bluuebunny/crossref_metadata_2025 huggingface dataset:
A path model for geodesics in Euclidean buildings and its applications to representation theory
Authors: Kapovich Michael , Millson John J.
Published: December 2009

In this paper we give a combinatorial characterization of projections of geodesics in Euclidean buildings to Weyl chambers. We apply these results to the representation theory of complex reductive Lie groups and to spherical Hecke rings associated with split nonarchimedean reductive Lie groups. Our main application is a generalization of the saturation theorem of Knutson and Tao for SLn to other complex semisimple Lie groups.
A minimal non-solvable group of homeomorphisms
Authors: Bleak Collin
Published: December 2009

Let

\mathrm{PL}_o(I)

represent the group of orientation-preserving piecewise-linear homeomorphisms of the unit interval which admit finitely many breaks in slope, under the operation of composition. We find a non-solvable group

W

and show that

W

embeds in every non-solvable subgroup of

\mathrm{PL}_o(I)

. We find mild conditions under which other non-solvable subgroups

B

,

(≀ℤ≀)^∞

,

(ℤ≀)^∞

, and

{}^∞(≀ℤ)

embed in subgroups of Let

\mathrm{PL}_o(I)

. We show that all solvable subgroups of

\mathrm{PL}_o(I)

embed in all non-solvable subgroups of PLo(I). These results continue to apply if we replace

\mathrm{PL}_o(I)

by any generalized Thompson group

F_n

.
Visual decompositions of Coxeter groups
Authors: Mihalik Michael , Tschantz Steven
Published: December 2009

A Coxeter system is an ordered pair (W,S) where S is the generating set in a particular type of presentation for the Coxeter group W. A subgroup of W is called special if it is generated by a subset of S. Amalgamated product decompositions of a Coxeter group having special factors and special amalgamated subgroup are easily recognized from the presentation of the Coxeter group. If a Coxeter group is a subgroup of the fundamental group of a given graph of groups, then the Coxeter group is also the fundamental group of a graph of special subgroups, where each vertex and edge group is a subgroup of a conjugate of a vertex or edge group of the given graph of groups. A vertex group of an arbitrary graph of groups decomposition of a Coxeter group is shown to split into parts conjugate to special groups and parts that are subgroups of edge groups of the given decomposition. Several applications of the main theorem are produced, including the classification of maximal FA subgroups of a finitely generated Coxeter group as all conjugates of certain special subgroups.
Most actions on regular trees are almost free
Authors: Abért Miklós , Glasner Yair
Published: December 2009

Let

T

be a

d

-regular tree (

d ≥ 3

) and

A = \mathrm{Aut}(T)

its automorphism group. Let

Γ

be the group generated by

n

independent Haar-random elements of

A

. We show that almost surely, every nontrivial element of

Γ

has finitely many fixed points on

T

.
Enumerating limit groups
Authors: Groves Daniel , Wilton Henry
Published: December 2009

We prove that the set of limit groups is recursively enumerable, answering a question of Delzant. One ingredient of the proof is the observation that a finitely presented group with local retractions ((à la Long and Reid) is coherent and, furthermore, there exists an algorithm that computes presentations for finitely generated subgroups. The other main ingredient is the ability to algorithmically calculate centralizers in relatively hyperbolic groups. Applications include the existence of recognition algorithms for limit groups and free groups.
On rigidity and the isomorphism problem for tree braid groups
Authors: Sabalka Lucas
Published: December 2009

We solve the isomorphism problem for braid groups on trees with n = 4 or 5 strands. We do so in three main steps, each of which is interesting in its own right. First, we establish some tools and terminology for dealing with computations using the cohomology of tree braid groups, couching our discussion in the language of differential forms. Second, we show that, given a tree braid group BnT on n = 4 or 5 strands, H*(BnT) is an exterior face algebra. Finally, we prove that one may reconstruct the tree T from a tree braid group BnT for n = 4 or 5. Among other corollaries, this third step shows that, when n = 4 or 5, tree braid groups BnT and trees T (up to homeomorphism) are in bijective correspondence. That such a bijection exists is not true for higher dimensional spaces, and is an artifact of the 1-dimensionality of trees. We end by stating the results for right-angled Artin groups corresponding to the main theorems, some of which do not yet appear in the literature.
Random subgroups of Thompson’s group $F$
Authors: Cleary Sean , Elder Murray , Rechnitzer Andrew , Taback Jennifer
Published: December 2009

We consider random subgroups of Thompson’s group F with respect to two natural stratifications of the set of all k-generator subgroups. We find that the isomorphism classes of subgroups which occur with positive density are not the same for the two stratifications. We give the first known examples of
persistent
subgroups, whose isomorphism classes occur with positive density within the set of k-generator subgroups, for all sufficiently large k. Additionally, Thompson’s group provides the first example of a group without a generic isomorphism class of subgroup. Elements of F are represented uniquely by reduced pairs of finite rooted binary trees. We compute the asymptotic growth rate and a generating function for the number of reduced pairs of trees, which we show is D-finite (short for differentiably finite) and not algebraic. We then use the asymptotic growth to prove our density results.
On semisimple representations of universal lattices
Authors: Shenfeld Daniel K.
Published: December 2009

We study finite-dimensional semisimple complex representations of the universal lattices

Γ_{n,k} = \mathrm{SL}_n(ℤ[x_1, \ldots, x_k])

(

n ≥ 3

). One may obtain such a representation by specializing

x_1, \ldots, x_k

to some complex values and composing the induced homomorphism

Γ_{n,k} → \mathrm{SL}_n(ℂ)

with a rational representation of

\mathrm{SL}_n(ℂ)

. We show that any semisimple representation coincides, on a subgroup of finite index, with a direct sum of tensor products of representations obtained in this way.
Multiscale analysis of a prototypical model for the interaction between microstructure and surface energy
Authors: Braides Andrea , Zeppieri Caterina Ida
Published: December 2009

The combined effect of fine heterogeneities and small gradient perturbations is analyzed by means of an asymptotic development by

\Gamma

-convergence for a family of energies related to (one-dimensional) phase transformations. We show that multi-scale effects add up to the usual sharp-interface limit, due to the homogenization of microscopic interfaces, internal and external boundary layers, optimal arrangements of microscopic oscillations, etc. Several regimes are analyzed depending on the “size” of the heterogeneity (
small or large perturbations
of a homogeneous situation) and their relative period as compared with the characteristic length of the phase transitions (
slow or fast oscillations
).
Level set approach for fractional mean curvature flows
Authors: Imbert Cyril
Published: December 2009

This paper is concerned with the study of a geometric flow whose law involves a singular integral operator. This operator is used to define a non-local mean curvature of a set. Moreover the associated flow appears in two important applications: dislocation dynamics and phasefield theory for fractional reaction-diffusion equations. It is defined by using the level set method. The main results of this paper are: on one hand, the proper level set formulation of the geometric flow; on the other hand, stability and comparison results for the geometric equation associated with the flow.
Stability of monolayers and bilayers in a copolymer-homopolymer blend model
Authors: van Gennip Yves , Peletier Mark A.
Published: December 2009

We study the stability of layered structures in a variational model for diblock copolymerhomopolymer blends with respect to perturbations of their interfaces. The main step consists of calculating the ﬁrst and second derivatives of a sharp-interface Ohta–Kawasaki energy for straight mono- and bilayers and determining the sign of the latter. By developing the interface perturbations in a Fourier series we fully characterise the stability of the structures in terms of the energy parameters. Both for the monolayer and for the bilayer there exist parameter regions where these structures are unstable. For strong repulsive interaction between the monomer types in the diblock copolymer the bilayer is always stable with respect to interface perturbations, irrespective of the domain size. The monolayer is only stable for small domain size.

In the course of our computations we also give a Green’s function for the Laplacian on a twodimensional periodic strip.
A semilinear equation for the American option in a general jump market
Authors: Karlsen Kenneth Hvistendahl , Wallin Olli
Published: December 2009

We study the pricing of American put and call options in a market with jumps. We extend and make rigorous previous work that characterizes the price as a solution of an integro-differential equation set on the whole domain. The equation closely resembles the equation for the corresponding European options, but involves an additional reaction term that depends on the American option value in a nonlinear, nonlocal and discontinuous manner. Thus standard theory for partial differential equations does not apply, and we give a proper deﬁnition of a viscosity solution of the equation. We then show that the characterization is well posed. In particular, we prove a strong comparison principle for the equation using an approach that overcomes some problems related to the appearance of integrals with respect to unbounded measures. In short, we extend the results in [20] to a general class of exponential additive models. The formulation constitutes a starting point for designing and analyzing “easy to implement” numerical algorithms for computing the value of an American option.
Two-component viscous ﬂows with density-dependent transition layer
Authors: Witterstein Gabriele
Published: December 2009

We present a model describing ﬂows which undergo a phase change caused by chemical reactions in the material ﬂow. In mathematical terms, we consider a free boundary problem for the onedimensional, compressible Navier–Stokes system coupled with an Allen–Cahn equation. One free boundary arises at the boundary of the ﬂow to a vacuum state with a jump in the density. Another approximate free boundary arises due to the Allen–Cahn equation modeling the phase transition of the ﬂow. We consider a density-dependent viscosity

µ = ρ^θ

and a density-dependent transition layer of thickness

δ = \tilde δρ^{−1}

. We establish the existence of a unique weak global solution, provided that

θ < 1/3

.
Optimal transportation networks and stations
Authors: Valkonen Tuomo
Published: December 2009

We study a model of optimal transportation networks that includes the cost of constructing stations. The model is expressed in terms of the Federer–Fleming language of currents.
Quasi-periodic solutions of nonlinear random Schrödinger equations
Authors: Bourgain Jean , Wang Wei-Min
Published: December 2009

In this paper, let

\Sigma\subset\R^{6}

be a compact convex hypersurface. We prove that if

\Sigma

carries only finitely many geometrically distinct closed characteristics, then at least two of them must possess irrational mean indices. Moreover, if

\Sigma

carries exactly three geometrically distinct closed characteristics, then at least two of them must be elliptic.
Semiclassical states for weakly coupled nonlinear Schrödinger systems
Authors: Montefusco Eugenio , Pellacci Benedetta , Squassina Marco
Published: December 2009

We consider systems of weakly coupled Schrödinger equations with nonconstant potentials and we investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials. We obtain sufficient and necessary conditions for a sequence of least energy solutions to concentrate.
Degree estimate for subalgebras generated by two elements
Authors: Makar-Limanov Leonid , Yu Jie-Tai
Published: December 2009

We develop a new combinatorial method to deal with a degree estimate for subalgebras generated by two elements in different environments. We obtain a lower bound for the degree of the elements in two-generated subalgebras of a free associative algebra over a field of zero characteristic. We also reproduce a somewhat refined degree estimate of Shestakov and Umirbaev for the polynomial algebra, that plays an essential role in the recent celebrated solution of the Nagata conjecture and the strong Nagata conjecture.
Analytic continuation and fixed points of the Poincaré mapping for a polynomial Abel equation
Authors: Francoise J.-P. , Roytvarf N. , Yomdin Y.
Published: December 2009

We consider an Abel differential equation

y'=p(x)y2+q(x)y3

with

p(x)

,

q(x)

polynomials in

x

. For two given points

a

and

b

in

\mathbb C

, the “Poincaré mapping” of the above equation transforms the values of its solutions at

a

into their values at

b

. In this paper we study global analytic properties of the Poincaré mapping, in particular, its analytic continuation, its singularities and its fixed points (which correspond to the “periodic solutions” such that

y(a)=y(b)

). On the one hand, we give a general description of singularities of the Poincaré mapping, and of its analytic continuation. On the other hand, we study in detail the structure of the Poincaré mapping for a local model near a simple fixed singularity, where an explicit solution can be written. Yet, the global analytic structure (in particular, the ramification) of the solutions and of the Poincaré mapping in this case is fairly complicated, and, in our view, highly instructive. For a given degree of the coefficients we produce examples with an infinite number of
complex
“periodic solutions” and analyze their mutual position and branching. Let us recall that Pugh's problem, which is closely related to the classical Hilbert's 16th problem, asks for the existence of a bound to the number of
real
isolated “periodic solutions”. New findings reported here lead us to propose new insights on the Poincaré mapping. If the “complexity” of the path in the

x

-plane between

a

and

b

is
a priori
bounded, the number of fixed points should be uniformly bounded. We think that, in some sense, this is close to the complex version of Khovansky's fewnomial theory.
On the topology of positively curved Bazaikin spaces
Authors: Florit Luis A. , Ziller Wolfgang
Published: December 2009

In this note we study the topology of the positively curved Bazaikin spaces. We show that the only Bazaikin space that is homotopically equivalent to a homogeneous space is the Berger space. Moreover, we compute their Pontryagin classes and linking form to conclude that there is no pair of positively curved Bazaikin spaces which are homeomorphic, at least if the order of the sixth cohomology group with integer coefficients is less than 108.
Three-space problems for the approximation property
Authors: Szankowski A.
Published: December 2009

It is shown that there is a subspace

Z_q

of

\ell_q

for

1<q<2

which is isomorphic to

\ell_q

such that

\ell_q/Z_q

does not have the approximation property. On the other hand, for

2<p<\infty

there is a subspace

Y_p

of

\ell_p

such that

Y_p

does not have the approximation property (AP) but the quotient space

\ell_p/Y_p

is isomorphic to

\ell_p

. The result is obtained by defining random "Enflo-Davie spaces"

Y_p

which with full probability fail AP for all

2 < p\leq\infty

and have AP for all

1 \leq p \leq 2

. For

1 < p \leq

2,

Y_p

are isomorphic to

\ell_p

.
The $K(\pi,1)$ problem for the afﬁne Artin group of type $\widetilde{B}_n$ and its cohomology
Authors: Callegaro Filippo , Moroni Davide , Salvetti Mario
Published: December 2009

We prove that the complement to the afﬁne complex arrangement of type

\widetilde B_n

is a

K(π,1)

space. We also compute the cohomology of the afﬁne Artin group

G_{\widetilde B_n}

(of type

\widetilde B_n

) with coefﬁcients in interesting local systems. In particular, we consider the module

ℚ[q^{±1} ,t^{±1}]

, where the ﬁrst

n

standard generators of

G_{\widetilde B_n}

act by

(−q)

-multiplication while the last generator acts by

(−t)

multiplication. Such a representation generalizes the analogous 1-parameter representation related to the bundle structure over the complement to the discriminant hypersurface, endowed with the monodromy action of the associated Milnor ﬁbre. The cohomology of

G_{\widetilde B_n}

with trivial coefﬁcients is derived from the previous one.
On real Kähler Euclidean submanifolds with non-negative Ricci curvature
Authors: Florit Luis A. , Hui Wing San , Zheng Fangyang
Published: December 2009

We show that any real Kähler Euclidean submanifold

f:M^{2n} \to \mathbb R^{2n+p}

with either non-negative Ricci curvature or non-negative holomorphic sectional curvature has index of relative nullity greater than or equal to

2n-2p

. Moreover, if equality holds everywhere, then the submanifold must be a product of Euclidean hypersurfaces almost everywhere, and the splitting is global provided that

M^{2n}

is complete. In particular, we conclude that the only real Kähler submanifolds

M^{2n}

in

\R^{3n}

that have either positive Ricci curvature or positive holomorphic sectional curvature are precisely products of

n

orientable surfaces in

\R^3

with positive Gaussian curvature. Further applications of our main result are also given.
Vortex rings for the Gross-Pitaevskii equation
Authors: Bethuel Fabrice , Orlandi Giandomenico , Smets Didier
Published: December 2009

We provide a mathematical proof of the existence of traveling vortex rings solutions to the Gross-Pitaevskii (GP) equation in dimension

N\geq 3.

We also extend the asymptotic analysis of the free field Ginzburg-Landau equation to a larger class of equations, including the Ginzburg-Landau equation for superconductivity as well as the traveling wave equation for GP. In particular we rigorously derive a curvature equation for the concentration set (i.e. line vortices if

N=3

).Ginzburg-Landau equation to a larger class of equations, including the Ginzburg-Landau equation for superconductivity as well as the traveling wave equation for GP. In particular we rigorously derive a curvature equation for the concentration set (i.e. line vortices if

N=3

).
A symmetry problem in the calculus of variations
Authors: Crasta Graziano
Published: December 2009

We consider a class of integral functionals defined in a Sobolev space of functions vanishing at the boundary of a nonempty bounded connected open n-dimensional set. We prove that, if the functional admits a minimizer depending only on the distance from the boundary, then that set must be a ball.
On the necessity of gaps
Authors: Matano Hiroshi , Rabinowitz Paul H.
Published: December 2009

Recent papers have studied the existence of phase transition solutions for model equations of Allen-Cahn type equations. These solutions are either single or multi-transition spatially heteroclinics or homoclinics between simpler equilibrium states. A sufficient condition of the construction of the multi-transition solutions is that there are gaps in the ordered set of single transition solutions. In this paper we explore the necessity of these gap conditions.
The inverse mean curvature flow and $p$-harmonic functions
Authors: Moser Roger
Published: December 2009

We consider the level set formulation of the inverse mean curvature flow. We establish a connection to the problem of

p

-harmonic functions and give a new proof for the existence of weak solutions.
Conformal structures in noncommutative geometry
Authors: Bär Christian
Published: December 2009

It is well known that a compact Riemannian spin manifold

(M, g)

can be reconstructed from its canonical spectral triple

(C^∞(M), L^2(M,ΣM), D)

where

ΣM

denotes the spinor bundle and

D

the Dirac operator. We show that

g

can be reconstructed up to conformal equivalence from

(C^∞(M), L^2(M,ΣM), \mathrm{sign}(D))

.
Spectral action on noncommutative torus
Authors: Essouabri Driss , Iochum Bruno , Levy Cyril , Sitarz Andrzej
Published: December 2009

The spectral action on the noncommutative torus is obtained using a Chamseddine–Connes formula via computations of zeta functions. The importance of a Diophantine condition is outlined. Several results on holomorphic continuation of series of holomorphic functions are obtained in this context.
Integration over complex manifolds via Hochschild homology
Authors: Ramadoss Ajay C.
Published: December 2009

Given a holomorphic vector bundle

\mathcal E

on a connected compact complex manifold

X

, in [FLS] a

ℂ

-linear functional

I_{\mathcal E}

on

H^{2n}(X, ℂ)

is constructed. This is done by producing a linear functional on the 0-th completed Hochschild homology

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

of the sheaf of holomorphic differential operators on

\mathcal E

using topological quantum mechanics. It is shown in [FLS] that this functional is

\int_X

if

\mathcal E

has non-zero Euler characteristic, and the conjecture is that it is

\int_X

for all

\mathcal E

.

In a subsequent work [Ram] the author proved that the linear functional

I_{\mathcal E}

is independent of the vector bundle

\mathcal E

. This article builds upon the work in [Ram] to prove that

I_{\mathcal E} = \int_X

for an arbitrary holomorphic vector bundle

\mathcal E

on an arbitrary connected compact complex manifold

X

. This is done using an argument that is very natural from the geometric point of view. Moreover, this argument enables one to make the approach to this conjecture developed first in [FLS] and subsequently in [Ram] independent of the Riemann–Roch–Hirzebruch theorem. This argument allows us to extend the construction in [FLS] to a construction of a linear functional

I_{\mathcal E}

on

H^{2n}_c(Y, ℂ)

for a holomorphic vector bundle

\mathcal E

with bounded geometry on an arbitrary connected complex manifold

Y

with bounded geometry, and to prove that

I_{\mathcal E} = \int_Y

. We also generalize a result of [Ram] pertaining to “cyclic homology analogs” of

I_{\mathcal E}

.
Local analytic classification of $q$-difference equations with $\lvert q \rvert= 1$
Authors: Di Vizio Lucia
Published: December 2009

In this article, we establish, under convenient diophantine assumptions, a complete analytic classification of

q

-difference modules over the field of germs of meromorphic functions at zero, proving some analytic analogs of the results by Soibelman–Vologodsky and by Baranovsky–Ginzburg.
A universal deformation formula for $\mathcal{H}_1$ without projectivity assumption
Authors: Tang Xiang , Yao Yi-Jun
Published: December 2009

We find a universal deformation formula for Connes–Moscovici's Hopf algebra

\mathcal{H}_1

without any projectivity assumption using Fedosov’s quantization of symplectic diffeomorphisms.
A noncommutative Bohnenblust–Spitzer identity for Rota–Baxter algebras solves Bogoliubov’s recursion
Authors: Ebrahimi-Fard Kurusch , Manchon Dominique , Patras Frédéric
Published: December 2009

The Bogoliubov recursion is a particular procedure appearing in the process of renormalization in perturbative quantum field theory. It provides convergent expressions for otherwise divergent integrals. We develop here a theory of functional identities for noncommutative Rota–Baxter algebras which is shown to encode, among others, this process in the context of Connes–Kreimer's Hopf algebra of renormalization. Our results generalize the seminal Cartier–Rota theory of classical Spitzer-type identities for commutative Rota–Baxter algebras. In the classical, commutative, case these identities can be understood as deriving from the theory of symmetric functions. Here we show that an analogous property holds for noncommutative Rota–Baxter algebras. That is, we show that functional identities in the noncommutative setting can be derived from the theory of noncommutative symmetric functions. Lie idempotents, and particularly the Dynkin idempotent, play a crucial role in the process. Their action on the pro-unipotent groups such as those of perturbative renormalization is described in detail along the way.
Noncommutative tori and the Riemann–Hilbert correspondence
Authors: Mahanta Snigdhayan , van Suijlekom Walter D.
Published: December 2009

We study the interplay between noncommutative tori and noncommutative elliptic curves through a category of equivariant differential modules on

ℂ^*

. We functorially relate this category to the category of holomorphic vector bundles on noncommutative tori as introduced by Polishchuk and Schwarz and study the induced map between the corresponding K-theories. In addition, there is a forgetful functor to the category of noncommutative elliptic curves of Soibelman and Vologodsky, as well as the forgetful functor to the category of vector bundles on

ℂ^*

with regular singular connections.

The category that we consider has the nice property of being a Tannakian category, hence it is equivalent to the category of representations of an affine group scheme. Via an equivariant version of the Riemann–Hilbert correspondence we determine this group scheme to be (the algebraic hull of)

ℤ^2

. We also obtain a full subcategory of the holomorphic vector bundles on the noncommutative torus which is equivalent to the category of representations of

ℤ

. This group is the proposed topological fundamental group of the noncommutative torus (understood as a degenerate elliptic curve) and we study Nori’s notion of étale fundamental group in this context.
Complete positivity of Rieffel’s deformation quantization by actions of $\mathbb{R}^d$
Authors: Kaschek Daniel , Neumaier Nikolai , Waldmann Stefan
Published: December 2009

In this paper we consider

\mathrm{C}^*

-algebraic deformations by actions of

\mathbb{R}^d

à la Rieffel and show that every state of the undeformed algebra can be deformed into a state of the deformed algebra in the sense of a continuous field of states. The construction is explicit and involves a convolution operator with a particular Gauß function.
Wave Motion
Authors: Constantin Adrian , Escher Joachim
Published: December 2009

The workshop
Wave Motion
that took place in the period January 25–31, 2004 was devoted to the study of nonlinear wave phenomena. The modelling of waves leads to a variety of difficult mathematical issues, involving several domains of mathematics: partial differential equations, harmonic analysis, dynamical systems, topological degree theory.

The progam of the workshop consisted in 18 talks, presented by international specialists in nonlinear waves coming from England, France, Germany, Japan, Norway, Sweden, Switzerland, U.S.A., and by three discussion sessions on the topics “Open Problems in PDEs”, “Stability Phenomena in the Theory of Nonlinear Waves”, and “Geodesic Flows and Fluid Mechanics”. Moreover, several doctoral and post-doctoral fellows participated in the workshop and did benefit from the unique academic atmosphere at the Oberwolfach Institute.

The proceedings of the workshop “Wave Motion” will appear as a special issue of the
Journal of Nonlinear Mathematical Physics
.
Mini-Workshop: Classification of Surfaces of General Type with Small Invariants
Authors: Catanese Fabrizio , Ciliberto Ciro
Published: December 2009

This mini-workshop has been organized by Fabrizio Catanese and Ciro Ciliberto. Unfortunately Catanese was unable to participate.

The classification of algebraic surfaces is a long-standing research subject in algebraic geometry, started by Castelnuovo and Enriques more than one hundred years ago, and continued by the Italian school (Severi, de Franchis, etc.) until about 1950.

In more recent times, fundamental contributions have been given by Kodaira in the 1950's and later in the 1970's by Bombieri, whose works on pluricanonical maps gave a strong impulse in studying surfaces of general type, and Mumford.

Adding important information to classical results by Noether and Castelnuovo, sharp bounds on the invariants have been given by Miyaoka and Bogomolov–Yau, allowing many authors to develop a systematic study of the “geography” of surfaces of general type.

Interesting investigations about the moduli space of surfaces of general type have been worked out in the last twenty years by Catanese, Manetti, and others.

Despite the intensive effort made in the last decades in order to make more precise our knowledge about surfaces of general type, their fine classification is still an open problem, even for small invariants. It is actually rather embarassing that, after more than one century of research on the subject, a complete classification of surfaces with geometric genus zero or one is still lacking.

This mini-workshop carried together 14 mathematicians actively working on this subject, and related arguments, with the idea of updating the state-of-the-art, exchanging information, discussing interesting open problems and stimulating collaborations. In this respect, the workshop has been very successful.

The atmosphere has been lively and very collaborative. During every talk, several questions have been posed and interesting problems pointed out. It has been especially remarkable the active presence of young participants.

During the week, 16 formal lectures have been given by the participants. This report contains extended abstracts of all the talks and also a contribution by Catanese, in collaboration with Pignatelli, about the lecture he was supposed to give.

The topics include: pluri-canonical maps for surfaces of general type (M. Mendes Lopes), canonical rings, projective embeddings and birational techniques (C. Böhning, F. Catanese, S. Papadakis, U. Persson, R. Pignatelli), irregular surfaces with low invariants (F. Polizzi, F. Zucconi), surfaces with

p_g=0

(A. Calabri, C. Ciliberto, K. Keum, M. Mendes Lopes, C. Werner), general techniques (V. Brînzănescu, K. Konno). Ulf Persson chaired an “open problem and discussions” session, which especially concerned surfaces with

p_g=0

.

The organizers thank the Institute staff for providing a comfortable environment to the participants.
Analysis and Design of Electoral Systems
Authors: Balinski Michel L. , Brams Steven J. , Pukelsheim Friedrich
Published: December 2009

Electoral systems transform numbers into numbers. In order not to be blinded by the intricacies of any particular case, as interesting as particular cases may be, it is imperative to view theses transformations as functions, or as relations, and to study their properties and structure in their full generality. Promoting the mathematical foundations of the topic cannot be a goal in and for itself, but must reflect the practical needs defined by the problems themselves that are traditionally treated by such non-mathematical fields as political science, economics, social choice theory, constitutional law, etc. This diversity was well reflected in the expert fields represented by the conference participants, as well as in the wide range of topics covered. They may broadly be summarized under three headings: (1) voting schemes that aggregate many individual preference rankings of a given set of alternatives into a single global ranking; (2) proportional representation schemes that map vote counts (or population counts) into parliamentary representation; and (3) the determination of electoral districts that reasonably reflect geographical, political, and social structures.While mathematics at large provides the appropriate language to investigate these problems, the tools that are used draw on specific mathematical fields, including:

The mathematical analysis of electoral systems has direct repercussions on current political issues. To mention but three, there is the question whether elections of the US president would be less likely to be disputed, and their outcomes considered more legitimate, were he to be elected directly. Another is the analysis of qualified majority rules in the EU Council of Ministers as stipulated in the draft constitution proposed by the European Convention 2003. A third relates to developing biproportional apportionment methods that simultaneously accommodate partisan and regional representation of the entire electorate.

Somewhat unusual for an Oberwolfach conference was the rather broad mix of participants, representing the fields of mathematics, physics, economics, political science, and psychology. It was this breadth that virtually all participants experienced as challenging and fruitful. As always, the atmosphere was congenial for scientific discussions, and the Institute in its Schwarzwald setting made people feel creative by just being there... particularly so, since the sky was blue and the sun was shining (most of the time).

M.L. Balinski

S.J. Brams

F. Pukelsheim
Geomathematik
Authors: Freeden Willi , Grafarend Erik W. , Sloan Ian H. , Svensson Leif
Published: December 2009

The present workshop gave an overview of recent results and current trends in geomathematics. The organizers and participants would like to take the opportunity to thank again the Mathematisches Forschungsinstitut Oberwolfach for having provided an inspiring environment for the meeting and the scientific work. The pleasant atmosphere contributed to the overall success of the workshop.
Probability Theory on Trees and Analysis of Algorithms
Authors: Alsmeyer Gerold , Devroye Luc , Rösler Uwe
Published: December 2009

The analysis of algorithms and data structures, started by D. Knuth, is a rapidly growing area at the interface of Mathematics and Theoretical Computer Science. Probability theory enters the subject in a natural way when studying an algorithm as to its performance over randomized inputs and/or if an algorithm itself takes randomization steps. The latter holds true for so-called divide and conquer algorithms, which were a central topic of this workshop. The sorting algorithm is presumably the most prominent example. While early work was mostly based on the generating function approach, the last two decades have seen an increasing number of contributions based on probabilistic methods involving martingales, random trees and other stochastic processes. The first article of this type was written in the eighties by L. Devroye, a Humboldt award winner of 2004 and also one of the organizers. Later another pioneering contribution came by U. Rösler who, by introducing weighted branching processes and the contraction method, determined the asymptotic distributional behavior of . One can say that much of the work presented at this workshop is to some extent spawn by these articles.

The aim and scope of the mini-workshop was to bring together leading junior and senior experts in the field with a strong probabilistic background and a focus on divide an conquer algorithms and related data structures. There were 16 one hour talks presented by 12 of the 14 participants from 7 countries. Main topics were branching processes, the contraction method, the asymptotic analysis of random trees, stochastic fixed point equations and randomized algorithms.
Komplexe Analysis
Authors: Demailly Jean-Pierre , Hulek Klaus , Peternell Thomas
Published: December 2009

The Komplexe Analysis workshop, organised by J.P. Demailly (Grenoble), K. Hulek (Hannover) and Th. Peternell (Bayreuth), was held August 22-27. The meeting was attended by over 40 participants from many European countries, USA, Japan and Korea, among them quite a number of young mathematicians. The spectrum of the meeting was very broad, ranging from analytic question, e.g. Levi flat hypersurfaces and non-Kaehler geometry to classification theory, classical algebraic geometry and numerical aspects of algebraic geometry.
Theory of the Riemann Zeta and Allied Functions
Authors: Huxley Martin N. , Jutila Matti , Motohashi Yoichi , Patterson Samuel J.
Published: December 2009

This meeting, the second Oberwolfach workshop devoted to zeta functions, was attended by 42 participants representing 16 countries. The scientific program consisted of 32 talks of various lengths and a problem session. In addition, social activities were organised: a hike in the mountains and piano recitals by Peter Elliott and Valentin Blomer.

Since the times of Dirichlet and Riemann, zeta functions and Dirichlet series have played a central role in analytic number theory, and in recent times connections have been found with other areas of mathematics and its applications, including theoretical physics. The talks represented the various aspects of the theory of zeta functions. In particular, the following topics were discussed, among others:
The History of Differential Equations, 1670–1950
Authors: Archibald Tom , Fraser Craig , Grattan-Guinness Ivor
Published: December 2009

Differential equations have been a major branch of pure and applied mathematics since their inauguration in the mid 17th century. While their history has been well studied, it remains a vital field of on-going investigation, with the emergence of new connections with other parts of mathematics, fertile interplay with applied subjects, interesting reformulation of basic problems and theory in various periods, new vistas in the 20th century, and so on. In this meeting we considered some of the principal parts of this story, from the launch with Newton and Leibniz up to around 1950.

‘Differential equations’ began with Leibniz, the Bernoulli brothers and others from the 1680s, not long after Newton's ‘fluxional equations’ in the 1670s. Applications were made largely to geometry and mechanics; isoperimetrical problems were exercises in optimisation.

Most 18th-century developments consolidated the Leibnizian tradition, extending its multi-variate form, thus leading to partial differential equations. Generalisation of isoperimetrical problems led to the calculus of variations. New figures appeared, especially Euler, Daniel Bernoulli, Lagrange and Laplace. Development of the general theory of solutions included singular ones, functional solutions and those by infinite series. Many applications were made to mechanics, especially to astronomy and continuous media.

In the 19th century: general theory was enriched by development of the understanding of general and particular solutions, and of existence theorems. More types of equation and their solutions appeared; for example, Fourier analysis and special functions. Among new figures, Cauchy stands out. Applications were now made not only to classical mechanics but also to heat theory, optics, electricity and magnetism, especially with the impact of Maxwell. Later Poincaré introduced recurrence theorems, initially in connection with the three-body problem.

In the 20th century: general theory was influenced by the arrival of set theory in mathematical analysis; with consequences for theorisation, including further topological aspects. New applications were made to quantum mathematics, dynamical systems and relativity theory.
Spectral Analysis of Partial Differential Equations
Authors: Sobolev Alexander V. , Weidl Timo
Published: December 2009

The goal of the workshop was to bring together specialists working in various branches of spectral theory with applications to solid state physics, superconductivity, quantum mechanics etc. The meeting was attended by more than 45 participants from Europe, Japan, Russia, South America and US. During the five days 26 talks were delivered. A special care was taken that apart from the recognized experts in the field, young participants also had an opportunity to speak about their results. The Wednesday morning session, preceding the traditional afternoon hike, consisted of talks of survey nature, which was appreciated by all. There were several major themes in the workshop. One was the study of discrete spectra, including Lieb-Thirring estimates, properties of resonances. A substantial number of talks was concerned with the connection between the continuous and discrete spectra. These include, in particular, the study of the so-called trace formulas. The investigation of various characteristics of the continuous spectra (e.g. density of states, spectral shift function) was featured in a number of talks in the context of random or magnetic operators. A variety of new results were also reported on the theory of periodic operators. They concerned the Liouville Theorem, the problem of absolute continuity, and the classical problem of homogenization. A relatively low number of talks gave the participants an opportunity for discussions in small groups outside the scheduled lecture time. It is hoped that these contacts will result in further collaboration. It is our pleasure to thank the administration and staff of the \textsl{Mathematisches Forschungsinstitut Oberwolfach} for creating comfortable and genuinely inspiring atmosphere, which facilitated the work of the organisers and contributed to the success of the workshop.
Topological and Variational Methods for Differential Equations
Authors: Bartsch Thomas , Dancer Andrew
Published: December 2009

Topological and variational methods have been at the core of nonlinear analysis for a long time and are still experiencing major new developments. They have had enormous new applications in the study of boundary value problems for nonlinear differential equations, in analyzing complicated (possibly infinite-dimensional) dynamics, phase transition and pattern formation, to name a few.

The workshop was mainly dedicated to variational methods for nonlinear elliptic and parabolic differential equations and systems with a special emphasis on

The workshop was attended by 47 mathematicians from 17 countries (from the Americas, Asia, Australia, Europe). During the five days 27 talks were delivered, both from leading researchers as well as young mathematicians. There was plenty of time for discussions and cooperative work in small groups outside the scheduled lecture time. This led to a fruitful and intensive scientific exchange between the participants. Many open problems were discussed and many collaborative projects were started or continued during this week. Many of the abstracts below list some open problems, thus guiding future research. We received a great deal of positive feedback from the participants and are sure that this workshop will lead to further collaboration between the participants.

It is our pleasure to thank the administration and staff of the MFO for their efficient work and their hospitality which was essential for the stimulating atmosphere during the workshop.
Dynamical Systems
Authors: Hofer Helmut W. , Yoccoz Jean-Christophe , Zehnder Eduard
Published: December 2009

This workshop, organised by Helmut Hofer (New York), Jean-Christophe Yoccoz (Paris), and Eduard Zehnder (Zürich), continued the biannual series at Oberwolfach on Dynamical Systems that started as the “Moser–Zehnder meeting” in 1981. The workshop was attended by more than 50 participants from 12 countries. The main theme of the workshop were the new results and developments in the area of classical dynamical systems, in particular in celestial mechanics and Hamiltonian systems. Among the main topics were new global results on the Reeb dynamics on

3

-manifolds, KAM theory in finite and infinite dimensions, as well as new developments in Floer homology and its applications. High points were the first complete existence proof of quasiperiodic solutions in the planetarian

N

-body problem, and the solution of a long-standing conjecture of Anosov about the number of closed geodesics on Finsler

2

-spheres. The meeting was held in a very informal and stimulating atmosphere.
Dynamics of Cocycles and One-Dimensional Spectral Theory
Authors: Damanik David , Johnson Russell , Lenz H. Daniel
Published: December 2009

The mini-workshop
Dynamics of Cocycles and One-Dimensional Spectral Theory
, organised by David Damanik (Pasadena), Russell Johnson (Firenze) and Daniel Lenz (Chemnitz), was held November 13–19, 2005.

There have been a number of recent breakthroughs in the spectral theory of one-dimensional Schrödinger operators with quasi-periodic potentials that were accomplished using sophisticated dynamical systems methods; especially by establishing reducibility properties of certain quasi-periodic

\mathrm{SL}(2,\R)

-valued cocycles.

The most popular example of a one-dimensional Schrödinger operators with quasi-periodic potential is given by the almost Mathieu operator,

[H u]_n = u_{n+1} + u_{n-1} + 2\lambda \cos (2\pi (n\alpha + \theta)) u_n,

where

\lambda \not= 0

and

\alpha

is irrational. Using the connection with dynamics, it was recently shown for all parameter values that the spectrum of

H

is a Cantor set of Lebesgue measure

|4 - 4 |\lambda||

.

It was the objective of the mini-workshop to bring together experts from both spectral theory and dynamical systems to learn from each other and to further explore potential applications of dynamical systems methods in the context of quasi-periodic Schrödinger operators. Special attention was paid to having many young participants. This was made easy by the fact that currently there are many excellent graduate students and postdocs working on problems located at the interface between the two areas. Consequently, about two thirds of the participants belonged to the age group 35 years or younger.

The talks presented by the participants reflected the current developments in this area. Among other things, there is now an improved understanding of analytic potentials with non-perturbatively small coupling, there are extensions of some results known for analytic potentials to certain classes of non-analytic potentials, while phenomena different from those in the analytic case may occur for potentials of low regularity, and there is improved understanding of the case of Liouville frequencies.

Among the highlights of the discussions outside the talks one could mention the “joining” of independent and closely related work of Bjerklöv and Jäger and the solution by Avila and Damanik of one of the few questions about the almost Mathieu operator that were still left open after the recent advances triggering the mini-workshop.
Heat Kernels, Stochastic Processes and Functional Inequalities
Authors: Coulhon Thierry , Franchi Bruno , Kumagai Takashi , Sturm Karl-Theodor
Published: December 2009

The workshop \emph{Heat kernels, stochastic processes and functional inequalities} was organized by Thierry Coulhon (Cergy), Bruno Franchi (Bologna), Takashi Kumagai (Kyoto) and Karl-Theodor Sturm (Bonn). It was held from November 27th to December 3nd. The meeting was attended by 56 participants from Australia, Austria, Canada, Finland, France, Germany, Italy, Japan, Poland, Switzerland, United Kingdom, and USA. This workshop was sponsored by the European Union, which allowed the invitation of 18 young people, who contributed positively to the atmosphere of the meeting.

The conference brought together mathematicians belonging to several fields, essentially analysis, probability and geometry. One of the main unifying topics was certainly the study of heat kernels in various contexts: fractals, manifolds, domains of the Euclidean space, percolation clusters, infinite dimensional spaces, metric measure spaces. Some related aspects of geometric analysis were also considered such as

L^p

-cohomology and mass transportation. There was a stimulating exchange between probabilistic and analytic points of view, together with a geometric emphasis in most of the problems. We had 5 one hour survey lectures and 21 thirty-five minutes talks. A lot of time was devoted to discussions and exchange of ideas.

Among the highlights were relations between mass transportation, generalized Ricci bounds and contraction properties, connections between heat kernel estimates and percolation clusters, non-linear aspects of diffusions, functional analytic approach to parabolic regularity, geometric and functional analytic aspects of infinite dimensional analysis.

This diversity of topics and mix of participants stimulated many extensive and fruitful discussions. It also helped initiate new collaborations, in particular for the younger researchers.