bluuebunny/crossref_metadata_2025 huggingface dataset:
Representation growth of linear groups
Authors: Larsen Michael , Lubotzky Alexander
Published: December 2009

Let

\Gamma

be a group and

r_n(\Gamma)

the number of its

n

-dimensional irreducible complex representations. We define and study the associated representation zeta function

\mathcal Z_\Gamma(s) = \sum^\infty_{n=1} r_n(\Gamma)n^{-s}

. When

\Gamma

is an arithmetic group satisfying the congruence subgroup property then

\mathcal Z_\Gamma(s)

has an “Euler factorization”. The “factor at infinity” is sometimes called the “Witten zeta function” counting the rational representations of an algebraic group. For these we determine precisely the abscissa of convergence. The local factor at a finite place counts the finite representations of suitable open subgroups

U

of the associated simple group

G

over the associated local field

K

. Here we show a surprising dichotomy: if

G(K)

is compact (i.e.

G

anisotropic over

K

) the abscissa of convergence goes to 0 when

\dim G

goes to infinity, but for isotropic groups it is bounded away from

0

. As a consequence, there is an unconditional positive lower bound for the abscissa for arbitrary finitely generated linear groups. We end with some observations and conjectures regarding the global abscissa.
Some examples of nil Lie algebras
Authors: Shestakov Ivan P. , Zelmanov Efim
Published: December 2009

Generalizing Petrogradsky’s construction, we give examples of infinite-dimensional nil Lie algebras of finite Gelfand–Kirillov dimension over any field of positive characteristic.
Numerical Campedelli surfaces with fundamental group of order 9
Authors: Mendes Lopes Margarida , Pardini Rita
Published: December 2009

We give explicit constructions of all the numerical Campedelli surfaces, i.e. the minimal surfaces of general type with

K^2=2

and

p_g=0

, whose fundamental group has order 9. There are three families, one with

\pi^{\mathrm{alg}}=\mathbb Z_9

and two with

\pi^{\mathrm{alg}}=\mathbb Z_3^2

.

We also determine the base locus of the bicanonical system of these surfaces. It turns out that for the surfaces with

\pi^{\mathrm{alg}}=\mathbb Z_9

and for one of the families of surfaces with

\pi^{\mathrm{alg}}=\mathbb Z_3^2

the base locus consists of two points. To our knowlegde, these are the only known examples of surfaces of general type with

K^2>1

whose bicanonical system has base points.
Sharp $L^1$ estimates for singular transport equations
Authors: Klainerman Sergiu , Rodnianski Igor
Published: December 2009

We provide _L_1 estimates for a transport equation which contains singular integral operators. The form of the equation was motivated by the study of Kirchhoff–Sobolev parametrices in a Lorentzian space-time verifying the Einstein equations. While our main application is for a specific problem in General Relativity we believe that the phenomenon which our result illustrates is of a more general interest.
Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in $\mathbb{R}^3$
Authors: Erdoğan M. Burak , Goldberg Michael , Schlag Wilhelm
Published: December 2009

We present a novel approach for bounding the resolvent of a Schroedinger operator with a large first-order perturbation. It is shown here that despite the size of the perturbation, its associated Born series is absolutely convergent at sufficiently high energies. This requires suitable smoothness and polynomial decay of the scalar and vector (magnetic) potentials.

Such a result is well known in the scalar case, but is more difficult for vector potentials because the gradient term exactly cancels the natural decay of the free resolvent. To control the individual terms of the Born series, we introduce a conical decomposition of the resolvent and then sum over all possible combinations of cones. Chains of cones that all point in the same direction lead to a factorial gain typical of iterated Volterra-type operators. On the other hand, cones that are not aligned contribute little due to the decay of Fourier transforms. We make no use of micro-local analysis, but instead rely only on classical phase-space techniques.

As a corollary, we show that the time evolution of the Schroedinger operator in

R^3

satisfies global Strichartz and smoothing estimates without any smallness assumptions. We require that zero energy is neither an eigenvalue nor a resonance for this application.
Estimates for $L^1$-vector fields under higher-order differential conditions
Authors: Van Schaftingen Jean
Published: December 2009

We prove that an

\mathrm{L}^1

vector field whose components satisfy some condition on

k

-th order derivatives induce linear functionals on the Sobolev space

\mathrm{W}^{1,n}(\R^n)

. Two proofs are provided, relying on the two distinct methods developed by Bourgain and Brezis (J.\ Eur.\ Math.\ Soc.\ (JEMS), to appear) and by the author (C.\ R.\ Math.\ Acad.\ Sci.\ Paris, 2004) to prove the same result for divergence-free vector fields and partial extensions to higher-order conditions.
Large data local solutions for the derivative NLS equation
Authors: Bejenaru Ioan , Tataru Daniel
Published: December 2009

We consider the Derivative NLS equation with general quadratic nonlinearities. In [2] the first author has proved a sharp small data local well-posedness result in Sobolev spaces with a decay structure at infinity in dimension

n = 2

. Here we prove a similar result for large initial data in all dimensions

n \geq 2

.
Metastability in Reversible Diffusion Processes I: Sharp Asymptotics for Capacities and Exit Times
Authors: Bovier Anton , Eckhoff Michael , Gayrard Véronique , Klein Markus
Published: December 2009

We develop a potential theoretic approach to the problem of metastability for reversible diffusion processes with generators of the form

-\epsilon \Delta +\nabla F(\cdot)\nabla

on

\mathbb R^d

or subsets of

\mathbb R^d

, where

F

is a smooth function with finitely many local minima. In analogy to previous work on discrete Markov chains, we show that
metastable exit times
from the attractive domains of the minima of

F

can be related, up to multiplicative errors that tend to one as

\epsilon\downarrow 0

, to the capacities of suitably constructed sets. We show that this capacities can be computed, again up to multiplicative errors that tend to one, in terms of local characteristics of

F

at the starting minimum and the relevant
saddle points
. As a result, we are able to give the first rigorous proof of the classical
Eyring–Kramers formula
in dimension larger than

1

. The estimates on capacities make use of their variational representation and monotonicity properties of Dirichlet forms. The methods developed here are extensions of our earlier work on discrete Markov chains to continuous diffusion processes.
Lieb–Thirring inequalities with improved constants
Authors: Dolbeault Jean , Laptev Ari , Loss Michael
Published: December 2009

Following Eden and Foias we obtain a matrix version of a generalised Sobolev inequality in one-dimension. This allows us to improve on the known estimates of best constants in Lieb-Thirring inequalities for the sum of the negative eigenvalues for multi-dimensional Schrödinger operators.
Liouville theorems for self-similar solutions of heat flows
Authors: Li Jiayu , Wang Meng
Published: December 2009

Let

N

be a compact Riemannian manifold. A quasi-harmonic sphere is a harmonic map from

({\bf R}^m, e^{-|x|^2/2(m-2)}ds_0^2)

to

N

(

m\geq 3

) with finite energy ([LnW]). Here

ds_0^2

is the Euclidean metric in

{\bf R}^m

. It arises from the blow-up analysis of the heat flow at a singular point. In this paper, we prove some kinds of Liouville theorems for the quasi-harmonic spheres. It is clear that the Liouville theorems imply the existence of the heat flow to the target

N

. We also derive gradient estimates and Liouville theorems for positive quasi-harmonic functions.
Cambrian fans
Authors: Reading Nathan , Speyer David E.
Published: December 2009

For a finite Coxeter group

W

and a Coxeter element

c

of

W,

the

c

-Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of

W\!

. Its maximal cones are naturally indexed by the

c

-sortable elements of

W\!

. The main result of this paper is that the known bijection

\operatorname{cl}_c

between

c

-sortable elements and

c

-clusters induces a combinatorial isomorphism of fans. In particular, the

c

-Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for

W\!

. The rays of the

c

-Cambrian fan are generated by certain vectors in the

W

-orbit of the fundamental weights, while the rays of the

c

-cluster fan are generated by certain roots. For particular (“bipartite”) choices of

c

, we show that the

c

-Cambrian fan is linearly isomorphic to the

c

-cluster fan. We characterize, in terms of the combinatorics of clusters, the partial order induced, via the map

\operatorname{cl}_c

, on

c

-clusters by the

c

-Cambrian lattice. We give a simple bijection from

c

-clusters to

c

-noncrossing partitions that respects the refined (Narayana) enumeration. We relate the Cambrian fan to well known objects in the theory of cluster algebras, providing a geometric context for

\mathbf{g}

-vectors and quasi-Cartan companions.
Representation of equilibrium solutions to the table problem of growing sandpiles
Authors: Cannarsa Piermarco , Cardaliaguet Pierre
Published: December 2009

In the dynamical theory of granular matter the so-called table problem consists instudying the evolution of a heap of matter poured continuously onto a bounded domain

\Omega\subset \R^2

. The mathematical description of the table problem, at an equilibrium configuration, can be reduced to a boundary value problem for a system of partial differential equations. The analysis of such a system, also connected with other mathematical models such as the Monge-Kantorovich problem, is the object of this paper. Our main result is an integral representation formula for the solution, in terms of the boundary curvature and of the normal distance to the cut locus of

\Omega

.
K3 surfaces with a symplectic automorphism of order 11
Authors: Dolgachev Igor V. , Keum JongHae
Published: December 2009

We classify possible ﬁnite groups of symplectic automorphisms of K3 surfaces of order divisible by 11. The characteristic of the ground ﬁeld must be equal to 11. The complete list of such groups consists of ﬁve groups: the cyclic group

C_{11}

of order 11,

C_{11} ⋊ C_5

,

\mathrm{PSL2}(\mathbb F_{11})

and the Mathieu groups

M_{11}

,

M_{22}

. We also show that a surface

X

admitting an automorphism

g

of order 11 admits a

g

-invariant elliptic ﬁbration with the Jacobian ﬁbration isomorphic to one of explicitly given elliptic K3 surfaces.
Skeletons, bodies and generalized $E(R)$-algebras
Authors: Göbel Rüdiger , Herden Daniel , Shelah Saharon
Published: December 2009

In this paper we want to solve a ﬁfty year old problem on
R
-algebras over cotorsionfree commutative rings
R
with 1. For simplicity (but only for the abstract) we will assume that
R
is any countable principal ideal domain, but not a ﬁeld. For example
R
can be the ring ℤ or the polynomial ring ℚ[
x
]. An
R
-algebra
A
is called a generalized
E
(
R
)-algebra if its algebra End_R_
A
of
R
-module endomorphisms of the underlying
R
-module
RA
is isomorphic to
A
(as an
R
-algebra). Properties, including the existence of such algebras are derived in various papers ([5, 6, 9, 10, 20, 22, 24, 25]). The study was stimulated by Fuchs [13], and specially by Schultz [26]. But due to [26] the investigation concentrated on
ordinary
commutative E(R)-algebras. A substantial part of problem 45 (p. 232) in the monograph [13] (repeated in later publications, e.g. [27]), which will be answered positively for all rings
R
above in this paper, remained open:

Can we ﬁnd non-commutative generalized E(R)-algebras?

In Theorem 1.5 we will show that for all countable, principal ideal domains
R
which are not ﬁelds and for any inﬁnite cardinal
κ
there is a non-commutative
R
-algebra
A
of cardinality |
A
| =
κ_ℵ0 with End_R
A
≅
A
, so
A
is a non-commutative generalized
E
(
R
)-algebra, and—not too surprisingly—there is a proper class of examples.

The new strategy should be interesting and useful for other problems as well: We will ﬁrst translate the heart of the algebraic question on the existence of certain monoids via model theory into geometric structures leading to a special class of (decorated) trees and solve this problem introducing products of trees etc. This can be compared with the well-known, but different process of translating group problems to small cancelations in groups via the van Kampen lemma. By small cancelation of trees we are able to ﬁnd a suitable monoid and thus a non-commutative algebra
A
with an important non-canonical embedding
A
→ End_R_
A
, our ∗-scalar multiplication. In a second part of this paper we must enlarge
A
to get rid of all undesired endomorphisms. This can be done more easily (thus ﬁrst) with the help of additional set theory (Jensen’s diamond predictions), which will support the reader to understand more quickly the last steps of the proof. In a ﬁnal chapter we will also give an argument (for removing unwanted endomorphisms) which is based on ordinary set theory of ZFC only (using our favored Black Box predictions; see [20]). Thus we get the result as stated above. Furthermore, this last chapter includes a construction for rigid systems of non-commutative generalized
E
(
R
)-algebras.
Intersection cohomology of reductive varieties
Authors: Brion Michel , Joshua Roy
Published: December 2009

We extend the methods developed in our earlier work to algorithmically compute the intersection cohomology Betti numbers of reductive varieties. These form a class of highly symmetric varieties that includes equivariant compactifications of reductive groups. Thereby, we extend a well-known algorithm for toric varieties.
Hypersurfaces in $\mathbb{H}^{n+1}$ and conformally invariant equations: the generalized Christoffel and Nirenberg problems
Authors: Espinar José M. , Gálvez José A. , Mira Pablo
Published: December 2009

Our ﬁrst objective in this paper is to give a natural formulation of the Christoffel problem for hypersurfaces in

\mathbb H^{n_+1}

, by means of the hyperbolic Gauss map and the notion of hyperbolic curvature radii for hypersurfaces. Our second objective is to provide an explicit equivalence of this Christoffel problem with the famous problem of prescribing scalar curvature on

\mathbb S^n

for conformal metrics, posed by Nirenberg and Kazdan–Warner. This construction lets us translate into the hyperbolic setting the known results for the scalar curvature problem, and also provides a hypersurface theory interpretation of such an intrinsic problem from conformal geometry. Our third objective is to place the above result in a more general framework. Speciﬁcally, we will show how the problem of prescribing the hyperbolic Gauss map and a given function of the hyperbolic curvature radii in

\mathbb H^{n_+1}

is strongly related to some important problems on conformally invariant PDEs in terms of the Schouten tensor. This provides a bridge between the theory of conformal metrics on

\mathbb S^n

and the theory of hypersurfaces with prescribed hyperbolic Gauss map in

\mathbb H^{n_+1}

. The fourth objective is to use the above correspondence to prove that for a wide family of Weingarten functionals

\mathcal W(\kappa_1, \dots ,\kappa_n)

, the only compact immersed hypersurfaces in

\mathbb H^{n_+1}

on which

\mathcal W

is constant are round spheres.
The sharp Sobolev inequality in quantitative form
Authors: Cianchi Andrea , Fusco Nicola , Maggi Francesco , Pratelli Aldo
Published: December 2009

A quantitative version of the sharp Sobolev inequality in

W^{1,p} (ℝ^n)

,

1 < p < n

, is established with a remainder term involving the distance from the family of extremals.
Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion
Authors: Marcus Moshe , Véron Laurent
Published: December 2009

Let

\Omega

be a bounded domain of class

C^2

in

\mathbb R^N

and let

K

be a compact subset of

\partial\Omega

. Assume that

q\geq (N+1)/(N-1)

and denote by

U_K

the maximal solution of

-\Delta u+u^q=0

in

\Omega

which vanishes on

\partial\Omega\setminus K

. We obtain sharp upper and lower estimates for

U_K

in terms of the Bessel capacity

C_{2/q,q'}

and prove that

U_K

is

\sigma

-moderate. In addition we describe the precise asymptotic behavior of

U_K

at points

\sigma\in K

, which depends on the “density” of

K

at

\sigma

, measured in terms of the capacity

C_{2/q,q'}

.
Separable $p$-harmonic functions in a cone and related quasilinear equations on manifolds
Authors: Porretta Alessio , Véron Laurent
Published: December 2009

In considering a class of quasilinear elliptic equations on a Riemannian manifold with nonnegative Ricci curvature, we give a new proof of Tolksdorf's result on the construction of separable
p
-harmonic functions in a cone.
Heegard Floer invariants of Legendrian knots in contact three-manifolds
Authors: Lisca Paolo , Ozsváth Peter S. , Stipsicz András I. , Szabó Zoltán
Published: December 2009

We deﬁne invariants of null-homologous Legendrian and transverse knots in contact 3manifolds. The invariants are determined by elements of the knot Floer homology of the underlying smooth knot. We compute these invariants, and show that they do not vanish for certain non-loose knots in overtwisted 3-spheres. Moreover, we apply the invariants to ﬁnd transversely non-simple knot types in many overtwisted contact 3-manifolds.
Pólya's conjecture in the presence of a constant magnetic field
Authors: Frank Rupert L. , Loss Michael , Weidl Timo
Published: December 2009

We consider the Dirichlet Laplacian with a constant magnetic field in a two-dimensional domain of finite measure. We determine the sharp constants in semi-classical eigenvalue estimates and show, in particular, that Pólya's conjecture is not true in the presence of a magnetic field.
Gardens of Eden and amenability on cellular automata
Authors: Bartholdi Laurent
Published: December 2009

We prove a converse to the “Garden-of-Eden” theorem by Ceccherini-Silberstein, Machì and Scarabotti, and to a theorem by Meyerovitch, yielding two new characterizations of amenable groups. The following are equivalent:

the group
G
is amenable;

all cellular automata living on
G
that admit mutually erasable patterns also admit gardens of Eden;

all cellular automata living on
G
that do not preserve Bernoulli measure admit gardens of Eden.

This solves in particular Conjecture 6.2 (1) in [2].
Anomalous large-time behaviour of the $p$-Laplacian flow in an exterior domain in low dimension
Authors: Iagar Razvan Gabriel , Vázquez Juan Luis
Published: December 2009

We study the large time behaviour of weak nonnegative solutions of the

p

-Laplace equation posed in an exterior domain in space dimension

N<p

with boundary conditions

u = 0

. The description is done in terms of matched asymptotics: the outer asymptotic proﬁle is a dipole-like self-similar solution with a singularity at

x = 0

and anomalous similarity exponents. The inner asymptotic behaviour is given by a separate-variable proﬁle. We gather both estimates in a global approximant and we also study the behaviour of the free boundary for compactly supported solutions. We complete in this way the analysis made in a previous work for high space dimensions

N ≥ p

, a range in which the large-time inﬂuence of the holes is less dramatic.
Metastability in reversible diffusion processes II: precise asymptotics for small eigenvalues
Authors: Bovier Anton , Gayrard Véronique , Klein Markus
Published: December 2009

We continue the analysis of the problem of metastability for reversible diffusion processes, initiated in [BEGK3], with a precise analysis of the low-lying spectrum of the generator. Recall that we are considering processes with generators of the form

-\epsilon \Delta +\nabla F(\cdot)\nabla

on

\R^d

or subsets of

\R^d

, where

F

is a smooth function with finitely many local minima. Here we consider only the generic situation where the depths of all local minima are different. We show that in general the exponentially small part of the spectrum is given, up to multiplicative errors tending to one, by the eigenvalues of the classical capacity matrix of the array of capacitors made of balls of radius

\epsilon

centered at the positions of the local minima of

F

. We also get very precise uniform control on the corresponding eigenfunctions. Moreover, these eigenvalues can be identified with the same precision with the inverse mean metastable exit times from each minimum. In [BEGK3] it was proven that these mean times are given, again up to multiplicative errors that tend to one, by the classical
Eyring–Kramers formula
.
Coxeter group actions on the complement of hyperplanes and special involutions
Authors: Felder Giovanni , Veselov Alexander P.
Published: December 2009

We consider both standard and twisted action of a (real) Coxeter group

G

on the complement

\mathcal M_G

to the complexified reflection hyperplanes by combining the reflections with complex conjugation. We introduce a natural geometric class of special involutions in

G

and give explicit formulae which describe both actions on the total cohomology

H^*(\mathcal M_G, {\mathbb C})

in terms of these involutions. As a corollary we prove that the corresponding twisted representation is regular only for the symmetric group

S_n

, the Weyl groups of type

D_{2m+1}

,

E_6

and dihedral groups

I_2 (2k+1).

We discuss also the relations with the cohomology of Brieskorn's braid groups.
On polynomials and surfaces of variously positive links
Authors: Stoimenow Alexander
Published: December 2009

It is known that the minimal degree of the Jones polynomial of a positive knot is equal to its genus, and the minimal coefficient is

1

, with a similar relation for links. We extend this result to almost positive links and partly identify the 3 following coefficients for special types of positive links. We also give counterexamples to the Jones polynomial-ribbon genus conjectures for a quasipositive knot. Then we show that the Alexander polynomial completely detects the minimal genus and fiber property of canonical Seifert surfaces associated to almost positive (and almost alternating) link diagrams.
Riemannian geometries on spaces of plane curves
Authors: Michor Peter W. , Mumford David B.
Published: December 2009

We study some Riemannian metrics on the space of regular smooth curves in the plane, viewed as the orbit space of maps from the circle to the plane modulo the group of diffeomorphisms of the circle, acting as reparameterizations. In particular we investigate the L^2 inner product with respect to 1 plus curvature squared times arclength as the measure along a curve, applied to normal vector field to the curve. The curvature squared term acts as a sort of geometric Tikhonov regularization because, without it, the geodesic distance between any 2 distinct curves is 0, while in our case the distance is always positive. We give some lower bounds for the distance function, derive the geodesic equation and the sectional curvature, solve the geodesic equation with simple endpoints numerically, and pose some open questions. The space has an interesting split personality: among large smooth curves, all its sectional curvatures are positive or 0, while for curves with high curvature or perturbations of high frequency, the curvatures are negative.
Stopping Markov processes and first path on graphs
Authors: Aletti Giacomo , Merzbach Ely
Published: December 2009

Given a strongly stationary Markov chain (discrete or continuous) and a finite set of stopping rules, we show a non combinatorial method to compute the law of stopping. Several applied examples are presented. The problem of embedding a graph into a larger but minimal graph under some constraints is studied. Given a connected graph, we show a non combinatorial manner to compute the law of a first given path among a set of stopping paths. We prove the existence of a minimal Markov chain without oversized information.
Submanifold averaging in riemannian and symplectic geometry
Authors: Zambon Marco
Published: December 2009

We give a construction to obtain canonically an “isotropic average” of given

C^1

-close isotropic submanifolds of a symplectic manifold. To do so we use an improvement of Weinstein's submanifold averaging theorem (obtained in collaboration with H. Karcher) and apply “Moser's trick”. We also present an application to Hamiltonian group actions.
Corrigendum to: Stable ergodicity and julienne quasi-conformality, J. Eur. Math. Soc. 2, 1-52
Authors: Pugh Charles , Shub Michael , Starkov Alexander
Published: December 2009

Abstract. In this paper we dramatically expand the domain of known stably ergodic, partially hyperbolic dynamical systems. For example, all partially hyperbolic affine diffeomorphisms of compact homogeneous spaces which have the accessibility property are stably ergodic. Our main tools are the new concepts - julienne density point and julienne quasi-conformality of the stable and unstable holonomy maps. Julienne quasi-conformal holonomy maps preserve all julienne density points.
Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity
Authors: Guedes de Figueiredo Djairo , Gossez Jean-Pierre , Ubilla Pedro
Published: December 2009

In this paper we study the existence, nonexistence and multiplicity of positive solutions for the family of problems

-\Delta u = f_\lambda (x,u)

,

u \in H^1_0(\Omega)

, where

\Omega

is a bounded domain in

\mathbb{R}^N

,

N\geq 3

and

\lambda>0

is a parameter. The results include the well-known nonlinearities of the Ambrosetti–Brezis–Cerami type in a more general form, namely

\lambda a (x)u^q + b(x) u^p

, where

0 \leq q<1<p\leq 2^{\ast}-1

. The coefficient

a(x)

is assumed nonnegative but

b(x)

is allowed to change sign, even in the critical case. The notions of local superlinearity and local sublinearity introduced in [9] are essential in this more general framework. The techniques used in the proofs are lower and upper solutions and variational methods.
A geometric problem and the Hopf Lemma. I
Authors: Li YanYan , Nirenberg Louis
Published: December 2009

A classical result of A. D. Alexandrov states that a connected compact smooth

n

-dimensional manifold without boundary, embedded in

{\mathbb R}^{n+1}

, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of

M

in a hyperplane

X_{n+1}={\rm const}

in case

M

satisfies: for any two points

(X', X_{n+1})

,

(X', \widehat X_{n+1})

on

M

, with

X_{n+1}>\widehat X_{n+1}

, the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional condition for

n=1

. Some variations of the Hopf Lemma are also presented. Part II [Y.Y. Li and L. Nirenberg, Chinese Ann. Math. Ser. B 27 (2006), 193–218] deals with corresponding higher dimensional problems. Several open problems for higher dimensions are described in this paper as well.
Existence of Gorenstein projective resolutions and Tate cohomology
Authors: Jørgensen Peter
Published: December 2009

Existence of proper Gorenstein projective resolutions and Tate cohomology is proved over rings with a dualizing complex. The proofs are based on Bousfield Localization which is originally a method from algebraic topology.
Ramsey partitions and proximity data structures
Authors: Mendel Manor , Naor Assaf
Published: December 2009

This paper addresses two problems lying at the intersection of geometric analysis and theoretical computer science: The non-linear isomorphic Dvoretzky theorem and the design of good approximate distance oracles for large distortion. We introduce the notion of Ramsey partitions of a finite metric space, and show that the existence of good Ramsey partitions implies a solution to the metric Ramsey problem for large distortion (a.k.a. the non-linear version of the isomorphic Dvoretzky theorem, as introduced by Bourgain, Figiel, and Milman in [8]). We then proceed to construct optimal Ramsey partitions, and use them to show that for every

\varepsilon \in (0,1)

, every

n

-point metric space has a subset of size

n^{1-\varepsilon}

which embeds into Hilbert space with distortion

O(1/\varepsilon)

. This result is best possible and improves part of the metric Ramsey theorem of Bartal, Linial, Mendel and Naor [5], in addition to considerably simplifying its proof. We use our new Ramsey partitions to design approximate distance oracles with a universal constant query time, closing a gap left open by Thorup and Zwick in [32]. Namely, we show that for every

n

point metric space

X

, and

k\geq 1

, there exists an

O(k)

-approximate distance oracle whose storage requirement is

O\left( n^{1+1/k}\right)

, and whose query time is a universal constant. We also discuss applications of Ramsey partitions to various other geometric data structure problems, such as the design of efficient data structures for approximate ranking.
New estimates for elliptic equations and Hodge type systems
Authors: Bourgain Jean , Brezis Haïm
Published: December 2009

We establish new estimates for the Laplacian, the div-curl system, and more general Hodge systems in arbitrary dimension
n
, with data in
L1
. We also present related results concerning differential forms with coefficients in the limiting Sobolev space
W1,n
.
Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations
Authors: Da Lio Francesca , Sirakov Boyan
Published: December 2009

We study uniformly elliptic fully nonlinear equations

F(D^2u, Du, u, x)=0,

and prove results of Gidas–Ni–Nirenberg type for positive viscosity solutions of such equations. We show that symmetries of the equation and the domain are reflected by the solution, both in bounded and unbounded domains.
Natural pseudodistances between closed surfaces
Authors: Donatini Pietro , Frosini Patrizio
Published: December 2009

Let us consider two closed surfaces

\mathcal{M}

,

\mathcal{N}

of class

C^1

and two functions

\varphi:{\mathcal{M}}\rightarrow \mathbb{R}

,

\psi:\mathcal{N}\rightarrow \mathbb{R}

of class

C^1

, called measuring functions. The natural pseudodistance

{d}

between the pairs

({\mathcal{M}},\varphi)

,

({\mathcal{N}},\psi)

is defined as the infimum of

\Theta(f)\stackrel{def}{=}\max_{P\in \mathcal{M}}|\varphi(P)-\psi(f(P))|

, as

f

varies in the set of all homeomorphisms from

\mathcal{M}

onto

\mathcal{N}

. In this paper we prove that the natural pseudodistance equals either

|c_1-c_2|

or

\frac{1}{2}|c_1-c_2|

, or

\frac{1}{3}|c_1-c_2|

, where

c_1

and

c_2

are two suitable critical values of the measuring functions. This equality shows that a previous relation between natural pseudodistance and critical values obtained in general dimension can be improved in the case of closed surfaces. Our result is based on a theorem by Jost and Schoen concerning harmonic maps between surfaces.
On the controllability of the 1-D isentropic Euler equation
Authors: Glass Olivier
Published: December 2009

We study the controllability problem for the one-dimensional Euler isentropic system, both in Eulerian and Lagrangian coordinates, by means of boundary controls, in the context of weak entropy solutions. We give a sufficient condition on the initial and final states under which the first one can be steered to the latter.
Enveloping algebras of Slodowy slices and the Joseph ideal
Authors: Premet Alexander
Published: December 2009

Let

G

be a simple algebraic group over an algebraically closed field

\mathbb k

of characteristic

0

, and

\mathfrak g=\text{Lie}\,G

. Let

(e,h,f)

be an

\mathfrak{sl}_2

-triple in

\mathfrak g

with

e

being a long root vector in

\mathfrak g

. Let

(\,\cdot\,,\,\cdot\,)

be the

G

-invariant bilinear form on

\mathfrak g

with

(e,f)=1

and let

\chi\in\mathfrak g^*

be such that

\chi(x)=(e,x)

for all

x\in\mathfrak g

. Let

{\mathcal S}

be the Slodowy slice at

e

through the adjoint orbit of

e

and let

H

be the enveloping algebra of

{\mathcal S}

; see [31]. In this note we give an explicit presentation of

H

by generators and relations. As a consequence we deduce that

H

contains an ideal of codimension

1

which is unique if

\mathfrak g

is not of type

\mathrm A

. Applying Skryabin's equivalence of categories we then construct an explicit Whittaker model for the Joseph ideal of

U(\mathfrak g)

. Inspired by Joseph's Preparation Theorem we prove that there exists a homeomorphism between the primitive spectrum of

H

and the spectrum of all primitive ideals of infinite codimension in

U(\mathfrak g)

which respects Goldie rank and Gelfand–Kirillov dimension. We study highest weight modules for the algebra

H

and apply earlier results of Miličić–Soergel and Backelin to express the composition multiplicities of the Verma modules for

H

in terms of some inverse parabolic Kazhdan–Lusztig polynomials. Our results confirm in the minimal nilpotent case the de Vos–van Driel conjecture on composition multiplicities of Verma modules for finite

{\mathcal W}

-algebras. We also obtain some general results on the enveloping algebras of Slodowy slices and determine the associated varieties of related primitive ideals of

U(\mathfrak g)

. A sequel to this paper will treat modular aspects of this theory.
Localization for Schrödinger operators with Poisson random potential
Authors: Germinet François , Hislop Peter D. , Klein Abel
Published: December 2009

We prove exponential and dynamical localization for the Schrödinger operator with a nonnegative Poisson random potential at the bottom of the spectrum in any dimension. We also conclude that the eigenvalues in that spectral region of localization have finite multiplicity. We prove similar localization results in a prescribed energy interval at the bottom of the spectrum provided the density of the Poisson process is large enough.
On the geometry of binary symmetric models of phylogenetic trees
Authors: Buczynska Weronika , Wisniewski Jaroslaw A.
Published: December 2009

We investigate projective varieties which are binary symmetric models of trivalent phylogenetic trees. We prove that they have Gorenstein terminal singularities and are Fano varieties of index 4 and dimension equal to the number of edges of the tree in question. Moreover any two such varieties which are of the same dimension are deformation equivalent, that is, they are in the same connected component of the Hilbert scheme of the projective space. As an application we provide a simple formula for computing their Hilbert-Ehrhart polynomial.
The Strong Anick Conjecture is true
Authors: Drensky Vesselin , Yu Jie-Tai
Published: December 2009

Recently Umirbaev has proved the long-standing Anick conjecture, that is, there exist wild automorphisms of the free associative algebra

K\langle x,y,z\rangle

over a field

K

of characteristic 0. In particular, the well-known Anick automorphism is wild. In this article we obtain a stronger result (the Strong Anick Conjecture that implies the Anick Conjecture). Namely, we prove that there exist wild coordinates of

K\langle x,y,z\rangle

. In particular, the two nontrivial coordinates in the Anick automorphism are both wild. We establish a similar result for several large classes of automorphisms of

K\langle x,y,z\rangle

. We also find a large new class of wild automorphisms of

K\langle x,y,z\rangle

which is not covered by the results of Umirbaev. Finally, we study the lifting problem for automorphisms and coordinates of polynomial algebras, free metabelian algebras and free associative algebras and obtain some interesting new results.
On the complexity of braids
Authors: Dynnikov Ivan , Wiest Bert
Published: December 2009

We define a measure of “complexity” of a braid which is natural with respect to both an algebraic and a geometric point of view. Algebraically, we modify the standard notion of the length of a braid by introducing generators

\Delta_{ij}

, which are Garside-like half-twists involving strings

i

through

j

, and by counting powered generators

\Delta_{ij}^k

as

\log(|k|+1)

instead of simply

|k|

. The geometrical complexity is some natural measure of the amount of distortion of the

n

times punctured disk caused by a homeomorphism. Our main result is that the two notions of complexity are comparable. This gives rise to a new combinatorial model for the Tei space of an

n+1

times punctured sphere. We also show how to recover a braid from its curve diagram in polynomial time. The key rôle in the proofs is played by a technique introduced by Agol, Hass, and Thurston.
The homotopy Gerstenhaber algebra of Hochschild cochains of a regular algebra is formal
Authors: Dolgushev Vasily , Tamarkin Dmitry , Tsygan Boris
Published: December 2009

The solution of Deligne's conjecture on Hochschild cochains and the formality of the operad of little disks provide us with a natural homotopy Gerstenhaber algebra structure on the Hochschild cochains of an associative algebra. In this paper we construct a natural chain of quasi-isomorphisms of homotopy Gerstenhaber algebras between the Hochschild cochain complex C•(A) of a regular commutative algebra A over a field

\mathbb{K}

of characteristic zero and the Gerstenhaber algebra of multiderivations of A. Unlike the original approach of the second author based on the computation of obstructions our method allows us to avoid the bulky Gelfand–Fuchs trick and prove the formality of the homotopy Gerstenhaber algebra structure on the sheaf of polydifferential operators on a smooth algebraic variety, a complex manifold, and a smooth real manifold.
Moduli space actions on the Hochschild co-chains of a Frobenius algebra I: cell operads
Authors: Kaufmann Ralph M.
Published: December 2009

This is the first of two papers in which we prove that a cell model of the moduli space of curves with marked points and tangent vectors at the marked points acts on the Hochschild co-chains of a Frobenius algebra. We also prove that a there is dg-PROP action of a version of Sullivan chord diagrams which acts on the normalized Hochschild co-chains of a Frobenius algebra. These actions lift to operadic correlation functions on the co-cycles. In particular, the PROP action gives an action on the homology of a loop space of a compact simply-connected manifold.

In this first part, we set up the topological operads/PROPs and their cell models. The main theorems of this part are: There is a cell model operad for the moduli space of genus g curves with n punctures and a tangent vector at each of these punctures, there exists a CW complex whose chains are isomorphic to a certain type of Sullivan chord diagrams and they form a PROP. Furthermore there exist weak versions of these structures on the topological level which all lie inside an all encompassing cyclic (rational) operad.
Phase transition in the Connes–Marcolli $\mathrm{GL}_2$-system
Authors: Laca Marcelo , Larsen Nadia S. , Neshveyev Sergey
Published: December 2009

We develop a general framework for analyzing KMS-states on

\mathrm{C}^*

-algebras arising from actions of Hecke pairs. We then specialize to the system recently introduced by Connes and Marcolli and classify its KMS-states for inverse temperatures

\beta \neq 0,1

. In particular, we show that for each

\beta \in (1,2]

there exists a unique KMS

_\beta

-state.
Pseudo-multiplicative unitaries on C*-modules and Hopf C*-families I
Authors: Timmermann Thomas
Published: December 2009

Pseudo-multiplicative unitaries on C*-modules generalize the multiplicative unitaries of Baaj and Skandalis [1], and are analogues of the pseudo-multiplicative unitaries on Hilbert spaces studied by Enock, Lesieur, Vallin [5], [10], [21]. We introduce Hopf C*-families on C*-bimodules and associate to special classes of pseudo-multiplicative unitaries two Hopf C*-families. Furthermore, we discuss dual pairings and counits on these Hopf C*-families, étalé and proper pseudo-multiplicative unitaries, and two classes of examples. In a later article, we will study regularity conditions on pseudo-multiplicative unitaries, coactions of Hopf C*-families on C*-algebras, and duality.
$N$-homogeneous superalgebras
Authors: Hai Phùng Hô , Kriegk Benoît , Lorenz Martin
Published: December 2009

We develop the theory of

N

-homogeneous algebras in a super-setting, with particular emphasis on the Koszul property. To any Hecke operator

\mathcal R

on a vector superspace, we associate certain superalgebras

S_{\mathcal{R},N}

and

\wedge_{\mathcal{R},N}

generalizing the ordinary symmetric and Grassmann algebra, respectively. We prove that these algebras are

N

-Koszul. For the special case where

\mathcal{R}

is the ordinary supersymmetry, we derive an

N

-generalized super-version of MacMahon’s classical “master theorem”.
The necklace Lie coalgebra and renormalization algebras
Authors: Gan Wee Liang , Schedler Travis
Published: December 2009

We give a natural monomorphism from the necklace Lie coalgebra, defined for any quiver, to Connes and Kreimer’s Lie coalgebra of trees, and extend this to a map from a certain quiver-theoretic Hopf algebra to Connes and Kreimer’s renormalization Hopf algebra as well as to pre-Lie versions. These results are direct analogues of Turaev’s results in 2004, by replacing algebras of loops on surfaces with algebras of paths on quivers. We also factor the morphism through an algebra of chord diagrams and explain the geometric version. We then describe how all of the Hopf algebras are uniquely determined by the pre-Lie structures and discuss noncommutative versions of the Hopf algebras.
Bordism, rho-invariants and the Baum–Connes conjecture
Authors: Piazza Paolo , Schick Thomas
Published: December 2009

Let

Γ

be a finitely generated discrete group. In this paper we establish vanishing results for rho-invariants associated to  (i) the spin Dirac operator of a spin manifold with positive scalar curvature and fundamental group

Γ

;  (ii) the signature operator of the disjoint union of a pair of homotopy equivalent oriented manifolds with fundamental group

Γ

.
The invariants we consider are more precisely

We prove that all these rho-invariants vanish if the group

Γ

is
torsion-free
and the Baum–Connes map for the maximal group

\mathrm C^*

-algebra is bijective. This condition is satisfied, for example, by torsion-free amenable groups or by torsion-free discrete subgroups of

\mathrm{SO}(n,1)

and

\mathrm{SU}(n,1)

. For the delocalized invariant we only assume the validity of the Baum–Connes conjecture for the reduced

\mathrm C^*

-algebra. In addition to the examples above, this condition is satisfied e.g. by Gromov hyperbolic groups or by cocompact discrete subgroups of

\mathrm{SL}(3,ℂ)

.

In particular, the three rho-invariants associated to the signature operator are, for such groups,
homotopy invariant
. For the APS and the Cheeger–Gromov rho-invariants the latter result had been established by Navin Keswani. Our proof reestablishes this result and also extends it to the delocalized eta-invariant of Lott. The proof exploits in a fundamental way results from bordism theory as well as various generalizations of the APS-index theorem; it also embeds these results in general vanishing phenomena for degree zero higher rho-invariants (taking values in

A/[A,A]

for suitable

\mathrm C^*

-algebras

A

). We also obtain precise information about the eta-invariants in question themselves, which are usually much more subtle objects than the rho-invariants.
Fourier analysis on the affine group, quantization and noncompact Connes geometries
Authors: Gayral Victor , Gracia-Bondí­a José M. , Várilly Joseph C.
Published: December 2009

We find the Stratonovich–Weyl quantizer for the nonunimodular affine group of the line. A noncommutative product of functions on the half-plane, underlying a noncompact spectral triple in the sense of Connes, is obtained from it. The corresponding Wigner functions reproduce the time-frequency distributions of signal processing. The same construction leads to scalar Fourier transformations on the affine group, simplifying and extending the Fourier transformation proposed by Kirillov.
Properly infinite $C(X)$-algebras and $K_1$-injectivity
Authors: Blanchard Étienne , Rohde Randi , Rørdam Mikael
Published: December 2009

We investigate if a unital

C(X)

-algebra is properly infinite when all its fibres are properly infinite. We show that this question can be rephrased in several different ways, including the question of whether every unital properly infinite

\mathrm{C}^*

-algebra is

K_1

-injective. We provide partial answers to these questions, and we show that the general question on proper infiniteness of

C(X)

-algebras can be reduced to establishing proper infiniteness of a specific

C([0,1])

-algebra with properly infinite fibres.
Moduli space actions on the Hochschild co-chains of a Frobenius algebra II: correlators
Authors: Kaufmann Ralph M.
Published: December 2009

This is the second of two papers in which we prove that a cell model of the moduli space of curves with marked points and tangent vectors at the marked points acts on the Hochschild co-chains of a Frobenius algebra. We also prove that there is dg-PROP action of a version of Sullivan chord diagrams which acts on the normalized Hochschild co-chains of a Frobenius algebra. These actions lift to operadic correlation functions on the co-cycles. In particular, the PROP action gives an action on the homology of a loop space of a compact simply connected manifold.

In this second part, we discretize the operadic and PROPic structures of the first part. We also introduce the notion of operadic correlation functions and use them in conjunction with operadic maps from the cell level to the discretized objects to define the relevant actions.
Some notes on the Feigin–Losev–Shoikhet integral conjecture
Authors: Ramadoss Ajay C.
Published: December 2009

Given a holomorphic vector bundle

\mathcal E

on a smooth connected compact complex manifold

X

, Feigin, Losev and Shoikhet [FLS] use a notion of completed Hochschild homology

\widehat{\mathrm{HH}}

of

\mathcal{D}\mathrm{iff}(\mathcal E)

such that

\widehat{\mathrm{HH}}_0(\mathcal{D}\mathrm{iff}(\mathcal E))

is isomorphic to

H^{2n}(X, ℂ)

. On the other hand, they construct a linear functional on

\widehat{\mathrm{HH}}_0 (\mathcal{D}\mathrm{iff}(ℰ))

. This therefore gives rise to a linear functional

I_{\mathcal E}

on

H^{2n}(X, ℂ)

. They show that this functional is

\int_X

if

\mathcal E

has non-zero Euler characteristic. They conjecture that this functional is

\int_X

for all

\mathcal E

.

In this article it is proved that

I_{\mathcal E} = I_{\mathcal F}

for any pair

(\mathcal E, \mathcal F)

of holomorphic vector bundles on

X

. In particular, if

X

has one vector bundle with non-zero Euler characteristic, then

I_{\mathcal E} = \int_X

for every vector bundle

\mathcal E

on

X

.

In [FLS] there is also used a notion of completed cyclic homology

\widehat{\mathrm{HC}}

of

\mathcal{D}\mathrm{iff}(\mathcal E)

such that

\widehat{\mathrm{HC}}_{−i}(\mathcal{D}\mathrm{iff}(\mathcal E)) \simeq H^{2n − i}(X, ℂ) \oplus H^{2n − i + 2}(X, ℂ) \oplus \cdots

. The construction yielding

I_{\mathcal E}

generalizes to give linear functionals on

\widehat{\mathrm{HC}}_{−2i}(\mathcal{D}\mathrm{iff}(\mathcal E))

for each

i ≥ 0

. The linear functional thus obtained on

\widehat{\mathrm{HC}}_{−2i}(\mathcal{D}\mathrm{iff}(\mathcal E))

yields a linear functional

I_{\mathcal E,2i,2k}

on

H^{2n − 2k}(X, ℂ)

for

0 ≤ k ≤ i

. It is conjectured in [FLS] that

I_{\mathcal E,2,0} = \int_X

, and a further conjecture about

I_{\mathcal E,2,2}

is made.

In this article we prove that

I_{\mathcal E,2i,0} = I_{\mathcal E}

for all

i ≥ 0

. In particular, if

X

has at least one vector bundle with non-zero Euler characteristic, then

I_{\mathcal E,2i,0} = \int_X

. We also prove that

I_{\mathcal E,2i,2k} = 0

for

k > 0

. The latter is stronger than what is expected in [FLS] when

i = k = 1

.