The Lefschetz property for barycentric subdivisions of shellable complexes

Abstract: We show that an 'almost strong Lefschetz' property holds for the barycentric subdivision of a shellable complex. From this we conclude that for the barycentric subdivision of a Cohen-Macaulay complex, the $h$-vector is unimodal, peaks in its middle degree (one of them if the dimension of the complex is even), and that its $g$-vector is an $M$-sequence. In particular, the (combinatorial) $g$-conjecture is verified for barycentric subdivisions of homology spheres. In addition, using the above algebraic result, we derive new inequalities on a refinement of the Eulerian statistics on permutations, where permutations are grouped by the number of descents and the image of 1.


Symplectic reflection algebras

Abstract: We survey recent results on the representation theory of symplectic reflection algebras, focusing particularly on connections with symplectic quotient singularities and their resolutions, spaces of representations of quivers, and on category O.


Stochastic Completeness of Graphs

Abstract: In this thesis, we analyze the stochastic completeness of a heat kernel on graphs which is a function of three variables: a pair of vertices and a continuous time, for infinite, locally finite, connected graphs. For general graphs, a sufficient condition for stochastic completeness is given in terms of the maximum valence on spheres about a fixed vertex. That this result is optimal is shown by studying a particular family of trees. We also prove a lower bound on the bottom of the spectrum for the discrete Laplacian and use this lower bound to show that in certain cases the Laplacian has empty essential spectrum.


Good Reductions of Shimura Varieties of Hodge Type in Arbitrary Unramified Mixed Characteristic, Part II

Abstract: We prove a conjecture of Milne pertaining to the existence of integral canonical models of Shimura varieties of abelian type in arbitrary unramified mixed characteristic $(0,p)$. As an application we prove for $p=2$ a motivic conjecture of Milne pertaining to integral canonical models of Shimura varieties of Hodge type.


Character sums to smooth moduli are small

Abstract: Recently, Granville and Soundararajan have made fundamental breakthroughs in the study of character sums. Building on their work and using estimates on short character sums developed by Graham-Ringrose and Iwaniec, we improve the Polya-Vinogradov inequality for characters with smooth conductor.


A simple proof for the existence of Zariski decompositions on surfaces

Abstract: In this note we give a quick and simple proof of the existence (and uniqueness) of Zariski decompositions on surfaces. While Zariski's original proof employs a rather sophisticated procedure to construct the negative part of the decomposition, the present approach is based on the idea that the positive part can be constructed from a maximality condition. It may also be useful that this approach yields a practical algorithm for the computation of the positive part.


Regular elliptic boundary-value problem in a two-sided refined scale of spaces

Abstract: A regular elliptic boundary-value problem over a bounded domain with a smooth boundary is studied. We prove that the operator of this problem is a Fredholm one in the two-sided refined scale of the functional Hilbert spaces and generates a complete collection of isomorphisms. Elements of this scale are the isotropic spaces of Hormander-Volevich-Paneah and some its modifications. A priori estimate for the solution is established and its regularity is investigated.


On fibering and splitting of 5-manifolds over the circle

Abstract: Our main result is a generalization of Cappell's 5-dimensional splitting theorem. As an application, we analyze, up to internal s-cobordism, the smoothable splitting and fibering problems for certain 5-manifolds mapping to the circle. For example, these maps may have homotopy fibers which are in the class of finite connected sums of certain geometric 4-manifolds. Most of these homotopy fibers have non-vanishing second mod 2 homology and have fundamental groups of exponential growth, which are not known to be tractable by Freedman--Quinn topological surgery. Indeed, our key technique is topological cobordism, which may not be the trace of surgeries.


Lines of Curvature on Surfaces, Historical Comments and Recent Developments

Abstract: This survey starts with the historical landmarks leading to the study of principal configurations on surfaces, their structural stability and further generalizations. Here it is pointed out that in the work of Monge, 1796, are found elements of the qualitative theory of differential equations (QTDE), founded by Poincar\'e in 1881. Here are also outlined a number of recent results developed after the assimilation into the subject of concepts and problems from the QTDE and Dynamical Systems, such as Structural Stability, Bifurcations and Genericity, among others, as well as extensions to higher dimensions. References to original works are given and open problems are proposed at the end of some sections.


Quivers and the Euclidean group

Abstract: We show that the category of representations of the Euclidean group of orientation-preserving isometries of two-dimensional Euclidean space is equivalent to the category of representations of the preprojective algebra of infinite type A. We also consider the moduli space of representations of the Euclidean group along with a set of generators. We show that these moduli spaces are quiver varieties of the type considered by Nakajima. Using these identifications, we prove various results about the representation theory of the Euclidean group. In particular, we prove it is of wild representation type but that if we impose certain restrictions on weight decompositions, we obtain only a finite number of indecomposable representations.


Around the Gysin triangle II

Abstract: We study the construction and properties of the Gysin triangle in an axiomatic framework which covers triangulated mixed motives and MGl-modules over an arbitrary base S. This allows to define the Gysin morphism associated to a projective morphism between smooth S-schemes and prove duality for projective smooth S-schemes. As part of the construction, cobordism classes are considered and we give a proof of the Myschenko theorem generalized in our context - this in fact gives another proof of the latter theorem in classical stable homotopy through complex realization. Finally, these constructions apply to rigid cohomology through the notion of a mixed Weil theory introduced by D.-C. Cisinski and the author in another work.


The structure of critical sets for F_p arithmetic progressions

Abstract: Fix a prime p and a density 0 < d <= 1. Among all functions f : F_p -> [0,1], what can one say about those which assign minimal weight to three-term arithmetic progressions -- that is, the sum of f(a)f(a+x)f(a+2x) is minimal as we sum over all a and x -- subject to the density constraint that the expected value of f equals d? In the present paper we show three things about them: 1) Such f are nearly indicator functions; 2) They enjoy a certain ``local minimal'' property; and, 3) They are approximately indicator functions for certain sumsets A+B.


Staggered sheaves on partial flag varieties

Abstract: Staggered $t$-structures are a class of $t$-structures on derived categories of equivariant coherent sheaves. In this note, we show that the derived category of coherent sheaves on a partial flag variety, equivariant for a Borel subgroup, admits an artinian staggered $t$-structure. As a consequence, we obtain a basis for its equivariant $K$-theory consisting of simple staggered sheaves.


James' Conjecture for Hecke algebras of exceptional type, I

Abstract: In this paper, and a second part to follow, we complete the programme (initiated more than 15 years ago) of determining the decomposition numbers and verifying James' Conjecture for Iwahori--Hecke algebras of exceptional type. The new ingredients which allow us to achieve this aim are: - the fact, recently proved by the first author, that all Hecke algebras of finite type are cellular in the sense of Graham--Lehrer, and - the explicit determination of $W$-graphs for the irreducible (generic) representations of Hecke algebras of type $E_7$ and $E_8$ by Howlett and Yin. Thus, we can reduce the problem of computing decomposition numbers to a manageable size where standard techniques, e.g., Parker's {\sf MeatAxe} and its variations, can be applied. In this part, we describe the theoretical foundations for this procedure.


Unique Tournaments and Radar Tracking

Abstract: The sequence counting the number of unique tournaments with n people is the same as the sequence counting non-tracking binary strings corresponding to n-2 radar observations with the tracking rule "3 out of 5 with loss 2." This fact allows us to build a bijection between unique tournaments and non-tracking binary strings.


Primitive decompositions of Johnson graphs

Abstract: A transitive decomposition of a graph is a partition of the edge set together with a group of automorphisms which transitively permutes the parts. In this paper we determine all transitive decompositions of the Johnson graphs such that the group preserving the partition is arc-transitive and acts primitively on the parts.


Bridge Number and Conway Products

Abstract: Schubert proved that, given a composite link $K$ with summands $K_{1}$ and $K_{2}$, the bridge number of $K$ satisfies the following equation: $$\beta(K)=\beta(K_{1})+\beta(K_{2})-1.$$ In ``Conway Produts and Links with Multiple Bridge Surfaces", Scharlemann and Tomova proved that, given links $K_{1}$ and $K_{2}$, there is a Conway product $K_{1}\times_{c}K_{2}$ such that $$\beta(K_{1}\times_{c} K_{2}) \leq \beta(K_{1}) + \beta(K_{2}) - 1$$ In this paper, we define the generalized Conway product $K_{1}\ast_{c}K_{2}$ and prove the lower bound $\beta(K_{1}\ast_{c}K_{2}) \geq \beta(K_{1})-1$ where $K_{1}$ is the distinguished factor of the generalized product. We go on to show this lower bound is tight for an infinite class of links with arbitrarily high bridge number.


Extension Theorems for Spheres in the Finite Field Setting

Abstract: In this paper we study the boundedness of extension operators associated with spheres in vector spaces over finite fields.In even dimensions, we estimate the number of incidences between spheres and points in the translated set from a subset of spheres. As a result, we improve the Tomas-Stein exponents, our previous results. The analytic approach and the explicit formula for Fourier transform of the characteristic function on spheres play an important role to get good bounds for exponential sums.


Cayley sum graphs and eigenvalues of $(3,6)$-fullerenes

Abstract: We determine the spectra of cubic plane graphs whose faces have sizes 3 and 6. Such graphs, "(3,6)-fullerenes", have been studied by chemists who are interested in their energy spectra. In particular we prove a conjecture of Fowler, which asserts that all their eigenvalues come in pairs of the form $\{\lambda,-\lambda\}$ except for the four eigenvalues $\{3,-1,-1,-1\}$. We exhibit other families of graphs which are "spectrally nearly bipartite" in this sense. Our proof utilizes a geometric representation to recognize the algebraic structure of these graphs, which turn out to be examples of Cayley sum graphs.


Evaluations of multiple Dirichlet $L$-values via symmetric functions

Abstract: We explicitly evaluate a special type of multiple Dirichlet $L$-values at positive integers in two different ways: One approach involves using symmetric functions, while the other involves using a generating function of the values. Equating these two expressions, we derive several summation formulae involving the Bernoulli and Euler numbers. Moreover, values at non-positive integers, called central limit values, are also studied.


The joint distribution of occupation times of skip-free Markov processes and a class of multivariate exponential distributions

Abstract: For a skip-free Markov process on non-negative integers with generator matrix Q, we evaluate the joint Laplace transform of the occupation times before hitting the state n (starting at 0). This Laplace transform has a very straightforward and familiar expression. We investigate the properties of this Laplace transform, especially the conditions under which the occupation times form a Markov chain.


Galois extensions ramified only at one prime

Abstract: This paper gives some restrictions on finite groups that can occur as Galois groups of extensions over $\Q$ and over $\F_q(t)$ ramified only at one finite prime. Over $\Q$, we strengthen results of Jensen and Yui about dihedral extensions and rule out some non-solvable groups. Over $\F_q(t)$ restrictions are given for symmetric groups and dihedral groups to occur as tamely ramified extension over $\F_q(t)$ ramified only at one prime.


Smoothing $\ell_1$-penalized estimators for high-dimensional time-course data

Abstract: When a series of (related) linear models has to be estimated it is often appropriate to combine the different data-sets to construct more efficient estimators. We use $\ell_1$-penalized estimators like the Lasso or the Adaptive Lasso which can simultaneously do parameter estimation and model selection. We show that for a time-course of high-dimensional linear models the convergence rates of the Lasso and of the Adaptive Lasso can be improved by combining the different time-points in a suitable way. Moreover, the Adaptive Lasso still enjoys oracle properties and consistent variable selection. The finite sample properties of the proposed methods are illustrated on simulated data and on a real problem of motif finding in DNA sequences.


Special Values of Generalized Polylogarithms

Abstract: We study values of generalized polylogarithms at various points and relationships among them. Polylogarithms of small weight at the points 1/2 and -1 are completely investigated. We formulate a conjecture about the structure of the linear space generated by values of generalized polylogarithms.


Efficient blind search: Optimal power of detection under computational cost constraints

Abstract: Some astronomy projects require a blind search through a vast number of hypotheses to detect objects of interest. The number of hypotheses to test can be in the billions. A naive blind search over every single hypothesis would be far too costly computationally. We propose a hierarchical scheme for blind search, using various "resolution" levels. At lower resolution levels, "regions" of interest in the search space are singled out with a low computational cost. These regions are refined at intermediate resolution levels and only the most promising candidates are finally tested at the original fine resolution. The optimal search strategy is found by dynamic programming. We demonstrate the procedure for pulsar search from satellite gamma-ray observations and show that the power of the naive blind search can almost be matched with the hierarchical scheme while reducing the computational burden by more than three orders of magnitude.


On the constant in the Mertens product for arithmetic progressions. II. Numerical values

Abstract: We give explicit numerical values with 100 decimal digits for the constant in the Mertens product over primes in the arithmetic progressions $a \bmod q$, for $q \in \{3$, ..., $100\}$ and $(a, q) = 1$.


Comparison of some solution concepts for linear first-order hyperbolic differential equations with non-smooth coefficients

Abstract: We discuss solution concepts for linear hyperbolic equations with coefficients of regularity below Lipschitz continuity. Thereby our focus is on theories which are based either on a generalization of the method of characteristics or on refined techniques concerning energy estimates. We provide a series of examples both as simple illustrations of the notions and conditions involved but also to show logical independence among the concepts.


Estimation in a class of nonlinear heteroscedastic time series models

Abstract: Parameter estimation in a class of heteroscedastic time series models is investigated. The existence of conditional least-squares and conditional likelihood estimators is proved. Their consistency and their asymptotic normality are established. Kernel estimators of the noise's density and its derivatives are defined and shown to be uniformly consistent. A simulation experiment conducted shows that the estimators perform well for large sample size.


Diamond-$\alpha$ Jensen's Inequality on Time Scales

Abstract: The theory and applications of dynamic derivatives on time scales has recently received considerable attention. The primary purpose of this paper is to give basic properties of diamond-$\alpha$ derivatives which are a linear combination of delta and nabla dynamic derivatives on time scales. We prove a generalized version of Jensen's inequality on time scales via the diamond-$\alpha$ integral and present some corollaries, including H\"{o}lder's and Minkowski's diamond-$\alpha$ integral inequalities.


Lemma Poincar\'e for L_infty,loc - forms

Abstract: We show that every closed L_infty,loc - form on R^n is exact. Differential is understood in the sense of currents. The proof does not use any explicit geometric constructions. De Rham theorem follows.


Random Cluster Tessellations

Abstract: This article describes, in elementary terms, a generic approach to produce discrete random tilings and similar random structures by using point process theory. The standard Voronoi and Delone tilings can be constructed in this way. For this purpose, convex polytopes are replaced by their vertex sets. Three explicit constructions are given to illustrate the concept.


Local tail bounds for functions of independent random variables

Abstract: It is shown that functions defined on $\{0,1,...,r-1\}^n$ satisfying certain conditions of bounded differences that guarantee sub-Gaussian tail behavior also satisfy a much stronger ``local'' sub-Gaussian property. For self-bounding and configuration functions we derive analogous locally subexponential behavior. The key tool is Talagrand's [Ann. Probab. 22 (1994) 1576--1587] variance inequality for functions defined on the binary hypercube which we extend to functions of uniformly distributed random variables defined on $\{0,1,...,r-1\}^n$ for $r\ge2$.


Random graph models of communication network topologies

Abstract: We consider a variant of so called power-law random graph. A sequence of expected degrees corresponds to a power-law degree distribution with finite mean and infinite variance. In previous works the asymptotic picture with number of nodes limiting to infinity has been considered. It was found that an interesting structure appears. It has resemblance with such graphs like the Internet graph. Some simulations have shown that a finite sized variant has similar properties as well. Here we investigate this case in more analytical fashion, and, with help of some simple lower bounds for large valued expectations of relevant random variables, we can shed some light into this issue. A new term, 'communication range random graph' is introduced to emphasize that some further restrictions are needed to have a relevant random graph model for a reasonable sized communication network, like the Internet. In this case a pleasant model is obtained, giving the opportunity to understand such networks on an intuitive level. This would be beneficial in order to understand, say, how a particular routing works in such networks.


Approximating Data with weighted smoothing Splines

Abstract: Given a data set (t_i, y_i), i=1,..., n with the t_i in [0,1] non-parametric regression is concerned with the problem of specifying a suitable function f_n:[0,1] -> R such that the data can be reasonably approximated by the points (t_i, f_n(t_i)), i=1,..., n. If a data set exhibits large variations in local behaviour, for example large peaks as in spectroscopy data, then the method must be able to adapt to the local changes in smoothness. Whilst many methods are able to accomplish this they are less successful at adapting derivatives. In this paper we show how the goal of local adaptivity of the function and its first and second derivatives can be attained in a simple manner using weighted smoothing splines. A residual based concept of approximation is used which forces local adaptivity of the regression function together with a global regularization which makes the function as smooth as possible subject to the approximation constraints.


PAC-Bayesian Bounds for Randomized Empirical Risk Minimizers

Abstract: The aim of this paper is to generalize the PAC-Bayesian theorems proved by Catoni in the classification setting to more general problems of statistical inference. We show how to control the deviations of the risk of randomized estimators. A particular attention is paid to randomized estimators drawn in a small neighborhood of classical estimators, whose study leads to control the risk of the latter. These results allow to bound the risk of very general estimation procedures, as well as to perform model selection.


Stokes matrices of hypergeometric integrals

Abstract: In this work we compute the Stokes matrices of the ordinary differential equation satisfied by the hypergeometric integrals associated to an arrangement of hyperplanes in generic position. This generalizes the computation done by Ramis and Duval for confluent hypergeometric functions, which correspond to the arrangement of two points on the line. The proof is based on an explicit description of a base of canonical solutions as integrals on the cones of the arrangement, and combinatorial relations between integrals on cones and on domains.


Explicit construction of manifolds realizing the prescribed homology classes

Abstract: We consider a classical N. Steenrod's problem on realization of homology classes by images of the fundamental classes of manifolds. It is well-known that each integral homology class can be realized with some multiplicity as an image of the fundamental class of a manifold. Our main result is an explicit purely combinatorial construction that for a given integral cycle provides a combinatorial manifold realizing a multiple of the homology class of this cycle. The construction is based on a local procedure for resolving singularities of a pseudo-manifold. We give an application of our result to the problem of constructing a combinatorial manifold with the prescribed set of links of vertices.


The \'Etale Homology and The Cycle Maps in Adic Coefficients

Abstract: In this article, we define the l-adic homology for a morphism of schemes satisfying certain finiteness conditions. This homology has these functors similar to the Chow groups: proper push-forward, flat pull-back, base change, cap-product, etc. In particular on singular varieties, this kind of l-adic homology behaves much better that the classical l-adic cohomology. As an application, we give an much easier approach to construct the cycle maps for arbitrary algebraic schemes over fields of finite cohomology dimension. And we prove these cycle maps kill the algebraic equivalences and commute with the Chern action of locally free sheaves.


Coset decomposition for semisimple Hopf algebras

Abstract: The notion of double coset for semisimple finite dimensional Hopf algebras is introduced. This is done by considering an equivalence relation on the set of irreducible characters of the dual Hopf algebra. As an application formulae for the restriction of the irreducible characters to normal Hopf subalgebras are given.


On small homotopies of loops

Abstract: Two natural questions are answered in the negative: (1) If a space has the property that small nulhomotopic loops bound small nulhomotopies, then are loops which are limits of nulhomotopic loops themselves nulhomotopic? (2) Can adding arcs to a space cause an essential curve to become nulhomotopic? The answer to the first question clarifies the relationship between the notions of a space being homotopically Hausdorff and $\pi_1$-shape injective.


Vinberg's \theta-groups in positive characteristic and Kostant-Weierstrass slices

Abstract: We generalize the basic results of Vinberg's \theta-groups, or periodically graded reductive Lie algebras, to fields of good positive characteristic. To this end we clarify the relationship between the little Weyl group and the (standard) Weyl group. We deduce that the ring of invariants associated to the grading is a polynomial ring. This approach allows us to prove the existence of a KW-section for a classical graded Lie algebra (in zero or good characteristic), confirming a conjecture of Popov in this case.


Generalizations of Swierczkowski's lemma and the arity gap of finite functions

Abstract: Swierczkowski's Lemma - as it is usually formulated - asserts that if f is an at least quaternary operation on a finite set A and every operation obtained from f by identifying a pair of variables is a projection, then f is a semiprojection. We generalize this lemma in various ways. First, it is extended to B-valued functions on A instead of operations on A and to essentially at most unary functions instead of projections. Then we characterize the arity gap of functions of small arities in terms of quasi-arity, which in turn provides a further generalization of Swierczkowski's Lemma. Moreover, we explicitly classify all pseudo-Boolean functions according to their arity gap. Finally, we present a general characterization of the arity gaps of B-valued functions on arbitrary finite sets A.


An Upper Bound of the Total Q-Curvature and Its Isoperimetric Deficit for Higher-dimensional Conformal Euclidean Metrics

Abstract: The aim of this paper is to give not only an explicit upper bound of the total Q-curvature but also an induced isoperimetric deficit formula for the complete conformal metrics on $\mathbb R^n$, $n\ge 3$ with scalar curvature being nonnegative near infinity and Q-curvature being absolutely convergent.


Euler-Mahonian Statistics On Ordered Set Partitions (II)

Abstract: We study statistics on ordered set partitions whose generating functions are related to $p,q$-Stirling numbers of the second kind. The main purpose of this paper is to provide bijective proofs of all the conjectures of \stein (Arxiv:math.CO/0605670). Our basic idea is to encode ordered partitions by a kind of path diagrams and explore the rich combinatorial properties of the latter structure. We also give a partition version of MacMahon's theorem on the equidistribution of the statistics inversion number and major index on words.


Lagrangian Klein bottles in R^{2n}

Abstract: It is shown that the n-dimensional Klein bottle admits a Lagrangian embedding into R^{2n} if and only if n is odd.


Irrationalit\'e aux entiers impairs positifs d'un q-analogue de la fonction zeta de Riemann

Abstract: In this paper, we focus on a q-analogue of the Riemann zeta function at positive integers, which can be written for s\in\N^* by \zeta_q(s)=\sum_{k\geq 1}q^k\sum_{d|k}d^{s-1}. We give a new lower bound for the dimension of the vector space over \Q spanned, for 1/q\in\Z\setminus\{-1;1\} and an even integer A, by 1,\zeta_q(3),\zeta_q(5),...,\zeta_q(A-1). This improves a recent result of Krattenthaler, Rivoal and Zudilin (\emph{S\'eries hyperg\'eom\'etriques basiques, q-analogues des valeurs de la fonction zeta et s\'eries d'Eisenstein}, J. Inst. Jussieu {\bf 5}.1 (2006), 53-79). In particular, a consequence of our result is that for 1/q\in\Z\setminus\{-1;1\}, at least one of the numbers \zeta_q(3),\zeta_q(5),\zeta_q(7),\zeta_q(9) is irrational.


Sur le centralisateur d'une involution de 2E6(2)

Abstract: In this paper we prove that $2^{20+1}.U_6(2)$, known as the centralizer of an involution in the group $2E_6(2)$ is a quotient of a Coxeter group. We obtain a presentation of $2^{20+1}.U_6(2)$ as a $Q_{222}$-group, which now resolve a long pending question.


Finite Sections of Weighted Hardy's Inequality

Abstract: We study finite sections of weighted Hardy's inequality following the approach of De Bruijn. Similar to the unweighted case, we obtain an asymptotic expression for the optimal constant.


Homogenization of variational problems in manifold valued Sobolev spaces

Abstract: Homogenization of integral functionals is studied under the constraint that admissible maps have to take their values into a given smooth manifold. The notion of tangential homogenization is defined by analogy with the tangential quasiconvexity introduced by Dacorogna, Fonseca, Maly and Trivisa \cite{DFMT}. For energies with superlinear or linear growth, a $\Gamma$-convergence result is established in Sobolev spaces, the homogenization problem in the space of functions of bounded variation being the object of \cite{BM}.


Existence of rational points on smooth projective varieties

Abstract: Fix a number field k. We prove that if there is an algorithm for deciding whether a smooth projective geometrically integral k-variety has a k-point, then there is an algorithm for deciding whether an arbitrary k-variety has a k-point and also an algorithm for computing X(k) for any k-variety X for which X(k) is finite. The proof involves the construction of a one-parameter algebraic family of Chatelet surfaces such that exactly one of the surfaces fails to have a k-point.


The set of non-squares in a number field is diophantine

Abstract: Fix a number field k. We prove that k* - k*^2 is diophantine over k. This is deduced from a theorem that for a nonconstant separable polynomial P(x) in k[x], there are at most finitely many a in k* modulo squares such that there is a Brauer-Manin obstruction to the Hasse principle for the conic bundle X given by y^2 - az^2 = P(x).


On deep Frobenius descent and flat bundles

Abstract: Let R be an integral domain of finite type over Z and let f:X --> Spec R be a smooth projective morphism of relative dimension d >= 1. We investigate, for a vector bundle E on the total space X, under what arithmetical properties of a sequence (p_n, e_n)_{n \in \NN}, consisting of closed points p_n in Spec R and Frobenius descent data E_{p_n} \cong F^{e_n}^*(F) on the closed fibers X_{p_n}, the bundle E_0 on the generic fiber X_0 is semistable.


Lens space surgeries & primitive/Seifert type constructions

Abstract: We show that lens space surgeries on knots in $S^3$ which arise from the primitive/Seifert type construction also arise from the primitive/primitive construction. This is the first step of a three step program to prove the Berge conjecture for tunnel number one knots.


Existence de points fixes enlac\'es \`a une orbite p\'eriodique d'un hom\'eomorphisme du plan

Abstract: Let f be an orientation-preserving homeomorphism of the plane such that f-Id is contracting. Under these hypotheses, we establish the existence, for every periodic orbit, of a fixed point which has nonzero linking number with this periodic orbit.


Simplicial complexes and Macaulay's inverse systems

Abstract: Let $\Delta$ be a simplicial complex on $V = \{x_1,...,x_n\}$, with Stanley-Reisner ideal $I_{\Delta}\subseteq R = k[x_1,...,x_n]$. The goal of this paper is to investigate the class of artinian algebras $A=A(\Delta,a_1,...,a_n)= R/(I_{\Delta},x_1^{a_1},...,x_n^{a_n})$, where each $a_i \geq 2$. By utilizing the technique of Macaulay's inverse systems, we can explicitly describe the socle of $A$ in terms of $\Delta$. As a consequence, we determine the simplicial complexes, that we will call {\em levelable}, for which there exists a tuple $(a_1,...,a_n)$ such that $A(\Delta,a_1,...,a_n)$ is a level algebra.


Scattering for H^1/2 bounded solutions to the cubic, defocusing NLS in 3 dimensions

Abstract: We show that if a solution of the defocusing cubic NLS in 3d remains bounded in the homogeneous Sobolev norm of order 1/2 in its maximal interval of existence, then the interval is infinite and the solution scatters. No radial assumption is made.


Geometry of Shimura varieties of Hodge type over finite fields

Abstract: We present a general and comprehensive overview of recent developments in the theory of integral models of Shimura varieties of Hodge type. The paper covers the following topics: construction of integral models, their possible moduli interpretations, their uniqueness, their smoothness, their properness, and basic stratifications of their special fibres.


A Note on Kasparov Product and Duality

Abstract: Using Paschke-Higson duality, we can get a natural index pairing $K_{i}(A) \times K_{i+1}(D_{\Phi}) \to \boldsymbol{Z} \quad (i=0,1) (\mbox{mod}2)$, where $A$ is a separable $C\sp*$-algebra, and $\Phi$ is a representation of $A$ on a separable infinite dimensional Hilbert space $H$. It is proved that this is a special case of the Kasparov Product. As a step, we show a proof of Bott-periodicity for KK-theory asserting that $\mathbb{C}_1$ and $S$ are $KK$-equivalent using the odd index pairing.


Betti Numbers of Graded Modules and Cohomology of Vector Bundles

Abstract: Mats Boij and Jonas Soederberg (math.AC/0611081) have conjectured that the Betti table of a Cohen-Macaulay module over a polynomial ring can be decomposed in a certain way as a positive linear combination of Betti tables of modules with pure resolutions. We prove, over any field, a strengthened form of their conjecture. Applications include a proof of the Multiplicity Conjecture of Huneke and Srinivasan and a proof of the convexity of a fan naturally associated to the Young lattice. We also characterize the rational cone of all cohomology tables of vector bundles on projective spaces in terms of the cohomology tables of "supernatural" bundles. This characterization is dual, in a certain sense, to our characterization of Betti tables.


A variation of multiple $L$-values arising from the spectral zeta function of the non-commutative harmonic oscillator

Abstract: A variation of multiple $L$-values, which arises from the description of the special values of the spectral zeta function of the non-commutative harmonic oscillator, is introduced. In some special cases, we show that its generating function can be written in terms of the gamma functions. This result enables us to obtain explicit evaluations of them.


On endomorphism algebras of separable monoidal functors

Abstract: We show that the (co)endomorphism algebra of a sufficiently separable "fibre" functor into Vect_k, for k a field of characteristic 0, has the structure of what we call a "unital" von Neumann core in Vect_k. For Vect_k, this particular notion of algebra is weaker than that of a Hopf algebra, although the corresponding concept in Set is again that of a group.


Coplanar k-unduloids are nondegenerate

Abstract: We prove each embedded, constant mean curvature (CMC) surface in Euclidean space with genus zero and finitely many coplanar ends is nondegenerate: there is no nontrivial square-integrable solution to the Jacobi equation, the linearization of the CMC condition. This implies that the moduli space of such coplanar surfaces is a real-analytic manifold and that a neighborhood of these in the full CMC moduli space is itself a manifold. Nondegeneracy further implies (infinitesimal and local) rigidity in the sense that the asymptotes map is an analytic immersion on these spaces, and also that the coplanar classifying map is an analytic diffeomorphism.


Poisson Matching

Abstract: Suppose that red and blue points occur as independent homogeneous Poisson processes in R^d. We investigate translation-invariant schemes for perfectly matching the red points to the blue points. For any such scheme in dimensions d=1,2, the matching distance X from a typical point to its partner must have infinite d/2-th moment, while in dimensions d>=3 there exist schemes where X has finite exponential moments. The Gale-Shapley stable marriage is one natural matching scheme, obtained by iteratively matching mutually closest pairs. A principal result of this paper is a power law upper bound on the matching distance X for this scheme. A power law lower bound holds also. In particular, stable marriage is close to optimal (in tail behavior) in d=1, but far from optimal in d>=3. For the problem of matching Poisson points of a single color to each other, in d=1 there exist schemes where X has finite exponential moments, but if we insist that the matching is a deterministic factor of the point process then X must have infinite mean.


Duality Theorem and Hom Functor in Braided Tensor Categories

Abstract: Blatter-Montgomery duality theorem is generalized into braided tensor categories. It is shown that $Hom(V,W)$ is a braided Yetter-Drinfeld module for any two braided Yetter-Drinfeld modules $V$ and $W$.


Supercritical general branching processes conditioned on extinction are subcritical

Abstract: It is well known that a simple, supercritical Bienaym\'e-Galton-Watson process turns into a subcritical such process, if conditioned to die out. We prove that the corresponding holds true for general, multi-type branching, where child-bearing may occur at different ages, life span may depend upon reproduction, and the whole course of events is thus affected by conditioning upon extinction.


On localizing subcategories of derived categories

Abstract: Let A be a commutative noetherian ring. In this paper, we interpret localizing subcategories of the derived category of A by using subsets of Spec A and subcategories of the category of A-modules. We unify theorems of Gabriel, Neeman and Krause.


Trigonometry in extended hyperbolic space and extended de Sitter space

Abstract: We study the hyperbolic cosine and sine laws in the extended hyperbolic space which contains hyperbolic space as a subset and is an analytic continuation of the hyperbolic space. And we also study the spherical cosine and sine laws in the extended de Sitter space which contains de Sitter Space $S^n_1$ as a subset and is also an analytic continuation of de Sitter space. In fact, the extended hyperbolic space and extended de Sitter space are the same space only differ by -1 multiple in the metric. Hence these two extended spaces clearly show and apparently explain that why many corresponding formulas in hyperbolic and spherical space are very similar each other. From these extended trigonometry laws, we can give a coherent and geometrically simple explanation for the various relations between the lengths and angles of hyperbolic polygons and relations on de Sitter polygons which lie on $S^2_1$.


Normal Forms, K3 Surface Moduli, and Modular Parametrizations

Abstract: The geometric objects of study in this paper are K3 surfaces which admit a polarization by the unique even unimodular lattice of signature (1,17). A standard Hodge-theoretic observation about this special class of K3 surfaces is that their polarized Hodge structures are identical with the polarized Hodge structures of abelian surfaces that are cartesian products of elliptic curves. Earlier work of the first two authors gives an explicit normal form and construction of the moduli space for these surfaces. In the present work, this normal form is used to derive Picard-Fuchs differential equations satisfied by periods of these surfaces. We also investigate the subloci of the moduli space on which the polarization is enhanced. In these cases, we derive information about the Picard-Fuchs differential equations satisfied by periods of these subfamilies, and we relate this information to the theory of genus zero quotients of the upper half-plane by Moonshine groups. For comparison, we also examine the analogous theory for elliptic curves in Weierstrass form.


A Criterion for Vertex Colorability of a Graph Stated in Terms of Edge Orientations

Abstract: L.M.Vitaver [1962] and G.I.Minty [1962] suggested criteria for vertex colorability of a graph in at most a given number of colors; these criteria are stated in terms of the orientation of the edges. One additional criterion of this kind is given here.


Determinants of finite-dimensional algebras

Abstract: To each associative unitary finite-dimensional algebra over a normal base, we associative a canonical multiplicative function called its determinant. We give various properties of this construction, as well as applications to the topology of the moduli stack of n-dimensional algebras.


Unitary processes with independent increments and representations of Hilbert tensor algebras

Abstract: The aim of this article is to characterize unitary increment process by a quantum stochastic integral representation on symmetric Fock space. Under certain assumptions we have proved its unitary equivalence to a Hudson-Parthasarathy flow.


Cycles of random permutations with restricted cycle lengths

Abstract: We prove some general results about the asymptotics of the distribution of the number of cycles of given length of a random permutation whose distribution is invariant under conjugation. These results were first established to be applied in a forthcoming paper (Cycles of free words in several random permutations with restricted cycles lengths), where we prove results about cycles of random permutations which can be written as free words in several independent random permutations. However, we also apply them here to prove asymptotic results about random permutations with restricted cycle lengths. More specifically, for $A$ a set of positive integers, we consider a random permutation chosen uniformly among the permutations of $\{1,..., n\}$ which have all their cycle lengths in $A$, and then let $n$ tend to infinity. Improving slightly a recent result of Yakymiv (Random A-Permutations: Convergence to a Poisson Process), we prove that under a general hypothesis on $A$, the numbers of cycles with fixed lengths of this random permutation are asymptotically independent and distributed according to Poisson distributions. In the case where $A$ is finite, we prove that the behavior of these random variables is completely different: cycles with length $\max A$ are predominant.


Binary linear forms as sums of two squares

Abstract: We revisit recent work of Heath-Brown on the average order of the quantity r(L_1)r(L_2)r(L_3)r(L_4), for suitable binary linear forms L_1,..., L_4, for integers ranging over quite general regions. In addition to improving the error term in Heath-Brown's estimate we generalise his result quite extensively.


Prediction of long memory processes on same-realisation

Abstract: For the class of stationary Gaussian long memory processes, we study some properties of the least-squares predictor of X_{n+1} based on (X_n, ..., X_1). The predictor is obtained by projecting X_{n+1} onto the finite past and the coefficients of the predictor are estimated on the same realisation. First we prove moment bounds for the inverse of the empirical covariance matrix. Then we deduce an asymptotic expression of the mean-squared error. In particular we give a relation between the number of terms used to estimate the coefficients and the number of past terms used for prediction, which ensures the L^2-sense convergence of the predictor. Finally we prove a central limit theorem when our predictor converges to the best linear predictor based on all the past.


Proofs of the martingale FCLT

Abstract: This is an expository review paper elaborating on the proof of the martingale functional central limit theorem (FCLT). This paper also reviews tightness and stochastic boundedness, highlighting one-dimensional criteria for tightness used in the proof of the martingale FCLT. This paper supplements the expository review paper Pang, Talreja and Whitt (2007) illustrating the ``martingale method'' for proving many-server heavy-traffic stochastic-process limits for queueing models, supporting diffusion-process approximations.


Semi-parametric estimation of shifts

Abstract: We observe a large number of functions differing from each other only by a translation parameter. While the main pattern is unknown, we propose to estimate the shift parameters using $M$-estimators. Fourier transform enables to transform this statistical problem into a semi-parametric framework. We study the convergence of the estimator and provide its asymptotic behavior. Moreover, we use the method in the applied case of velocity curve forecasting.


Designs, groups and lattices

Abstract: We study the Grassmannian 4-designs contained in lattices, in connection with the local property of the Rankin constant. We prove that the sequence of Barnes-Wall lattices contain Grassmannian 6-designs.


Improving population-specific allele frequency estimates by adapting supplemental data: an empirical Bayes approach

Abstract: Estimation of the allele frequency at genetic markers is a key ingredient in biological and biomedical research, such as studies of human genetic variation or of the genetic etiology of heritable traits. As genetic data becomes increasingly available, investigators face a dilemma: when should data from other studies and population subgroups be pooled with the primary data? Pooling additional samples will generally reduce the variance of the frequency estimates; however, used inappropriately, pooled estimates can be severely biased due to population stratification. Because of this potential bias, most investigators avoid pooling, even for samples with the same ethnic background and residing on the same continent. Here, we propose an empirical Bayes approach for estimating allele frequencies of single nucleotide polymorphisms. This procedure adaptively incorporates genotypes from related samples, so that more similar samples have a greater influence on the estimates. In every example we have considered, our estimator achieves a mean squared error (MSE) that is smaller than either pooling or not, and sometimes substantially improves over both extremes. The bias introduced is small, as is shown by a simulation study that is carefully matched to a real data example. Our method is particularly useful when small groups of individuals are genotyped at a large number of markers, a situation we are likely to encounter in a genome-wide association study.


Scaling limit and aging for directed trap models

Abstract: We consider one-dimensional directed trap models and suppose that the trapping times are heavy-tailed. We obtain the inverse of a stable subordinator as scaling limit and prove an aging phenomenon expressed in terms of the generalized arcsine law. These results confirm the status of universality described by Ben Arous and \v{C}ern\'y for a large class of graphs.


Brauer-Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms

Abstract: An integer may be represented by a quadratic form over each ring of p-adic integers and over the reals without being represented by this quadratic form over the integers. More generally, such failure of a local-global principle may occur for the representation of one integral quadratic form by another integral quadratic form. We show that many such examples may be accounted for by a Brauer-Manin obstruction for the existence of integral points on schemes defined over the integers. For several types of homogeneous spaces of linear algebraic groups, this obstruction is shown to be the only obstruction to the existence of integral points. ----- Une forme quadratique enti\`ere peut \^etre repr\'esent\'ee par une autre forme quadratique enti\`ere sur tous les anneaux d'entiers p-adiques et sur les r\'eels, sans l'\^etre sur les entiers. On en trouve de nombreux exemples dans la litt\'erature. Nous montrons qu'une partie de ces exemples s'explique au moyen d'une obstruction de type Brauer-Manin pour les points entiers. Pour plusieurs types d'espaces homog\`enes de groupes alg\'ebriques lin\'eaires, cette obstruction est la seule obstruction \`a l'existence d'un point entier.


The BARISTA: A model for bid arrivals in online auctions

Abstract: The arrival process of bidders and bids in online auctions is important for studying and modeling supply and demand in the online marketplace. A popular assumption in the online auction literature is that a Poisson bidder arrival process is a reasonable approximation. This approximation underlies theoretical derivations, statistical models and simulations used in field studies. However, when it comes to the bid arrivals, empirical research has shown that the process is far from Poisson, with early bidding and last-moment bids taking place. An additional feature that has been reported by various authors is an apparent self-similarity in the bid arrival process. Despite the wide evidence for the changing bidding intensities and the self-similarity, there has been no rigorous attempt at developing a model that adequately approximates bid arrivals and accounts for these features. The goal of this paper is to introduce a family of distributions that well-approximate the bid time distribution in hard-close auctions. We call this the BARISTA process (Bid ARrivals In STAges) because of its ability to generate different intensities at different stages. We describe the properties of this model, show how to simulate bid arrivals from it, and how to use it for estimation and inference. We illustrate its power and usefulness by fitting simulated and real data from eBay.com. Finally, we show how a Poisson bidder arrival process relates to a BARISTA bid arrival process.


Foundations for abstract forcing

Abstract: The foundations of forcing theory are reworked to streamline the presentation and to show how the most basic results are applicable in very general contexts.


The Jiang-Su algebra does not always embed

Abstract: We exhibit a unital simple nuclear non-type-I C*-algebra into which the Jiang-Su algebra does not embed unitally. This answers a question of M. R{\o}rdam.


Hypergeometric D-modules and twisted Gauss-Manin systems

Abstract: The Euler-Koszul complex is the fundamental tool in the homological study of A-hypergeometric differential systems and functions. We compare Euler-Koszul homology with D-module direct images from the torus to the base space through orbits in the corresponding toric variety. Our approach generalizes a result by Gel'fand et al. and yields a simpler, more algebraic proof. In the process we extend the Euler-Koszul functor a category of infinite toric modules and describe multigraded localizations of Euler-Koszul homology.


Constructing elliptic curves of prime order

Abstract: We present a very efficient algorithm to construct an elliptic curve E and a finite field F such that the order of the point group E(F) is a given prime number N. Heuristically, this algorithm only takes polynomial time Otilde((\log N)^3), and it is so fast that it may profitably be used to tackle the related problem of finding elliptic curves with point groups of prime order of prescribed size. We also discuss the impact of the use of high level modular functions to reduce the run time by large constant factors and show that recent gonality bounds for modular curves imply limits on the time reduction that can be obtained.


Where to place a spherical obstacle so as to maximize the second Dirichlet eigenvalue

Abstract: We prove that among all doubly connected domains of $\mathbb{R}^n$ bounded by two spheres of given radii, the second eigenvalue of the Dirichlet Laplacian achieves its maximum when the spheres are concentric (spherical shell). The corresponding result for the first eigenvalue has been established by Hersch in dimension 2, and by Harrell, Kr\"oger and Kurata and Kesavan in any dimension. We also prove that the same result remains valid when the ambient space $\mathbb{R}^n$ is replaced by the standard sphere $\mathbb{S}^n$ or the hyperbolic space $\mathbb{H}^n$ .


Diviseur Theta et Formes Differentielles

Abstract: In this papers, we study the geometric and arithmetic properties of the theta divisor associated to the sheaf of locally exact differential forms over a curve in positive characteristic. In this published version, we prove a stronger version of the main result of chapiter 5.


Moving codimension-one subvarieties over finite fields

Abstract: We give the first examples of nef line bundles on smooth projective varieties over finite fields which are not semi-ample. More concretely, we find smooth curves on smooth projective surfaces over finite fields such that the normal bundle has degree zero, but no positive multiple of the curve moves in a family of disjoint curves. This answers questions by Keel and Mumford. The proof uses an obstruction theory, in the spirit of homotopy theory, which links the infinitely many obstructions to moving higher and higher multiples of a given codimension-one subvariety. On 3-folds over a finite field, we find nef and big line bundles which are not semi-ample. Finally, we reprove some of the known positive results about semi-ampleness over finite fields.


Isogenies of supersingular elliptic curves over finite fields and operations in elliptic cohomology

Abstract: We investigate stable operations in supersingular elliptic cohomology using isogenies of supersingular elliptic curves over finite fields. Our main results provide a framework in which we give a conceptually simple proof of an elliptic cohomology version of the Morava change of rings theorem and also gives models for explicit stable operations in terms of isogenies and morphisms in certain enlarged isogeny categories. We relate our work to that of G. Robert on the Hecke algebra structure of the ring of supersingular modular forms.


Equations of the moduli of Higgs pairs and infinite Grassmannian

Abstract: In this paper the moduli space of Higgs pairs over a fixed smooth projective curve with extra formal data is defined and it is endowed with a scheme structure. We introduce a relative version of the Krichever map using a fibration of Sato Grassmannians and show that this map is injective. This fact and the characterization of the points of the image of the Krichever map allow us to prove that this moduli space is a closed subscheme of the product of the moduli of vector bundles (with formal extra data) and a formal anologue of the Hitchin base. This characterization also provide us the method to compute explicitely KP-type equations which describe the moduli space of Higgs pairs. Finally, for the case where the spectral cover is totally ramified at a fixed point of the curve, these equations are given in terms of the characteristic coefficients of the Higgs field.


A Not-so-Characteristic Equation: the Art of Linear Algebra

Abstract: Can the cross product be generalized? Why are the trace and determinant so important in matrix theory? What do all the coefficients of the characteristic polynomial represent? This paper describes a technique for `doodling' equations from linear algebra that offers elegant solutions to all these questions. The doodles, known as trace diagrams, are graphs labeled by matrices which have a correspondence to multilinear functions. This correspondence permits computations in linear algebra to be performed using diagrams. The result is an elegant theory from which standard constructions of linear algebra such as the determinant, the trace, the adjugate matrix, Cramer's rule, and the characteristic polynomial arise naturally. Using the diagrams, it is easy to see how little structure gives rise to these various results, as they all can be `traced' back to the definition of the determinant and inner product.


On Character Amenability of Banach Algebras

Abstract: Associated to a nonzero homomorphism $\varphi$ of a Banach algebra $A$, we regard special functionals, say $m_\varphi$, on certain subspaces of $A^\ast$ which provide equivalent statements to the existence of a bounded right approximate identity in the corresponding maximal ideal in $A$. For instance, applying a fixed point theorem yields an equivalent statement to the existence of a $m_\varphi$ on $A^\ast$; and, in addition we expatiate the case that if a functional $m_\varphi$ is unique, then $m_\varphi$ belongs to the topological center of the bidual algebra $A^{\ast\ast}$. An example of a function algebra, surprisingly, contradicts a conjecture that a Banach algebra $A$ is amenable if $A$ is $\varphi$-amenable in every character $\varphi$ and if functionals $m_\varphi$ associated to the characters $\varphi$ are uniformly bounded. Aforementioned are also elaborated on the direct sum of two given Banach algebras.


($\ell,0)$-Carter partitions, a generating function, and their crystal theoretic interpretation

Abstract: In this paper we give an alternate combinatorial description of the "$(\ell,0)$-JM partitions" (see \cite{F}) that are also $\ell$-regular. Our main theorem is the equivalence of our combinatoric and the one introduced by James and Mathas (\cite{JM}). The condition of being an $(\ell,0)$-JM partition is fundamentally related to the hook lengths of the partition. The representation-theoretic significance of their combinatoric on an $\ell$-regular partition is that it indicates the irreducibility of the corresponding specialized Specht module over the finite Hecke algebra (see \cite{JM}). We use our result to find a generating series which counts the number of such partitions, with respect to the statistic of a partition's first part. We then apply our description of these partitions to the crystal graph $B(\Lambda_0)$ of the basic representation of $\hat{\mathfrak{sl}_{\ell}}$, whose nodes are labeled by $\ell$-regular partitions. Here we give a fairly simple crystal-theoretic rule which generates all $\ell$-regular $(\ell,0)$-JM partitions in the graph $B(\Lambda_0)$. Finally, we mention how our construction can be generalized to recent results of M. Fayers (see \cite{F}) and S. Lyle (see \cite{L}) to count the total number of (not necessarily $\ell$-regular) Specht modules which stay irreducible at a primitive $\ell$th root of unity (for $\ell >2$).


On the irreducible representations of a finite semigroup

Abstract: Work of Clifford, Munn and Ponizovski{\u\i} parameterized the irreducible representations of a finite semigroup in terms of the irreducible representations of its maximal subgroups. Explicit constructions of the irreducible representations were later obtained independently by Rhodes and Zalcstein and by Lallement and Petrich. All of these approaches make use of Rees's theorem characterizing 0-simple semigroups up to isomorphism. Here we provide a short modern proof of the Clifford-Munn-Ponizovski{\u\i} result based on a lemma of J. A. Green, which allows us to circumvent the theory of 0-simple semigroups. A novelty of this approach is that it works over any base ring.


Homotopy groups of the spaces of self-maps of Lie groups

Abstract: We compute the homotopy groups of the spaces of self maps of Lie groups of rank 2, SU(3), Sp(2), and G_2. We use the cell structures of these Lie groups and the standard methods of homotopy theory.


Riesz transforms in one dimension

Abstract: We study the boundedness on $L^p$ of the Riesz transform $\nabla L^{-1/2}$, where $L$ is one of several operators defined on $\R$ or $\R_+$, endowed with the measure $r^{d-1} dr$, $d > 1$, where $dr$ is Lebesgue measure. For integer $d$, this mimics the measure on Euclidean $d$-dimensional space, and in this case our setup is equivalent to looking at the Laplacian acting on radial functions on Euclidean space or variations of Euclidean space such as the exterior of a sphere (with either Dirichlet or Neumann boundary conditions), or the connected sum of two copies of $\R^d$. In this way we illuminate some recent results on the Riesz transform on asymptotically Euclidean manifolds. We are however interested in all real values of $d > 1$, and another goal of our analysis is to study the range of boundedness as a function of $d$; it is particularly interesting to see the behaviour as $d$ crosses 2. For example, in one of our cases which models radial functions on the connected sum of two copies of $\R^d$, the upper threshold for $L^p$ boundedness is $p=d$ for $d \ge 2$ and $p=d/(d-1)$ for $d < 2$. Only in the case $d=2$ is the Riesz transform actually bounded on $L^p$ when $p$ is equal to the upper threshold. We also study the Riesz transform when we have an inverse square potential, or a delta function potential; these cases provide a simple model for recent results of the first author and Guillarmou. Finally we look at the Hodge projector in a slightly more general setup.


On the Berstein-Svarc Theorem in dimension 2

Abstract: We prove that for any group of the cohomological dimension $n$ the $n$th power of the Berstein class of the group is nontrivial. This allows to prove the following Berstein-Svarc theorem for all $n$: Theorem. For a connected complex $X$ with $\dim X=\cat X=n$, the $n$th power of the Berstein class of $X$ is nontrivial. Previously it was known for $n\ge 3$. We also prove that, for every map $f: M \to N$ of degree $\pm 1$ of closed orientable manifolds, the fundamental group of $N$ is free provided that the fundamental group of $M$ is.


Asymptotic distributions and chaos for the supermarket model

Abstract: In the supermarket model there are n queues, each with a unit rate server. Customers arrive in a Poisson process at rate \lambda n, where 0<\lambda <1. Each customer chooses d > 2 queues uniformly at random, and joins a shortest one. It is known that the equilibrium distribution of a typical queue length converges to a certain explicit limiting distribution as n -> oo. We quantify the rate of convergence by showing that the total variation distance between the equilibrium distribution and the limiting distribution is essentially of order n^{-1}; and we give a corresponding result for systems starting from quite general initial conditions (not in equilibrium). Further, we quantify the result that the systems exhibit chaotic behaviour: we show that the total variation distance between the joint law of a fixed set of queue lengths and the corresponding product law is essentially of order at most n^{-1}.


Bessel models for $GSp(4)$

Abstract: Methods of theta correspondence are used to analyze local and global Bessel models for $GSp(4)$ proving a conjecture of Gross and Prasad which describes these models in terms of local epsilon factors in the local case, and the non-vanishing of central critical $L$-value in the global case.


An Example of Constructing Versal Deformation for Leibniz Algebras

Abstract: In this work we compute a versal deformation of the three dimensional nilpotent Leibniz algebra over $\mathbb{C}$, defined by the nontrivial brackets $[e_1,e_3]=e_2$ and $[e_3,e_3]=e_1$.


Prime spectrum and primitive Leavitt path algebras

Abstract: In this paper a bijection between the set of prime ideals of a Leavitt path algebra $L_K(E)$ and a certain set which involves maximal tails in $E$ and the prime spectrum of $K[x,x^{-1}]$ is established. Necessary and sufficient conditions on the graph $E$ so that the Leavitt path algebra $L_K(E)$ is primitive are also found.


Symplectic Heegaard splittings and linked abelian groups

Abstract: Let $f$ be the gluing map of a Heegaard splitting of a 3-manifold $W$. The goal of this paper is to determine the information about $W$ contained in the image of $f$ under the symplectic representation of the mapping class group. We prove three main results. First, we show that the first homology group of the three manifold together with Seifert's linking form provides a complete set of stable invariants. Second, we give a complete, computable set of invariants for these linking forms. Third, we show that a slight augmentation of Birman's determinantal invariant for a Heegaard splitting gives a complete set of unstable invariants.


Non-special scrolls with general moduli

Abstract: In this paper we study smooth, non-special scrolls S of degree d, genus g, with general moduli. In particular, we study the scheme of unisecant curves of a given degree on S. Our approach is mostly based on degeneration techniques.


Brill-Noether theory and non-special scrolls

Abstract: In this paper we study the Brill-Noether theory of sub-line bundles of a general, stable rank-two vector bundle on a curve C with general moduli. We relate this theory to the geometry of unisecant curves on smooth, non-special scrolls with hyperplane sections isomorphic to C. Most of our results are based on degeneration techniques.


Dirichlet sets and Erdos-Kunen-Mauldin theorem

Abstract: By a theorem proved by Erdos, Kunen and Mauldin, for any nonempty perfect set $P$ on the real line there exists a perfect set $M$ of Lebesgue measure zero such that $P+M=\mathbb{R}$. We prove a stronger version of this theorem in which the obtained perfect set $M$ is a Dirichlet set. Using this result we show that for a wide range of familes of subsets of the reals, all additive sets are perfectly meager in transitive sense. We also prove that every proper analytic subgroup $G$ of the reals is contained in an F-sigma set $F$ such that $F+G$ is a meager null set.


A statistical framework for the analysis of microarray probe-level data

Abstract: In microarray technology, a number of critical steps are required to convert the raw measurements into the data relied upon by biologists and clinicians. These data manipulations, referred to as preprocessing, influence the quality of the ultimate measurements and studies that rely upon them. Standard operating procedure for microarray researchers is to use preprocessed data as the starting point for the statistical analyses that produce reported results. This has prevented many researchers from carefully considering their choice of preprocessing methodology. Furthermore, the fact that the preprocessing step affects the stochastic properties of the final statistical summaries is often ignored. In this paper we propose a statistical framework that permits the integration of preprocessing into the standard statistical analysis flow of microarray data. This general framework is relevant in many microarray platforms and motivates targeted analysis methods for specific applications. We demonstrate its usefulness by applying the idea in three different applications of the technology.


On the existence of homomorphisms between principal series of complex semisimple Lie groups

Abstract: We determine when there exists a nonzero homomorphism between principal series representations of a complex semisimple Lie group. We also determines the existence of homomorphisms between twisted Verma modules.


Conformal Metrics with Constant Q-Curvature

Abstract: We consider the problem of varying conformally the metric of a four dimensional manifold in order to obtain constant $Q$-curvature. The problem is variational, and solutions are in general found as critical points of saddle type. We show how the problem leads naturally to consider the set of formal barycenters of the manifold.


Describing disability through individual-level mixture models for multivariate binary data

Abstract: Data on functional disability are of widespread policy interest in the United States, especially with respect to planning for Medicare and Social Security for a growing population of elderly adults. We consider an extract of functional disability data from the National Long Term Care Survey (NLTCS) and attempt to develop disability profiles using variations of the Grade of Membership (GoM) model. We first describe GoM as an individual-level mixture model that allows individuals to have partial membership in several mixture components simultaneously. We then prove the equivalence between individual-level and population-level mixture models, and use this property to develop a Markov Chain Monte Carlo algorithm for Bayesian estimation of the model. We use our approach to analyze functional disability data from the NLTCS.


On an identity by Chaundy and Bullard. I

Abstract: An identity by Chaundy and Bullard writes 1/(1-x)^n (n=1,2,...) as a sum of two truncated binomial series. This identity was rediscovered many times. Notably, a special case was rediscovered by I. Daubechies, while she was setting up the theory of wavelets of compact support. We discuss or survey many different proofs of the identity, and also its relationship with Gauss hypergeometric series. We also consider the extension to complex values of the two parameters which occur as summation bounds. The paper concludes with a discussion of a multivariable analogue of the identity, which was first given by Damjanovic, Klamkin and Ruehr. We give the relationship with Lauricella hypergeometric functions and corresponding PDE's. The paper ends with a new proof of the multivariable case by splitting up Dirichlet's multivariable beta integral.


Distances in random Apollonian network structures

Abstract: In this paper, we study the distribution of distances in random Apollonian network structures (RANS), a family of graphs which has a one-to-one correspondence with planar ternary trees. Using multivariate generating functions that express all information on distances, and singularity analysis for evaluating the coefficients of these functions, we describe the distribution of distances to an outermost vertex, and show that the average value of the distance between any pair of vertices in a RANS of order n is asymptotically square root of n.


Diverse correlation structures in gene expression data and their utility in improving statistical inference

Abstract: It is well known that correlations in microarray data represent a serious nuisance deteriorating the performance of gene selection procedures. This paper is intended to demonstrate that the correlation structure of microarray data provides a rich source of useful information. We discuss distinct correlation substructures revealed in microarray gene expression data by an appropriate ordering of genes. These substructures include stochastic proportionality of expression signals in a large percentage of all gene pairs, negative correlations hidden in ordered gene triples, and a long sequence of weakly dependent random variables associated with ordered pairs of genes. The reported striking regularities are of general biological interest and they also have far-reaching implications for theory and practice of statistical methods of microarray data analysis. We illustrate the latter point with a method for testing differential expression of nonoverlapping gene pairs. While designed for testing a different null hypothesis, this method provides an order of magnitude more accurate control of type 1 error rate compared to conventional methods of individual gene expression profiling. In addition, this method is robust to the technical noise. Quantitative inference of the correlation structure has the potential to extend the analysis of microarray data far beyond currently practiced methods.


Jacobi osculating rank and isotropic geodesics on naturally reductive 3-manifolds

Abstract: We study the Jacobi osculating rank of geodesics on naturally reductive homogeneous manifolds and we apply this theory to the 3-dimensional case. Here, each non-symmetric, simply connected naturally reductive 3-manifold can be given as a principal bundle over a surface of constant curvature, such that the curvature of its horizontal distribution is also a constant. Then, we prove that the Jacobi osculating rank of every geodesic is two except for the Hopf fibers, where it is zero. Moreover, we determine all isotropic geodesics and the isotropic tangent conjugate locus.


The Div-Curl Lemma Revisited

Abstract: The Div-Curl Lemma, which is the basic result of the compensated compactness theory in Sobolev spaces, was introduced by F. Murat (1978) with distinct proofs for the $L^2(\Omega)$ and $L^p(\Omega)$, $p \neq 2$, cases. In this note we present a slightly different proof, relying only on a Green-Gauss integral formula and on the usual Rellich-Kondrachov compactness properties.


Complexes of Injective Words and Their Commutation Classes

Abstract: Let $S$ be a finite alphabet. An injective word over $S$ is a word over $S$ such that each letter in $S$ appears at most once in the word. We study Boolean cell complexes of injective words over $S$ and their commutation classes. This generalizes work by Farmer and by Bj\"orner and Wachs on the complex of all injective words.


An example of noncommutative deformations

Abstract: We compute the noncommutative deformations of a family of modules over the first Weyl algebra. This example shows some important properties of noncommutative deformation theory that separates it from commutative deformation theory.


Sobolev homeomorphisms and Poincare inequality

Abstract: We study global regularity properties of Sobolev homeomorphisms on $n$-dimensional Riemannian manifolds under the assumption of $p$-integrability of its first weak derivatives in degree $p\geq n-1$. We prove that inverse homeomorphisms have integrable first weak derivatives. For the case $p>n$ we obtain necessary conditions for existence of Sobolev homeomorphisms between manifolds. These necessary conditions based on Poincar\'e type inequality: $$ \inf_{c\in \mathbb R} \|u-c\mid L_{\infty}(M)\|\leq K \|u\mid L^1_{\infty}(M)\|. $$ As a corollary we obtain the following geometrical necessary condition: {\em If there exists a Sobolev homeomorphisms $\phi: M \to M'$, $\phi\in W^1_p(M, M')$, $p>n$, $J(x,\phi)\ne 0$ a. e. in $M$, of compact smooth Riemannian manifold $M$ onto Riemannian manifold $M'$ then the manifold $M'$ has finite geodesic diameter.}}


Chemical and forensic analysis of JFK assassination bullet lots: Is a second shooter possible?

Abstract: The assassination of President John Fitzgerald Kennedy (JFK) traumatized the nation. In this paper we show that evidence used to rule out a second assassin is fundamentally flawed. This paper discusses new compositional analyses of bullets reportedly to have been derived from the same batch as those used in the assassination. The new analyses show that the bullet fragments involved in the assassination are not nearly as rare as previously reported. In particular, the new test results are compared to key bullet composition testimony presented before the House Select Committee on Assassinations (HSCA). Matches of bullets within the same box of bullets are shown to be much more likely than indicated in the House Select Committee on Assassinations' testimony. Additionally, we show that one of the ten test bullets is considered a match to one or more assassination fragments. This finding means that the bullet fragments from the assassination that match could have come from three or more separate bullets. Finally, this paper presents a case for reanalyzing the assassination bullet fragments and conducting the necessary supporting scientific studies. These analyses will shed light on whether the five bullet fragments constitute three or more separate bullets. If the assassination fragments are derived from three or more separate bullets, then a second assassin is likely, as the additional bullet would not easily be attributable to the main suspect, Mr. Oswald, under widely accepted shooting scenarios [see Posner (1993), Case Closed, Bantam, New York].


Small index subgroups of the mapping class group

Abstract: We prove that the mapping class group of a closed oriented surface of genus $\rho \ge 3$ has no proper subgroup of index $\le 4 \rho +4$.


Non-existence of polar factorisations and polar inclusion of a vector-valued mapping

Abstract: This paper proves some results concerning the polar factorisation of an integrable vector-valued function u into the composition of the gradient of a convex function with a measure-preserving mapping. Not every integrable function has a polar factorisation; we extend the class of counterexamples. We introduce a generalisation: u has a polar inclusion if u(x) belongs to the subdifferential of the convex function at y for almost every pair (x,y) with respect to a measure-preserving plan. Given a regularity assumption, we show that such measure-preserving plans are exactly the minimisers of a Monge-Kantorovich optimisation problem.


Existence of positive definite noncoercive sums of squares

Abstract: Positive definite forms $f$ which are sums of squares are constructed to have the additional property that the members of any collection of forms whose squares sum to $f$ must share a nontrivial complex root.


Integration on valuation fields over local fields

Abstract: We present elements of a theory of translation-invariant integration, measure, and harmonic analysis on a valuation field with local field as residue field. This extends the work of Fesenko. Applications to zeta integrals for two-dimensional local fields are then considered.


Integration on product spaces and GL_n of a valuation field over a local field

Abstract: We present elements of a theory of translation-invariant integration on finite dimensional vector spaces and on GL_n over a valuation field with local field as residue field. We then discuss the case of an arbitrary algebraic group. This extends the work of Fesenko.


Fubini's theorem and non-linear change of variables over a two-dimensional local field

Abstract: We consider non-linear changes of variables and Fubini's theorem for certain integrals over a two-dimensional local field. An interesting example is presented in which imperfectness of a finite characteristic local field causes Fubini's theorem to unexpectedly fail. The relationship to ramification theory is discussed.


The ordered distribution of natural numbers on the square root spiral

Abstract: Natural numbers divisible by the same prime factor lie on defined spiral graphs which are running through the Square Root Spiral (also named as the Spiral of Theodorus or Wurzel Spirale or Einstein Spiral). Prime Numbers also clearly accumulate on such spiral graphs. And the square numbers 4, 9, 16, 25, 36,... form a highly three-symmetrical system of three spiral graphs, which divides the square-root-spiral into three equal areas. A mathematical analysis shows that these spiral graphs are defined by quadratic polynomials. Fibonacci number sequences also play a part in the structure of the Square Root Spiral. Fibonacci Numbers divide the Square Root Spiral into areas and angle sectors with constant proportions. These proportions are linked to the golden mean (or golden section), which behaves as a self-avoiding-walk-constant in the lattice-like structure of the square root spiral.


Neumann problems associated to nonhomogeneous differential operators in Orlicz--Sobolev spaces

Abstract: We study a nonlinear Neumann boundary value problem associated to a nonhomogeneous differential operator. Taking into account the competition between the nonlinearity and the bifurcation parameter, we establish sufficient conditions for the existence of nontrivial solutions in a related Orlicz--Sobolev space.


Harmonic functions on R-covered foliations and group actions on the circle

Abstract: Let (M, F) be a compact codimension-one foliated manifold whose leaves are equipped with Riemannian metrics, and consider continuous functions on M that are harmonic along the leaves of F . If every such function is constant on leaves we say that (M, F) has the Liouville property. Our main result is that codimension-one foliated bundles over compact negatively curved manifolds satisfy the Liouville property. Related results for R-covered foliations, as well as for discrete group actions and discrete harmonic functions, are also established.


Noncommutative Analogs of Monomial Symmetric Functions, Cauchy Identity, and Hall Scalar Product

Abstract: This paper introduces noncommutative analogs of monomial symmetric functions and fundamental noncommutative symmetric functions. The expansion of ribbon Schur functions in both of these basis is nonnegative. With these functions at hand, one can derive a noncommutative Cauchy identity as well as study a noncommutative scalar product implied by Cauchy identity. This scalar product seems be the noncommutative analog of Hall scalar product in the commutative theory.


Wrinkled fibrations on near-symplectic manifolds

Abstract: Motivated by the programmes initiated by Taubes and Perutz, we study the geometry of near-symplectic 4-manifolds, i.e., manifolds equipped with a closed 2-form which is symplectic outside a union of embedded 1-dimensional submanifolds, and broken Lefschetz fibrations on them. We present a set of four moves which allow us to pass from any given fibration to any other broken fibration which is deformation equivalent to it. Moreover, we study the change of the near-symplectic geometry under each of these moves. The arguments rely on the introduction of a more general class of maps, which we call wrinkled fibrations and which allow us to rely on classical singularity theory.Finally, we illustrate these constructions by showing how one can merge components of the zero-set of the near-symplectic form. We also disprove a conjecture of Gay and Kirby by showing that any achiral broken Lefschetz fibration can be turned into a broken Lefschetz fibration by applying a sequence of our moves.


Real and integral structures in quantum cohomology I: toric orbifolds

Abstract: We study real and integral structures in the space of solutions to the quantum differential equations. First we show that, under mild conditions, any real structure in orbifold quantum cohomology yields a pure and polarized tt^*-geometry near the large radius limit. Secondly, we use mirror symmetry to calculate the "most natural" integral structure in quantum cohomology of toric orbifolds. We show that the integral structure pulled back from the singularity B-model is described only in terms of topological data in the A-model; K-group and a characteristic class. Using integral structures, we give a natural explanation why the quantum parameter should specialize to a root of unity in Ruan's crepant resolution conjecture.


Chern classes in Deligne cohomology for coherent analytic sheaves

Abstract: In this article, we construct Chern classes in rational Deligne cohomology for coherent sheaves on a smooth complex compact manifold. We prove that these classes verify the functoriality property under pullbacks, the Whitney formula and the Grothendieck-Riemann-Roch theorem for projective morphisms between smooth complex compact manifolds.


On the Bakry-Emery criterion for linear diffusions and weighted porous media equations

Abstract: The goal of this paper is to give a non-local sufficient condition for generalized Poincar\'e inequalities, which extends the well-known Bakry-Emery condition. Such generalized Poincar\'e inequalities have been introduced by W. Beckner in the gaussian case and provide, along the Ornstein-Uhlenbeck flow, the exponential decay of some generalized entropies which interpolate between the $L^2$ norm and the usual entropy. Our criterion improves on results which, for instance, can be deduced from the Bakry-Emery criterion and Holley-Stroock type perturbation results. In a second step, we apply the same strategy to non-linear equations of porous media type. This provides new interpolation inequalities and decay estimates for the solutions of the evolution problem. The criterion is again a non-local condition based on the positivity of the lowest eigenvalue of a Schr\"odinger operator. In both cases, we relate the Fisher information with its time derivative. Since the resulting criterion is non-local, it is better adapted to potentials with, for instance, a non-quadratic growth at infinity, or to unbounded perturbations of the potential.


Large scale geometry of certain solvable groups

Abstract: In this paper we provide the final steps in the proof, announced by Eskin-Fisher-Whyte, of quasi-isometric rigidity of a class of non-nilpotent polycyclic groups. To this end, we prove a rigidity theorem on the boundaries of certain negatively curved homogeneous spaces and combine it with work of Eskin-Fisher-Whyte and Peng on the structure of quasi-isometries of certain solvable Lie groups.


On coherent systems of type (n,d,n+1) on Petri curves

Abstract: We study coherent systems of type $(n,d,n+1)$ on a Petri curve $X$ of genus $g\ge2$. We describe the geometry of the moduli space of such coherent systems for large values of the parameter $\alpha$. We determine the top critical value of $\alpha$ and show that the corresponding ``flip'' has positive codimension. We investigate also the non-emptiness of the moduli space for smaller values of $\alpha$, proving in many cases that the condition for non-emptiness is the same as for large $\alpha$. We give some detailed results for $g\le5$ and applications to higher rank Brill-Noether theory and the stability of kernels of evaluation maps, thus proving Butler's conjecture in some cases in which it was not previously known.


Representation theorems for backward doubly stochastic differential equations

Abstract: In this paper we study the class of backward doubly stochastic differential equations (BDSDEs, for short) whose terminal value depends on the history of forward diffusion. We first establish a probabilistic representation for the spatial gradient of the stochastic viscosity solution to a quasilinear parabolic SPDE in the spirit of the Feynman-Kac formula, without using the derivatives of the coefficients of the corresponding BDSDE. Then such a representation leads to a closed-form representation of the martingale integrand of BDSDE, under only standard Lipschitz condition on the coefficients.


On the cuspidality of pullbacks of Siegel Eisenstein series and applications to the Bloch-Kato conjecture

Abstract: Let $k > 3$ be an integer and $p$ a prime with $p > 2k-2$. Let $f$ be a newform of weight $2k-2$ and level 1 so that $f$ is ordinary at $p$ and $\bar{\rho}_{f}$ is irreducible. Under some additional hypotheses we prove that $ord_{p}(L_{alg}(k,f)) \leq ord_{p}(# S)$ where $S$ is the Pontryagin dual of the Selmer group associated to $\rho_{f} \otimes \epsilon^{1-k}$ with $\epsilon$ the $p$-adic cyclotomic character. We accomplish this by first constructing a congruence between the Saito-Kurokawa lift of $f$ and a non-CAP Siegel cusp form. Once this congruence is established, we use Galois representations to obtain the lower bound on the Selmer group.


Conway classification of alternating knots

Abstract: The alternating knots, links and twists projected on the $S_2$ sphere were identified with the phase space of a Hamiltonian dynamic system of one degree of freedom. The saddles of the system correspond to the crossings, the edges correspond to the stable and unstable manifolds connecting the saddles. Each face is then oriented in one of two different senses determined by the direction of these manifolds. This correspondence can be also realized between the knot and the Poincar\'e section of a two degrees of freedom integrable dynamical system. The crossings corresponding to unstable orbits, and the faces to foliated torus, around a stable orbit. The associated matrix to that connected graph was decomposed in two permutations. The separation was shown unique for knots not for links. The characteristic polynomial corresponding to some knot, link or twist families was explicitly computed in terms of Chebyschev polynomials. A classification of rational knots was formulated in terms of the first derivative of the polynomial of a knot computed in $x=2$, equal to the number of crossings of the knot multiplying the same number used previously by Conway for tabulation of knot properties. This leads to a classification of knots exemplified for the families having up to five ribbons. We subdivide the families of $N$ ribbons in subfamilies related to the prime knots of $N$ crossings.


Eta forms and determinant lines

Abstract: We show that there is a canonical construction of a zeta (Bismut-Quillen) connection on the determinant line bundle of a family of APS elliptic boundary problems and that it has curvature equal to the 2-form part of a relative eta form.


How to Create a New Integer Sequence

Abstract: There are several standard procedures used to create new sequences from a given sequence or from a given pair of sequences. In this paper I discuss the most popular of these procedures. For each procedure, I give a definition and provide examples based on three famous sequences: the natural numbers, the prime numbers and the Fibonacci numbers. I also add my thoughts on what makes a sequence interesting. My goal is to help my readers invent new sequences, differentiate interesting sequences from boring ones, and better understand sequences they encounter.


Non-commutative Schur-Horn theorems and extended majorization for hermitian matrices

Abstract: Let $\mathcal A\subseteq \mat$ be a unital $*$-subalgebra of the algebra $\mat$ of all $n\times n$ complex matrices and let $B$ be an hermitian matrix. Let $\U_n(B)$ denote the unitary orbit of $B$ in $\mat$ and let $\mathcal E_\mathcal A$ denote the trace preserving conditional expectation onto $\mathcal A$. We give an spectral characterization of the set $$ \mathcal E_\mathcal A(\U_n(B))=\{\mathcal E_\mathcal A(U^* B U): U\in \mat,\ \text{unitary matrix}\}.$$ We obtain a similar result for the contractive orbit of a positive semi-definite matrix $B$. We then use these results to extend the notions of majorization and submajorization between self-adjoint matrices to spectral relations that come together with extended (non-commutative) Schur-Horn type theorems.


Multigraded regularity and the Koszul property

Abstract: We give a criterion for the section ring of an ample line bundle to be Koszul in terms of multigraded regularity. We discuss an application to polytopal semigroup rings.


Large Deviations for Random Trees

Abstract: We consider large random trees under Gibbs distributions and prove a Large Deviation Principle (LDP) for the distribution of degrees of vertices of the tree. The LDP rate function is given explicitly. An immediate consequence is a Law of Large Numbers for the distribution of vertex degrees in a large random tree. Our motivation for this study comes from the analysis of RNA secondary structures.


On free profinite subgroups of free profinite monoids

Abstract: We answer a question of Margolis from 1997 by establishing that the maximal subgroup of the minimal ideal of a finitely generated free profinite monoid is a free profinite group. More generally if $\mathbf H$ is variety of finite groups closed under extension and containing $\mathbb Z/p\mathbb Z$ for infinitely may primes $p$, the corresponding result holds for free pro-$\bar{\mathbf H}$ monoids.


Subspace correction methods for total variation and $\ell_1-$minimization

Abstract: This paper is concerned with the numerical minimization of energy functionals in Hilbert spaces involving convex constraints coinciding with a semi-norm for a subspace. The optimization is realized by alternating minimizations of the functional on a sequence of orthogonal subspaces. On each subspace an iterative proximity-map algorithm is implemented via \emph{oblique thresholding}, which is the main new tool introduced in this work. We provide convergence conditions for the algorithm in order to compute minimizers of the target energy. Analogous results are derived for a parallel variant of the algorithm. Applications are presented in domain decomposition methods for singular elliptic PDE's arising in total variation minimization and in accelerated sparse recovery algorithms based on $\ell_1$-minimization. We include numerical examples which show efficient solutions to classical problems in signal and image processing.


A Realization of Measurable Sets as Limit Points

Abstract: Starting with a sigma finite measure on an algebra, we define a pseudometric and show how measurable sets from the Caratheodory Extension Theorem can be thought of as limit points of Cauchy sequences in the algebra.


A Cheerful Introduction to Forcing and the Continuum Hypothesis

Abstract: This is an introduction to the set-theoretic method of forcing, including its application in proving the independence of the Continuum Hypothesis from the Zermelo-Fraenkel axioms of set theory. I presuppose no particular mathematical background beyond some familiarity with set theory and mathematical logic - in particular, no algebra is presupposed, though it can be useful. The goal is to have a document that makes this material accessible to mathematics graduate students in all fields, and to philosophers with an interest in set theory and mathematical logic but no other mathematical background.


Irreducible plane sextics with large fundamental groups

Abstract: We compute the fundamental group of the complement of each irreducible sextic of weight eight or nine (in a sense, the largest groups for irreducible sextics), as well as of 169 of their derivatives (both of and not of torus type). We also give a detailed geometric description of sextics of weight eight and nine and of their moduli spaces and compute their Alexander modules; the latter are shown to be free over an appropriate ring.


Representation theory of the $\alpha$-determinant and zonal spherical functions

Abstract: We prove that the multiplicity of each irreducible component in the $\mathcal{U}(\mathfrak{gl}_n)$-cyclic module generated by the $l$-th power $\det^{(\alpha)}(X)^l$ of the $\alpha$-determinant is given by the rank of a matrix whose entries are given by a variation of the spherical Fourier transformation for $(\mathfrak{S}_{nl},\mathfrak{S}_l^n)$. Further, we calculate the matrix explicitly when $n=2$. This gives not only another proof of the result by Kimoto-Matsumoto-Wakayama (2007) but also a new aspect of the representation theory of the $\alpha$-determinants.


Lusztig's conjecture for finite classical groups with even characteristic

Abstract: The determination of scalars involved in Lusztig's conjecture for finite reductive groups $G(F_q)$ was achieved by Waldspurger in the case of symplectic groups or orthogonal groups, under the condition that $p,q$ are large enough. Here $p$ is the characteristic of the finite field $F_q$. In this paper, we determine the scalars in the case of symplectic groups with $p = 2$, by applying the theory of symmetric spaces over a finite field due to Kawanaka and Lusztig. We also obtain a partial result in the case of special orthogonal groups with $p = 2$.


What is a superrigid subgroup?

Abstract: This is an expository paper. It is well known that a linear transformation can be defined to have any desired action on a basis. From this fact, one can show that every group homomorphism from Z^k to R^d extends to a homomorphism from R^k to R^d, and we will see other examples of discrete subgroups H of connected groups G, such that the homomorphisms defined on $H$ can ("almost") be extended to homomorphisms defined on all of G. This is related to a very classical topic in geometry, the study of linkages.


On simultaneous rational approximations to a real number, its square, and its cube

Abstract: We provide an upper bound on the uniform exponent of approximation to a triple (xi, xi^2, xi^3) by rational numbers with the same denominator, valid for any transcendental real number xi. This upper bound refines a previous result of Davenport and Schmidt. As a consequence, we get a sharper lower bound on the exponent of approximation of such a number xi by algebraic integers of degree at most 4.


How to compute the Stanley depth of a monomial ideal

Abstract: Let $J\subset I$ be monomial ideals. We show that the Stanley depth of $I/J$ can be computed in a finite number of steps. We also introduce the $\fdepth$ of a monomial ideal which is defined in terms of prime filtrations and show that it can also be computed in a finite number of steps. In both cases it is shown that these invariants can be determined by considering partitions of suitable finite posets into intervals.


The $n$-Queens Problem in Higher Dimensions

Abstract: A well-known chessboard problem is that of placing eight queens on the chessboard so that no two queens are able to attack each other. (Recall that a queen can attack anything on the same row, column, or diagonal as itself.) This problem is known to have been studied by Gauss, and can be generalized to an (n \times n) board, where (n \geq 4). We consider this problem in $d$-dimensional chess spaces, where (d \geq 3), and obtain the result that in higher dimensions, $n$ queens do not always suffice (in any arrangement) to attack all board positions. Our methods allow us to obtain the first lower bound on the number of queens that are necessary to attack all positions in a $d$-dimensional chess space of size $n$, and further to show that for any $k$, there are higher-dimensional chess spaces in which not all positions can be attacked by (n^k) queens.


Conformal maps from a 2-torus to the 4-sphere

Abstract: We study the space of conformal immersions of a 2-torus into the 4-sphere. The moduli space of generalized Darboux transforms of such an immersed torus has the structure of a Riemann surface, the spectral curve. This Riemann surface arises as the zero locus of the determinant of a holomorphic family of Dirac type operators parameterized over the complexified dual torus. The kernel line bundle of this family over the spectral curve describes the generalized Darboux transforms of the conformally immersed torus. If the spectral curve has finite genus the kernel bundle can be extended to the compactification of the spectral curve and we obtain a linear 2-torus worth of algebraic curves in projective 3-space. The original conformal immersion of the 2-torus is recovered as the orbit under this family of the point at infinity on the spectral curve projected to the 4-sphere via the twistor fibration.


Real forms and finite order automorphisms of affine Kac-Moody algebras - an outline of a new approach

Abstract: We outline a new approach to classify real forms and automorphisms of finite order of affine Kac-Moody algebras.


$L^2$-stability of explicit schemes for incompressible Euler equations

Abstract: We present an original study on the numerical stabiliy of explicit schemes solving the incompressible Euler equations on an open domain with slipping boundary conditions. Relying on the skewness property of the non-linear term, we demonstrate that some explicit schemes are numerically stable for small perturbations under the condition $\delta t\leq C \delta x^{2r/(2r-1)}$ where $r$ is an integer, $\delta t$ the time step and $\delta x$ the space step.


Fixed points of circle actions on spaces with rational cohomology of $S^n V S^{2n} V S^{3n}$ or $P^2(n) V S^{3n}$

Abstract: Let $X$ be a finitistic space with its rational cohomology isomorphic to that of the wedge sum $P^2(n)\vee S^{3n} $ or $S^{n} \vee S^{2n}\vee S^{3n}$. We study continuous $\mathbb{S}^1$ actions on $X$ and determine the possible fixed point sets up to rational cohomology depending on whether or not $X$ is totally non-homologous to zero in $X_{\mathbb{S}^1}$ in the Borel fibration $X\hookrightarrow X_{\mathbb{S}^1} \longrightarrow B_{\mathbb{S}^1}$. We also give examples realizing the possible cases.