an unbounded sum of squares hierarchy integrality gap for a polynomially solvable problem in this paper we study the complexity of the min sum single machine scheduling problem under algorithms from the sum of squares lasserre hierarchy we prove the first lower bound for this model by showing that the integrality gap is unbounded at level omega sqrt n even for a variant of the problem that is solvable in o n log n time by the moore hodgson algorithm namely min number of late jobs we consider a natural formulation that incorporates the objective function as a constraint and prove the result by partially diagonalizing the matrix associated with the relaxation and exploiting this characterization to the best of our knowledge our result provides the first example where the sum of squares hierarchy exhibits an unbounded integrality gap for a polynomially solvable problem after non constant number of levels

the integral monodromy of hyperelliptic and trielliptic curves we compute the mathbb z ell and mathbb z ell monodromy of every irreducible component of the moduli spaces of hyperelliptic and trielliptic curves in particular we provide a proof that the mathbb z ell monodromy of the moduli space of hyperelliptic curves of genus g is the symplectic group rm s p2g mathbb z ell we prove that the mathbb z ell monodromy of the moduli space of trielliptic curves with signature r s is the special unitary group rm su r s mathbb z ell otimes mathbb z zeta 3

parametric galois extensions abstract given a field k and a finite group h an h parametric extension over k is a finite galois extension of k t of galois group containing h which is regular over k and has all the galois extensions of k of group h among its specializations we are mainly interested in producing non h parametric extensions which relates to classical questions in inverse galois theory like the beckmann black problem and the existence of one parameter generic polynomials we develop a general approach started in a preceding paper and provide new non parametricity criteria and new examples

formulation of elastostatic hybrid trefftz stress elements abstract the hybrid trefftz stress hts element is derived from the fundamental conditions of linear elastostatics the stress and boundary displacement fields are directly approximated consistent finite element variables are so defined as to obtain boundary integral expressions for the finite element equilibrium and compatibility conditions encoded by dual transformations independent of the constitutive relations reciprocity is preserved in the finite element elasticity relations which are also found to be described by boundary integral arrays when the trefftz constraint is enforced consequent upon duality and reciprocity the resulting boundary integral solving system is symmetric and sparse the energy statements associated with the hybrid trefftz stress element are established and the conditions for the existence uniqueness multiplicity and static and kinematic admissibility of the finite element solutions are presented the hts clement is compared with the displacement frame based hybrid trefftz ht d element of jirousek it is shown that the ht d element is in fact an hybrid stress element the numerical tests performed to assess the relative performances of the hts and ht d elements show that better accuracy levels are obtained using discontinuous boundary approximation bases

limits of galois representations in fundamental groups along maximal degeneration of marked curves i we study gal inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i actions on the fundamental groups arising in deformation of maximally degenerate marked curves and give explicit descriptions of them in terms of the principal parameters of the grothendieck teichmuller group inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i as a special application we present galois theoretic connections between the moduli spaces m 0 4 m 0 5 m 1 1 and m 1 2 which lead to a newtype equation satisfied by the image of gal inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i

the geometry of scheduling we consider the following general scheduling problem the input consists of n jobs each with an arbitrary release time size and monotone function specifying the cost incurred when the job is completed at a particular time the objective is to find a preemptive schedule of minimum aggregate cost this problem formulation is general enough to include many natural scheduling objectives such as total weighted flow time total weighted tardiness and sum of flow time squared we give an o log log p approximation for this problem where p is the ratio of the maximum to minimum job size we also give an o 1 approximation in the special case of identical release times these results are obtained by reducing the scheduling problem to a geometric capacitated set cover problem in two dimensions

large normal extension of hilbertian fields the goal of this note is to consider a certain natural family of closed normal subgroups of g q and to prove that each group in this family is free more generally consider a countable separably hilbertian field k denote the absolute galois group of k by g k then for almost all g k e the field ks is pac and e free fj2 thms 16 13 and 16 18 here ks is the separable closure of k and ks is the fixed field of in ks being pac means that every nonvoid absolutely irreducible variety defined over ks has a ks rational point we say that ks is e free if g ks i e the closed subgroup 1 e of g k generated by 1 e is free on e generators denote the largest galois extension of k which is contained in ks by ks it is the intersection of all k conjugates of ks and also the fixed field of the smallest closed normal subgroup of g k which contains 1 e if char k 0 then for almost all g k e the field ks is pac fj2 thm 16 47 lemma 1 2 below generalizes this result to arbitrary characteristic if we knew that ks is separably hilbertian then a theorem of fried volklein and pop would imply that ks is free that is g ks is isomorphic to the free profinite group fw on countably many generators unfortunately it is not clear how to prove the hilbertianity of almost all ks directly so we use instead a forerunner to the above mentioned theorem of fried volklein pop and a recent theorem of neumann neu to prove directly that g ks f for almost all g k e a theorem of roquette then implies that ks is also separably hilbertian let k be the algebraic closure of k denote the maximal purely inseparable extension of ks by k then for almost all g k e the field k

rational rigidity and the sporadic groups the question of which finite groups can or do occur as the galois group of a galois extension of the rational numbers is one which has been studied since early in the twentieth century it is conjectured that all finite groups do occur as galois groups over the rationals this is known to be true for all solvable groups see jacobson 73 for brief historical notes in a breakthrough paper thompson 9 has defined the new concept of rigidity of a finite group he then proves that if a group g with trivial centre is rigid then an associated group g is a galois group over the rationals in this paper it is shown that 14 of the 26 sporadic simple groups are galois groups over the rationals the method used involves the complex character table and knowledge of maximal subgroups of the group in some cases where the maximal subgroups are not known the classification of finite simple groups is used section 1 contains the results section 2 describes the notation used and section 3 details the proof for each of the 14 groups separately

the irreducibility of certain pure cycle hurwitz spaces we study pure cycle hurwitz spaces parametrizing covers of the projective line having only one ramified point over each branch point we start with the case of genus 0 covers using a combination of limit linear series theory and group theory to show that these spaces are always irreducible in the case of four branch points we also compute the associated hurwitz numbers finally we give a conditional result in the higher genus case requiring at least 3g simply branched points these results have equivalent formulations in group theory and in this setting complement results of conway fried parker v olklein

on fields of moduli of curves the field of moduli k of a curve x a priori defined over the separable closure k5 of k need not be a field of definition this paper shows that the obstruction is essentially the same as the obstruction to k being a field of definition of the cover x x aut x using previous results of debes douai we then obtain a cohomological measure of the obstruction this yields concrete criteria for the field of moduli to be a field of definition an interesting application is the following local global principle if a curve x together with all of its automorphisms is defined over p for all primes p then it is defined over

wilson s map operations on regular dessins and cyclotomic fields of definition dessins d enfants can be seen as bipartite graphs embedded in compact orientable surfaces 5 according to grothendieck and others a dessin uniquely determines a complex structure on the surface even an algebraic structure as a projective algebraic curve defined over a number field combinatorial properties of the dessin should therefore determine the equations and also structural properties of the curve such as the field of moduli or the field of definition however apart from a few series of examples very few general results concerning such correspondences 10 are known as a step in this direction we present a graph theoretic characterisation of certain quasiplatonic curves defined over cyclotomic fields based on wilson s operations on maps these leave invariant the graph but change the cyclic ordering of edges around the vertices therefore they change the embeddings and hence the dessins and hence the conformal and algebraic structure of the underlying curves under suitable assumptions satisfied by many series of regular 15 dessins these changes coincide with the effect of galois conjugation this coincidence allows one to draw conclusions about galois orbits and fields of definition of dessins the possibilities afforded by these techniques and their limitations are illustrated by a new look at some known examples and a study of dessins based on the regular embeddings of complete graphs 1 the mathematical framework 20 for the last twenty years embeddings of graphs into surfaces have become important for the theory of algebraic curves as an essential tool in the theory of dessins d enfants in this first section we briefly recall the necessary definitions and give an outline of the problems and results presented in this paper one may introduce dessins d enfants as hypermaps on compact oriented 2 manifolds these 25 objects first studied by cori 5 have several equivalent algebraic or topological definitions topologically they are a generalisation of maps whose underlying graphs are replaced with hypergraphs so that the hyperedges are now allowed to be incident with any finite number of hypervertices and hyperfaces one way to visualise this concept is to represent a hypermap d by its walsh map w d a connected bipartite graph b embedded in a compact oriented 30 surface dividing it into simply connected cells 26 in the language of hypermaps the white and black vertices represent the hypervertices and hyperedges of d the edges represent incidences between them and the cells represent the hyperfaces algebraic hypermaps are an alternative and equivalent way to describe dessins these are triples g x y consisting of a permutation group g sn the monodromy or hypercarto 35 graphic group of the dessin acting transitively on the n edges of the walsh map w d a nd generated by two elements x and y the generators x and y describe the anticlockwise cyclic permutations of the edges around their incident white and black vertices we call d a dessin

hurwitz spaces of galois coverings of p1 whose galois groups are weyl groups abstract we prove the irreducibility of the hurwitz spaces which parametrize equivalence classes of galois coverings of p 1 whose galois group is an arbitrary weyl group and the local monodromies are reflections this generalizes a classical theorem due to luroth clebsch and hurwitz

the embedding problem over a hilbertian pac field we show that the absolute galois group of a countable hilbertian p seudo a lgebraically c losed field of characteristic 0 is a free profinite group of countably infinite rank theorem a as a consequence g q q is the extension of groups with a fairly simple structure e g 2 sn by a countably free group in addition we characterize those pac fields over which every finite group is a galois group as those with the rg hilbertian property

endomorphism algebras of jacobians very little is known about the existence of curves and families of curves whose jacobians are acted on by large rings of endomorphisms in this paper we show the existence of curves x with an injection k hom jac x jac x z q where k is a subfield of even index at most 10 in a primitive cyclotomic field q p or a subfield of index 2 in q pq or q p this result generalizes previous work of brumer mestre and tautz top and verberkmoes our curves are constructed as branched covers of the projective line and the endomorphisms arise as quotients of double coset algebras of the galois groups of these coverings in certain cases we show the existence of continuous families of jacobians admitting the desired endomorphism algebras at the end we raise some questions about upper bounds for endomorphism algebras of the type described here and ask about a non galois version of hurwitz s theorem

resolution of singularities and modular galois theory i shall sketch a brief history of the desingularization problem from riemann thru zariski to hironaka including the part i played in it and the work on galois theory which this led me to and how that caused me to search out many group theory gurus i shall also formulate several conjectures and suggest numerous thesis problems

stability of the homology of the mapping class groups of orientable surfaces the mapping class group of f fgs r is f rgs wo a where a is the topological group of orientation preserving diffeomorphisms of f which are the identity on df and fix the s punctures when r 0 and 2g s 3 an important feature of f is that it acts properly discontinuously on the teichmiller space us the quotient is the moduli space f of isometry classes of complete hyperbolic metrics of finite area on f since us is homeomorphic to euclidean space and f is virtually torsion free we have

embedding problems over large fields in this paper we study galois theoretic properties of a large class of fields a class which includes all fields satisfying a universal local global principle for the existence of rational points on varieties as applications we give a positive answer to a long standing conjecture which originates in an unpublished note of roquette and asserts that the absolute galois group of a countable hilbertian pac field is profinite free see f j problem 24 41 secondly we give new evidence for the conjecture of shafarevich which asserts that the absolute galois group of qab is profinite free finally one of the most interesting applications of the theory we develop here is the insight in the galois structure of the totally eg adic numbers

tchebotarev theorems for function fields abstract the central theme of the paper is the specialization of algebraic function field extensions our main results are tchebotarev type theorems for galois function field extensions finite or infinite over various base fields under some conditions we extend the classical finite field case to number fields p adic fields pac fields function fields x etc we also compare the tchebotarev conclusion existence of unramified local specializations with galois group any cyclic subgroup of the generic galois group up to conjugation to the hilbert specialization property for a function field extension with the tchebotarev property the exponent of the galois group is bounded by the l c m of the local specialization degrees local global questions arise for which we provide answers examples and counter examples

algebraic curves uniformized by congruence subgroups of triangle groups we construct certain subgroups of hyperbolic triangle groups which we call congruence subgroups these groups include the classical congruence subgroups of sl2 z hecke triangle groups and 19 families of arithmetic triangle groups associated to shimura curves we determine the eld of moduli of the curves associated to these groups and thereby realize the groups psl2 fq and pgl2 fq regularly as galois groups

some remarks on cohen lenstra heuristics cohen and lenstra have given a heuristic model which predicts the fraction of imaginary quadratic fields with class number divisible by a given odd prime p and of those whose class groups have a given p rank we show that these numbers also arise by

galois groups of unramified covers of projective curves in characteristicp abstract we show several results concerning the finite groups that occur as galois groups of unramified covers of projective curves in characteristic p in particular we prove that every finite group with g generators occurs over some curve of genus g this implies for example that every finite simple group occurs in genus 2 by similar methods we obtain several other families of groups which occur in genus 2 in addition we show that if a group g occurs over some curve of genus g then it must occur over almost all curves of genus g or greater the results are obtained using formal patching

three dimensional hybrid trefftz stress elements the stress model of the hybrid trefftz finite element formulation is applied to the linear elastostatic analysis of solids the stresses are approximated in the domain of the element and displacements on its boundary complete linearly independent hierarchical polynomial approximation functions are used in both domain and boundary approximations the displacement basis is defined independently on each inter element surface continuity at the edges and on the corners of the elements is not enforced a priori the stress basis is constrained to solve locally the beltrami governing differential equation it is derived from the associated papkovitch neuber elastic displacement solution generalized variables are used to ensure that the approximations are independent of the geometric description of the elements the solving system is derived directly from the fundamental relations of elastostatics the solving system is symmetric when the same property applies to the local elasticity condition sparse described by boundary integral arrays and well suited to p refinement and parallel processing the numerical implementation of these equations is discussed and numerical tests are presented to illustrate the performance of the finite element formulation copyright 2000 john wiley sons ltd

computation of a cover of shimura curves using a hurwitz space using a hurwitz space computation we determine the canonical model of the cover of shimura curves x 0 2 to x 1 associated to the quaternion algebra over the cubic field of discriminant 13 2 which is ramified at exactly two real places and unramified at finite places then we list the coordinates of some rational cm points on x 1

finite covers of random 3 manifolds a 3 manifold is haken if it contains a topologically essential surface the virtual haken conjecture posits that every irreducible 3 manifold with infinite fundamental group has a finite cover which is haken in this paper we study random 3 manifolds and their finite covers in an attempt to shed light on this difficult question in particular we consider random heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface for this model of random 3 manifold we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds our results contrast with the analogous probabilities for groups coming from random balanced presentations giving quantitative theorems to the effect that 3 manifold groups have many more finite quotients than random groups the next natural question is whether these covers have positive betti number for abelian covers of a fixed type over 3 manifolds of heegaard genus 2 we show that the probability of positive betti number is 0

homological stability for hurwitz spaces and the cohen lenstra conjecture over function fields we prove a homological stabilization theorem for hurwitz spaces moduli spaces of branched covers of the complex projective line this has the following arithmetic consequence let 2 be prime and a a nite abelian group then there exists q q a such that for q greater than q a positive fraction of quadratic extensions of fq t have the part of their class group isomorphic to a

study and computation of a hurwitz space and totally real psl2 f8 extensions of q abstract developing on works by fried volklein matzat malle debes wewers we give a method for computing a hurwitz space and illustrate it on some example of number theoretic interest we study and compute a family of degree 9 covers of p c 1 with monodromy group psl 2 f 8 and having four branch points we deduce explicit regular psl 2 f 8 extensions of the rational function field q with totally real fibers this gives rise to totally real polynomials over q with galois group psl 2 f 8

the deligne simpson problem a survey abstract the deligne simpson problem dsp respectively the weak dsp is formulated like this give necessary and sufficient conditions for the choice of the conjugacy classes c j gl n c or c j gl n c so that there exist irreducible respectively with trivial centralizer p 1 tuples of matrices m j c j or a j c j satisfying the equality m 1 m p 1 i or a 1 a p 1 0 the matrices m j and a j are interpreted as monodromy operators of regular linear systems and as matrices residua of fuchsian ones on riemann s sphere the present paper offers a survey of the results known up to now concerning the dsp

fake congruence modular curves and subgroups of the modular group abstract we construct a special class of noncongruence modular subgroups and curves analogous in some ways to the usual congruence ones the subgroups are obtained via the burau representation and the associated quotient curves have a natural moduli space interpretation in fact they are reduced hurwitz spaces corresponding to covers with 4 branch points and monodromy group equal to semi direct products of a cyclic and an abelian group furthermore they form a modular tower in the sense of fried we study representations on the cohomology of these fake congruence modular curves and also calculate the genera of certain quotient curves

harbater mumford subvarieties of moduli spaces of covers we define harbater mumford subvarieties which are special kinds of closed subvarieties of hurwitz moduli spaces obtained by fixing some of the branch points we show that for many finite groups finding geometrically irreducible hm subvarieties defined over open image in new window is always possible this provides information on the arithmetic of hurwitz spaces and applies in particular to the regular inverse galois problem with almost all fixed branch points profinite versions of our results can also be stated providing new tools to study the geometry of modular towers and the regular inverse galois problem for profinite groups

extending families of curves over log regular schemes in this paper we generalize to the log regular case a result of de jong and oort which states that any morphism satisfying certain conditions from the complement of a divisor with normal crossings in a regular scheme to a moduli stack of stable curves extends over the entire regular scheme the proof uses the theory of regular log schemes i e schemes with singularities like those of toric varieties due to k kato 9 we then use this extension theorem to prove that under certain natural conditions any scheme which is a successive fibration of smooth hyperbolic curves may be compactified to a successive fibration of stable curves 1991 mathematics subject classification primary subject 14h10 secondary subject 14e15

split embedding problems over function fields over complete domains let r be a krull domain complete with respect to a nonzero ideal let k be the quotient field of r we prove that every finite split embedding problem is solvable over every function field in one variable over k if dim r 1 then every finite split embedding problem over k is solvable

constant factor approximation algorithm for weighted flow time on a single machine in pseudopolynomial time in the weighted flow time problem on a single machine we are given a set of n jobs where each job has a processing requirement p j release date r j and weight w j the goal is to find a

irreducibility of moduli spaces of cyclic unramified covers of genus g curves let c1 cr g c g be an r tuple consisting of a transitive subgroup g of sm and r conjugacy classes c1 cr of g we consider the concept of the moduli space w c g of compact riemann surface covers of the riemann sphere of nielsen class c g the irreducibility of c g is equivalent to the transitivity of a specific permutation representation of the hurwitz monodromy group 1 but there are few general tools to decide questions about this representation theorem 2 gives a class of examples of c g for which c g is irreducible as an immediate corollary this gives an elementary proof and generalization of the irreduciblity of the moduli space of cyclic unramified covers of genus g curves for which deligne and mumford dm theorem 5 15 applied teichmuller theory and dehn s theorem this contrasts with the examples of c g in bfr for which c g is reducible these kinds of questions combined with the study of the existence of rational subvarieties of w c g have application to the realization of a group g as the galois group of a regular extension of 12 t fr3 4j

three dimensional hybrid trefftz stress finite elements for plates and shells three dimensional hybrid trefftz stress finite elements for plates and shells are proposed two independent fields are approximated stresses within the element and displacement on their boundary the required stress field derived from the papkovitch neuber solution of the navier equation which a priori satisfies the trefftz constraint is generated using homogeneous harmonic polynomials restriction on the polynomial degree in the coordinate measured along the thickness direction is imposed to reduce the number of independent terms explicit expressions of the corresponding independent polynomials are listed up to the fifth order illustrative applications to evaluate displacements and stresses are carried out by hexahedral hybrid trefftz stress element models the hierarchical p refinement and h refinement strategy are exploited in the numerical tests

connected components of affine primitive permutation groups abstract for a finite group g the hurwitz space h r g i n g is the space of genus g covers of the riemann sphere with r branch points and the monodromy group g in this paper we give a complete list of primitive genus one systems of affine type that is we assume that g is a primitive group of affine type under this assumption we determine the braid orbits on the suitable nielsen classes which is equivalent to finding connected components in h r 1 i n g furthermore we give a new algorithm for computing large braid orbits on nielsen classes this algorithm utilizes a correspondence between the components of h r 1 i n g and h r 1 i n m where m is the point stabilizer in g

the inverse galois problem and rational points on moduli spaces we reduce the regular version of the inverse galois problem for any finite group g to finding one rational point on an infinite sequence of algebraic varieties as a consequence any finite group g is the galois group of an extension l p x with l regular over any pac field p of characteristic zero a special case of this implies that g is a galois group over fp x for almost all primes p