modular algorithm of Ash-Rudolph to a larger class of groups, and provides techniques to compute the Hecke-module structure of previously inaccessible cohomology groups.Comment: 25 pages, 10 figures, uses psfrag.sty; authors names omitted from first versio

Tannaka duality and stable infinity-categories

We introduce a notion of fine Tannakian infinity-categories and prove Tannakian characterization results for symmetric monoidal stable infinity-categories over a field of characteristic zero. It connects derived quotient stacks with symmetric monoidal stable infinity-categories which satisfy a certain simple axiom. We also discuss several applications to examples.Comment: The final version. Published in Journal of Topology, Wiley 201

On the reductive Borel-Serre compactification: LpL^p-cohomology of arithmetic groups (for large pp)

The $L^2$-cohomology of a locally symmetric variety is known to have the topological interpretation as the intersection homology of its Baily-Borel Satake compactification. In this article, we observe that even without the Hermitian hypothesis, the $L^p$-cohomology of an arithmetic quotient, for $p$ finite and sufficiently large, is isomorphic to the ordinary cohomology of its reductive Borel-Serre compactification. We use this to generalize a theorem of Mumford concerning homogeneous vector bundles, their invariant Chern forms and the canonical extensions of the bundles; here, though, we are referring to canonical extensions to the reductive Borel-Serre compactification of any arithmetic quotient. To achieve that, we give a systematic discussion of vector bundles and Chern classes on stratifiedComment: 32 page

The Essential Dimension of Congruence Covers

Consider the algebraic function $\Phi_{g,n}$ that assigns to a general $g$-dimensional abelian variety an $n$-torsion point. A question first posed by Kronecker and Klein asks: What is the minimal $d$ such that, after a rational change of variables, the function $\Phi_{g,n}$ can be written as an algebraic function of $d$ variables? Using techniques from the deformation theory of $p$-divisible groups and finite flat group schemes, we answer this question by computing the essential dimension and $p$-dimension of congruence covers of the moduli space of principally polarized abelian varieties. We apply this result to compute the essential $p$-dimension of congruence covers of the moduli space of genus $g$ curves, as well as its hyperelliptic locus, and of certain locally symmetric varieties.Comment: 26 pages. Minor revisions. Comments welcome

Representation theory in complex rank, II

This paper is a sequel to arXiv:1401.6321. We define and study representation categories based on Deligne categories Rep(GL_t), Rep(O_t), Rep(Sp_2t), where t is any (non-integer) complex number. Namely, we define complex rank analogs of the parabolic category O and the representation categories of real reductive Lie groups and supergroups, affine Lie algebras, and Yangians. We develop a framework and language for studying these categories, prove basic results about them, and outline a number of directions of further research.Comment: 33 pages, late

On the Hodge theory of stratified spaces

This article is a survey of recent work of the author, together with Markus Banagl, Eric Leichtnam, Rafe Mazzeo, and Paolo Piazza, on the Hodge theory of stratified spaces. We discuss how to resolve a Thom-Mather stratified space to a manifold with corners with an iterated fibration structure and the generalization of a perversity in the sense of Goresky-MacPherson to a mezzoperversity. We define Cheeger spaces and their signatures and describe how to carry out the analytic proof of the Novikov conjecture on these spaces. Finally we review the reductive Borel-Serre compactification of a locally symmetric space to a stratified space and describe its resolution to a manifold with corners.Comment: Dedicated to Steven Zucker on the occasion of his 65th birthda

The topological trace formula

The Lefschetz formula for the action of a Hecke correspondence on the weighted cohomology of a locally symmetric space is derived. It is also proven that each Hecke correspondence on the reductive Borel-Serre compactification of the locally symmetric space is weakly hyperbolic

Representations associated to small nilpotent orbits for complex Spin groups

This paper provides a comparison between the $K$-structure of unipotent representations and regular sections of bundles on nilpotent orbits for complex groups of type $D$. Precisely, let $G_ 0 =Spin(2n,\mathbb C)$ be the Spin complex group viewed as a real group, and $K\cong G_0$ be the complexification of the maximal compact subgroup of $G_0$. We compute $K$-spectra of the regular functions on some small nilpotent orbits $\mathcal O$ transforming according to characters $\psi$ of $C_{ K}(\mathcal O)$ trivial on the connected component of the identity $C_{ K}(\mathcal O)^0$. We then match them with the ${K}$-types of the genuine (i.e. representations which do not factor to $SO(2n,\mathbb C)$) unipotent representations attached to $\mathcal O$.Comment: arXiv admin note: text overlap with arXiv:1702.0484

Computing discrete equivariant harmonic maps

We present effective methods to compute equivariant harmonic maps from the universal cover of a surface into a nonpositively curved space. By discretizing the theory appropriately, we show that the energy functional is strongly convex and derive convergence of the discrete heat flow to the energy minimizer, with explicit convergence rate. We also examine center of mass methods, after showing a generalized mean value property for harmonic maps. We feature a concrete illustration of these methods with Harmony, a computer software that we developed in C++, whose main functionality is to numerically compute and display equivariant harmonic maps.Comment: 57 pages, 14 figure