Advances in mathematics

Vol. 350, p. 396-439

DOI: 10.1016/j.aim.2019.04.048

Date of publication: 2019-07-09

Advances in mathematics

Vol. 334, p. 544-584

DOI: 10.1016/j.aim.2018.03.018

Date of publication: 2018-08-20

Abstract:

Decomposition spaces are simplicial 8-groupoids subject to a certain exactness condition, needed to induce a coalgebra structure on the space of arrows. Conservative ULF functors (CULF) between decomposition spaces induce coalgebra homomorphisms. Suitable added finiteness conditions define the notion of Möbius decomposition space, a far-reaching generalisation of the notion of Möbius category of Leroux. In this paper, we show that the Lawvere–Menni Hopf algebra of Möbius intervals, which contains the universal Möbius function (but is not induced by a Möbius category), can be realised as the homotopy cardinality of a Möbius decomposition space U of all Möbius intervals, and that in a certain sense U is universal for Möbius decomposition spaces and CULF functors.]]>

Advances in mathematics

Vol. 333, p. 1242-1292

DOI: 10.1016/j.aim.2018.03.017

Date of publication: 2018-07-31

Abstract:

This is the second in a trilogy of papers introducing and studying the notion of decomposition space as a general framework for incidence algebras and Möbius inversion, with coefficients in 8-groupoids. A decomposition space is a simplicial 8-groupoid satisfying an exactness condition weaker than the Segal condition. Just as the Segal condition expresses composition, the new condition expresses decomposition. In this paper, we introduce various technical conditions on decomposition spaces. The first is a completeness condition (weaker than Rezk completeness), needed to control simplicial nondegeneracy. For complete decomposition spaces we establish a general Möbius inversion principle, expressed as an explicit equivalence of 8-groupoids. Next we analyse two finiteness conditions on decomposition spaces. The first, that of locally finite length, guarantees the existence of the important length filtration for the associated incidence coalgebra. We show that a decomposition space of locally finite length is actually the left Kan extension of a semi-simplicial space. The second finiteness condition, local finiteness, ensures we can take homotopy cardinality to pass from the level of 8-groupoids to the level of -vector spaces. These three conditions — completeness, locally finite length, and local finiteness — together define our notion of Möbius decomposition space, which extends Leroux's notion of Möbius category (in turn a common generalisation of the locally finite posets of Rota et al. and of the finite decomposition monoids of Cartier–Foata), but which also covers many coalgebra constructions which do not arise from Möbius categories, such as the Faà di Bruno and Connes–Kreimer bialgebras.]]>

Advances in mathematics

Vol. 332, p. 34-56

DOI: 10.1016/j.aim.2018.04.006

Date of publication: 2018-07-09

Abstract:

In [7] Monod introduced examples of groups of piecewise projective homeomorphisms which are not amenable and which do not contain free subgroups, and in [6] Lodha and Moore introduced examples of finitely presented groups with the same property. In this article we examine the normal subgroup structure of these groups. Two important cases of our results are the groups H and . We show that the group H of piecewise projective homeomorphisms of has the property that is simple and that every proper quotient of H is metabelian. We establish simplicity of the commutator subgroup of the group , which admits a presentation with 3 generators and 9 relations. Further, we show that every proper quotient of is abelian. It follows that the normal subgroups of these groups are in bijective correspondence with those of the abelian (or metabelian) quotient.

© 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/]]>

Advances in mathematics

Vol. 331, p. 952-1015

DOI: 10.1016/j.aim.2018.03.016

Date of publication: 2018-06-20

Abstract:

This is the first in a series of papers devoted to the theory of decomposition spaces, a general framework for incidence algebras and Möbius inversion, where algebraic identities are realised by taking homotopy cardinality of equivalences of 8-groupoids. A decomposition space is a simplicial 8-groupoid satisfying an exactness condition, weaker than the Segal condition, expressed in terms of active and inert maps in Image 1. Just as the Segal condition expresses composition, the new exactness condition expresses decomposition, and there is an abundance of examples in combinatorics. After establishing some basic properties of decomposition spaces, the main result of this first paper shows that to any decomposition space there is an associated incidence coalgebra, spanned by the space of 1-simplices, and with coefficients in 8-groupoids. We take a functorial viewpoint throughout, emphasising conservative ULF functors; these induce coalgebra homomorphisms. Reduction procedures in the classical theory of incidence coalgebras are examples of this notion, and many are examples of decalage of decomposition spaces. An interesting class of examples of decomposition spaces beyond Segal spaces is provided by Hall algebras: the Waldhausen -construction of an abelian (or stable infinity) category is shown to be a decomposition space. In the second paper in this series we impose further conditions on decomposition spaces, to obtain a general Möbius inversion principle, and to ensure that the various constructions and results admit a homotopy cardinality. In the third paper we show that the Lawvere–Menni Hopf algebra of Möbius intervals is the homotopy cardinality of a certain universal decomposition space. Two further sequel papers deal with numerous examples from combinatorics.]]>

Advances in mathematics

Vol. 331, p. 941-951

DOI: 10.1016/j.aim.2018.04.003

Date of publication: 2018-05-20

Abstract:

We study a notion of pre-quantization for b-symplectic manifolds. We use it to construct a formal geometric quantization of b-symplectic manifolds equipped with Hamiltonian torus actions with nonzero modular weight. We show that these quantizations are finite dimensional T-modules.]]>

Advances in mathematics

Vol. 321, p. 298-325

DOI: 10.1016/j.aim.2017.09.019

Date of publication: 2017-12-01

Abstract:

We study the structure of D -modules over a ring R which is a direct sum- mand of a polynomial or a power series ring S with coefficients over a field. We relate properties of D -modules over R to D -modules over S . We show that the localization R f and the local cohomology module H i I ( R ) have finite length as D -modules over R . Furthermore, we show the existence of the Bernstein-Sato polynomial for elements in R . In positive characteristic, we use this relation between D -modules over R and S to show that the set of F -jumping numbers of an ideal I ¿ R is contained in the set of F -jumping numbers of its extension in S . As a consequence, the F -jumping numbers of I in R form a]]>

Advances in mathematics

Vol. 315, p. 285-323

DOI: 10.1016/j.aim.2017.05.003

Date of publication: 2017-07-31

Abstract:

Phylogenetic varieties related to equivariant substitution models have been studied largely in the last years. One of the main objectives has been finding a set of generators of the ideal of these varieties, but this has not yet been achieved in some cases (for example, for the general Markov model this involves the open “salmon conjecture”, see [2]) and it is not clear how to use all generators in practice. Motivated by applications in biology, we tackle the problem from another point of view. The elements of the ideal that could be useful for applications in phylogenetics only need to describe the variety around certain points of no evolution (see [13]). We produce a collection of explicit equations that describe the variety on a Zariski open neighborhood of these points (see Theorem 5.4). Namely, for any tree T on any number of leaves (and any degrees at the interior nodes) and for any equivariant model on any set of states ¿, we compute the codimension of the corresponding phylogenetic variety. We prove that this variety is smooth at general points of no evolution and, if a mild technical condition is satisfied (“d-claw tree hypothesis”), we provide an algorithm to produce a complete intersection that describes the variety around these points.]]>

Advances in mathematics

Vol. 312, p. 286-314

DOI: 10.1016/j.aim.2017.03.022

Date of publication: 2017-05-25

Abstract:

A clutter is a family of mutually incomparable sets. The set of circuits of a matroid, its set of bases, and its set of hyperplanes are examples of clutters arising from matroids. In this paper we address the question of determining which are the matroidal clutters that best approximate an arbitrary clutter ¿. For this, we first define two orders under which to compare clutters, which give a total of four possibilities for approximating ¿ (i.e., above or below with respect to each order); in fact, we actually consider the problem of approximating ¿ with clutters from any collection of clutters S, not necessarily arising from matroids. We show that, under some mild conditions, there is a finite non-empty set of clutters from S that are the closest to ¿ and, moreover, that ¿ is uniquely determined by them, in the sense that it can be recovered using a suitable clutter operation. We then particularize these results to the case where S is a collection of matroidal clutters and give algorithmic procedures to compute these clutters.]]>

Advances in mathematics

Vol. 304, p. 769-792

Date of publication: 2017-01-02

Abstract:

We study the multiplicity of the jumping numbers of an mm-primary ideal a in a two-dimensional local ring with a rational singularity. The formula we provide for the multiplicities leads to a very simple and efficient method to detect whether a given rational number is a jumping number. We also give an explicit description of the Poincare series of multiplier ideals associated to a proving, in particular, that it is a rational function.]]>

Advances in mathematics

Vol. 301, p. 615-692

DOI: 10.1016/j.aim.2016.06.018

Date of publication: 2016-10-01

Abstract:

We consider the completely resonant non-linear Schrödinger equation on the two dimensional torus with any analytic gauge invariant nonlinearity. Fix s>1. We show the existence of solutions of this equation which achieve arbitrarily large growth of Hs Sobolev norms. We also give estimates for the time required to attain this growth.]]>

Advances in mathematics

Vol. 294, p. 689-755

DOI: 10.1016/j.aim.2015.11.010

Date of publication: 2016-05-14

Abstract:

We consider models given by Hamiltonians of the form View the MathML sourceH(I,f,p,q,t;e)=h(I)+¿j=1n±(12pj2+Vj(qj))+eQ(I,f,p,q,t;e) Turn MathJax on where I¿I¿Rd,f¿TdI¿I¿Rd,f¿Td, p,q¿Rnp,q¿Rn, t¿T1t¿T1. These are higher dimensional analogues, both in the center and hyperbolic directions, of the models studied in , and and are usually called “a-priori unstable Hamiltonian systems”. All these models present the large gap problem. We show that, for 0

Advances in mathematics

Vol. 287, p. 788

DOI: 10.1016/j.aim.2015.10.011

Date of publication: 2016-01-10

Abstract:

This paper addresses a very classical topic that goes back at least to Plücker: how to understand a plane curve singularity using its polar curves. Here, we explicitly construct the singular points of a plane curve singularity directly from the weighted cluster of base points of its polars. In particular, we determine the equisingularity class (or topological equivalence class) of a germ of plane curve from the equisingularity class of generic polars and combinatorial data about the non-singular points shared by them.]]>

Advances in mathematics

Vol. 283, p. 130-142

DOI: 10.1016/j.aim.2015.07.007

Date of publication: 2015-10-01

Abstract:

This article examines the Fourier expansions of certain non-classical p-adic modular forms of weight one: the eponymous generalised eigertforms of the title, so called because they lie in a generalised eigenspace for the Hecke operators. When this generalised eigenspace contains the theta series attached to a character of a real quadratic field K in which the prime p splits, the coefficients of the attendant generalised eigenform are expressed as p-adic logarithms of algebraic numbers belonging to an idoneous ring class field of K. This suggests an approach to "explicit class field theory" for real quadratic fields which is simpler than the one based on Stark's conjecture or its p-adic variants, and is perhaps closer in spirit to the classical theory of singular moduli. (C) 2015 Elsevier Inc. All rights

This article examines the Fourier expansions of certain non-classical p-adic modular forms of weight one: the eponymous generalised eigertforms of the title, so called because they lie in a generalised eigenspace for the Hecke operators. When this generalised eigenspace contains the theta series attached to a character of a real quadratic field K in which the prime p splits, the coefficients of the attendant generalised eigenform are expressed as p-adic logarithms of algebraic numbers belonging to an idoneous ring class field of K. This suggests an approach to]]>

Advances in mathematics

Vol. 270, num. --, p. 97-137

DOI: 10.1016/j.aim.2014.09.026

Date of publication: 2015-01-01

Abstract:

We show that the conformally invariant fractional powers of the sub-Laplacian on the Heisenberg group are given in terms of the scattering operator for an extension problem to the Siegel upper halfspace. Remarkably, this extension problem is dierent from the one studied, among others, by Caarelli and Silvestre.

We show that the conformally invariant fractional powers of the sub-Laplacian on the Heisenberg group are given in terms of the scattering operator for an extension problem to the Siegel upper halfspace. Remarkably, this extension problem is di erent from the one studied, among others, by Ca arelli and Silvestre.]]>

Advances in mathematics

Vol. 258, p. 286-350

DOI: 10.1016/j.aim.2014.03.011

Date of publication: 2014-06-20

Abstract:

By a 2-group we mean a groupoid equipped with a weakened group structure. It is called split when it is equivalent to the semidirect product of a discrete 2-group and a one-object 2-group. By a permutation 2-group we mean the 2-group Sym(G) of self-equivalences of a groupoid G and natural isomorphisms between them, with the product given by composition of self-equivalences. These generalize the symmetric groups S-n, n >= 1, obtained when G is a finite discrete groupoid.; After introducing the wreath 2-product S-n (sic) G of the symmetric group S-n with an arbitrary 2-group G, it is shown that for any (finite type) groupoid G the permutation 2-group Sym(G) is equivalent to a product of wreath 2-products of the form S-n (sic) Sym(G) for a group G thought of as a one-object groupoid. This is next used to compute the homotopy invariants of Sym(G) which classify it up to equivalence. Using a previously shown splitness criterion for strict 2-groups, it is then proved that Sym(G) can be non-split, and that the step from the trivial groupoid to an arbitrary one-object groupoid is the only source of non-splitness. Various examples of permutation 2-groups are explicitly computed, in particular the permutation 2-group of the underlying groupoid of a (finite type) 2-group. It also follows from well known results about the symmetric groups that the permutation 2-group of the groupoid of all finite sets and bijections between them is equivalent to the direct product 2-group Z(2)[1] x Z(2)[0] where Z(2)[0] and Z(2)[1] stand for the group Z(2) thought of as a discrete and a one-object 2-group, respectively. (C) 2014 Elsevier Inc. All rights reserved.]]>

Advances in mathematics

Vol. 254, p. 233-250

DOI: 10.1016/j.aim.2013.12.006

Date of publication: 2014-03-20

Abstract:

We characterize the centers of the quasi-homogeneous planar polynomial differential systems of degree three. Such systems do not admit isochronous centers. At most one limit cycle can bifurcate from the periodic orbits of a center of a cubic homogeneous polynomial system using the averaging theory of first order]]>

Advances in mathematics

Vol. 254, p. 79-117

DOI: 10.1016/j.aim.2013.12.015

Date of publication: 2014-03-20

Abstract:

We prove a Faa di Bruno formula for the Green function in the bialgebra of P-trees, for any polynomial endofunctor P. The formula appears as relative homotopy cardinality of an equivalence of groupoids. (C) 2013 Elsevier Inc. All rights reserved.

We prove a Faà di Bruno formula for the Green function in the bialgebra of P-trees, for any polynomial endofunctor P. The formula appears as relative homotopy cardinality of an equivalence of groupoids.]]>

Advances in mathematics

Vol. 264, p. 864-896

DOI: 10.1016/j.aim.2014.07.032

Date of publication: 2014

Abstract:

Let M2nM2n be a Poisson manifold with Poisson bivector field ¿ . We say that M is b -Poisson if the map ¿n:M¿¿2n(TM)¿n:M¿¿2n(TM) intersects the zero section transversally on a codimension one submanifold Z¿MZ¿M. This paper will be a systematic investigation of such Poisson manifolds. In particular, we will study in detail the structure of (M,¿)(M,¿) in the neighborhood of Z and using symplectic techniques define topological invariants which determine the structure up to isomorphism. We also investigate a variant of de Rham theory for these manifolds and its connection with Poisson cohomology.]]>

Advances in mathematics

Vol. 229, num. 2, p. 1136-1179

DOI: 10.1016/j.aim.2011.09.013

Date of publication: 2011-11-08

Advances in mathematics

Vol. 227, num. 1, p. 170-209

DOI: 10.1016/j.aim.2011.01.016

Date of publication: 2011-05-01

Abstract:

The regular representation of an essentially finite 2-group $\mathbb{G}$ in the 2-category $\mathbf{2Vect}_k$ of (Kapranov and Voevodsky) 2-vector spaces is defined and cohomology invariants classifying it computed. It is next shown that all hom-categories in $\mathbf{Rep}_{\mathbf{2Vect}_k}(\mathbb{G})$ are 2-vector spaces under quite standard assumptions on the field $k$, and a formula giving the corresponding "intertwining numbers" is obtained which proves they are symmetric. Finally, it is shown that the forgetful 2-functor ${\boldmath$\omega$}:\mathbf{Rep}_{\mathbf{2Vect}_k}(\mathbb{G})\To\mathbf{2Vect}_k$ is representable with the regular representation as representing object. As a consequence we obtain a $k$-linear equivalence between the 2-vector space $\mathbf{Vect}_k^{\mathcal{G}}$ of functors from the underlying groupoid of $\mathbb{G}$ to $\mathbf{Vect}_k$, on the one hand, and the $k$-linear category $\mathcal{E} nd({\boldmath$\omega$})$ of pseudonatural endomorphisms of ${\boldmath$\omega$}$, on the other hand. We conclude that $\mathcal{E} nd({\boldmath$\omega$})$ is a 2-vector space, and we (partially) describe a basis of it.]]>

Advances in mathematics

Vol. 226, num. 2, p. 1410-1432

DOI: 10.1016/j.aim.2010.07.016

Date of publication: 2011-01-30

Abstract:

In this note, we study the connection between the fractional Laplacian operator that appeared in the recent work of Caffarelli and Silvestre and a class of conformally covariant operators in conformal geometry.]]>

Advances in mathematics

Vol. 4, num. 4, p. 423-453

DOI: 10.3934/amc.2010.4.453

Date of publication: 2010-11

Advances in mathematics

Vol. 224, num. 5, p. 2052-2093

DOI: 10.1016/j.aim.2010.01.025

Date of publication: 2010-08-01

Advances in mathematics

Vol. 217, num. 3, p. 1096-1153

Date of publication: 2008-02

Advances in mathematics

Vol. 216, num. 2, p. 753-770

Date of publication: 2007-12

Advances in mathematics

Vol. 215, num. 1, p. 379-426

Date of publication: 2007-10

Advances in mathematics

Vol. 213, num. 1, p. 53-92

Date of publication: 2007-08

Advances in mathematics

Vol. 211, num. 1, p. 94-104

DOI: 10.1016/j.aim.2006.07.011

Date of publication: 2007-05

Advances in mathematics

Vol. 202, num. 1, p. 64-188

Date of publication: 2006-04

Advances in mathematics

Vol. 197, num. 2, p. 499-522

Date of publication: 2005-10

Advances in mathematics

Vol. 189, num. 1, p. 68-87

Date of publication: 2004-12

Advances in mathematics

Vol. 182, p. 204-277

Date of publication: 2004-03

Advances in mathematics

Vol. 180, num. 1, p. 104-111

Date of publication: 2003-12

Advances in mathematics

Vol. 1, num. 174, p. 35-53

Date of publication: 2003-03