Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups

Abstract: We consider stochastic equations in Hilbert spaces with singular drift in the framework of [Da Prato, R\"ockner, PTRF 2002]. We prove a Harnack inequality (in the sense of [Wang, PTRF 1997]) for its transition semigroup and exploit its consequences. In particular, we prove regularizing and ultraboundedness properties of the transition semigroup as well as that the corresponding Kolmogorov operator has at most one infinitesimally invariant measure $\mu$ (satisfying some mild integrability conditions). Finally, we prove existence of such a measure $\mu$ for non-continuous drifts.


Functoriality of the BGG Category O

Abstract: This article aims to contribute to the study of algebras with triangular decomposition over a Hopf algebra, as well as the BGG Category O. We study functorial properties of O across various setups. The first setup is over a skew group ring, involving a finite group $\Gamma$ acting on a regular triangular algebra $A$. We develop Clifford theory for $A \rtimes \Gamma$, and obtain results on block decomposition, complete reducibility, and enough projectives. O is shown to be a highest weight category when $A$ satisfies one of the "Conditions (S)"; the BGG Reciprocity formula is slightly different because the duality functor need not preserve each simple module. Next, we turn to tensor products of such skew group rings; such a product is also a skew group ring. We are thus able to relate four different types of Categories O; more precisely, we list several conditions, each of which is equivalent in any one setup, to any other setup - and which yield information about O.


On universal estimates for binary renewal processes

Abstract: A binary renewal process is a stochastic process $\{X_n\}$ taking values in $\{0,1\}$ where the lengths of the runs of 1's between successive zeros are independent. After observing ${X_0,X_1,...,X_n}$ one would like to predict the future behavior, and the problem of universal estimators is to do so without any prior knowledge of the distribution. We prove a variety of results of this type, including universal estimates for the expected time to renewal as well as estimates for the conditional distribution of the time to renewal. Some of our results require a moment condition on the time to renewal and we show by an explicit construction how some moment condition is necessary.


Axiomatic framework for the BGG Category O

Abstract: The main goal of this paper is to show that a wide variety of infinite-dimensional algebras all share a common structure, including a triangular decomposition and a theory of weights. This structure allows us to define and study the BGG Category O, generalizing previous definitions of it. Having presented our axiomatic framework, we present sufficient conditions that guarantee finite length, enough projectives, and a block decomposition into highest weight categories. The framework is strictly more general than the usual theory of O; this is needed to accommodate (quantized or higher rank) infinitesimal Hecke algebras, in addition to semisimple Lie algebras and their quantum groups. We then present numerous examples, two families of which are studied in detail. These are quantum groups defined using not necessarily the root or weight lattices (for these, we study the center and central characters), and infinitesimal Hecke algebras.


On a length preserving curve flow

Abstract: In this paper, we consider a new length preserving curve flow for convex curves in the plane. We show that the global flow exists, the area of the region bounded by the evolving curve is increasing, and the evolving curve converges to the circle in C-infinity topology as t goes to infinity.


A characterisation of compact, fragmentable linear orders

Abstract: We give a characterisation of fragmentable, compact linearly order spaces. In particular, we show that if $K$ is a compact, fragmentable, linearly ordered space then $K$ is a Radon-Nikod\'{y}m compact. In addition, we obtain some corollaries in topology and renorming theory.


Growth Estimates for a Class of Subharmonic Functions in a Half Space

Abstract: A class of subharmonic functions represented by the modified kernels are proved to have the growth estimates u(x) =o(x_{n}^{1-alpha}|x|^{m+alpha})at infinity in the upper half space of Rn, which generalizes the growth properties of analytic functions and harmonic functions.


A Central Limit Theorem, and related results, for a two-color randomly reinforced urn

Abstract: We prove a Central Limit Theorem for the sequence of random compositions of a two-color randomly reinforced urn. As a consequence, we are able to show that the distribution of the urn limit composition has no point masses.


On transfer inequalities in Diophantine approximation, II

Abstract: We refine Khintchine Transference Principle which relates the measure of simultaneous rational approximation of an $n$ real numbers with the measure of linear independence of these $n$ numbers. Khintchine's inequalities are known to be optimal. However, they may be sharpened by taking into account two further uniform exponents.


On the Minimum Many-Valued Modal Logic over a Finite Residuated Lattice

Abstract: This paper deals with many-valued modal logics, based only on the necessity operator, over a residuated lattice. We focus on three basic classes, according to the accessibility relation, of Kripke frames: the full class of frames evaluated in the residuated lattice (and so defining the minimum modal logic), the ones evaluated in the idempotent elements and the ones evaluated in 0 and 1. We show how to expand an axiomatization, with canonical constants in the language, of a finite residuated lattice into one of the modal logic, for each one of the three basic classes of Kripke frames, over the very lattice. And we also give axiomatizations for the case of a finite MV chain but this time without canonical constants.


Homology stability for the special linear group of a field and Milnor-Witt K-theory

Abstract: Let F be a field of characteristic zero and let f(t,n) be the stabilization homomorphism from the n-th integral homology of SL(t,F) to the n-th homology of SL(t+1,F). We prove the following results: For all n, f(t,n) is an isomorphism if t is at least n+1, and is surjective for t=n, confirming a conjecture of C-H. Sah. Furthermore if n is odd, then f(n,n) is an isomorphism and when n is even the kernel of f(n,n) is the (n+1)st power of the fundamental ideal of the Witt Ring of the field.. If n is even, then the cokernel of f(n-1,n) is naturally isomorphic to the n-th Milnor-Witt K-group of F, MWK(n,F) and when n>2 is odd the cokernel of f(n-1,n) is the square of the nth Milnor K-group of F.


On the entropy and letter frequencies of powerfree words

Abstract: We review the recent progress in the investigation of powerfree words, with particular emphasis on binary cubefree and ternary squarefree words. Besides various bounds on the entropy, we provide bounds on letter frequencies and consider their empirical distribution obtained by an enumeration of binary cubefree words up to length 80.


Asymptotic Behaviour for a Class of Subharmonic Functions in a Half Space

Abstract: A class of subharmonic functions are proved to have the growth estimates $u(x)= o(x_n^{1-\frac{\alpha}{p}}|x|^{\frac{\gamma}{p}+\frac{n-1}{q}-n+\frac{\alpha}{p}})$ at infinity in the upper half space of ${\bf R}^{n}$, which generalizes the growth properties of analytic functions and harmonic functions.


Growth Estimates for a Class of Subharmonic Functions in a Half Plane

Abstract: A class of subharmonic functions represented by the modified kernels are proved to have the growth estimates $u(z)= o(y^{1-\alpha}|z|^{m+\alpha})$ at infinity in the upper half plane ${\bf C}_{+}$, which generalizes the growth properties of analytic functions and harmonic functions.


Maximal rationally connected fibrations and movable curves

Abstract: A well known result of Miyaoka asserts that a complex projective manifold is uniruled if its cotangent bundle restricted to a general complete intersection curve is not nef. Using the Harder-Narasimhan filtration of the tangent bundle, it can moreover be shown that the choice of such a curve gives rise to a rationally connected foliation of the manifold. In this note we show that, conversely, a movable curve can be found so that the maximal rationally connected fibration of the manifold may be recovered as a term of the associated Harder-Narasimhan filtration of the tangent bundle.


Morphisms fixing words associated with exchange of three intervals

Abstract: We consider words coding exchange of three intervals with permutation (3,2,1), here called 3iet words. Recently, a characterization of substitution invariant 3iet words was provided. We study the opposite question: what are the morphisms fixing a 3iet word? We reveal a narrow connection of such morphisms and morphisms fixing Sturmian words using the new notion of amicability.


Existence of weak solutions to the Cauchy problem of a semilinear wave equation with supercritical interior source and damping

Abstract: In this paper we show existence of finite energy solutions for the Cauchy problem associated with a semilinear wave equation with interior damping and supercritical source terms. The main contribution consists in dealing with super-supercritical source terms (terms of the order of $|u|^p$ with $p\geq 5$ in $n=3$ dimensions), an open and highly recognized problem in the literature on nonlinear wave equations.


On the existence of star products on quotient spaces of linear Hamiltonian torus actions

Abstract: We discuss BFV deformation quantization of singular symplectic quotient spaces in the special case of linear Hamiltonian torus actions. In particular, we show that the Koszul complex on the moment map of an effective linear Hamiltonian torus action is acyclic. We rephrase the nonpositivity condition of Arms, Gotay and Jennings for linear Hamiltonian torus actions. It follows that reduced spaces of such actions admit continuous star products.


Lois pr\'e-Lie en interaction

Abstract: D. Calaque, K. Ebrahimi-Fard and D. Manchon have recently defined a Hopf algebra by introducing a new coproduct on a commutative algebra of rooted forests. The space of primitive elements of the graded dual is endowed with a left pre-Lie product defined in terms of insertion of a tree inside another. In this work we prove a ``derivation'' relation between this pre-Lie structure and the left pre-Lie product defined by grafting.


A survey of combinatorial aspects in the topology of complex hyperplane arrangements

Abstract: We survey interactions between the topology and the combinatorics of complex hyperplane arrangements. Without claiming to be exhaustive, we examine in this setting combinatorial aspects of fundamental groups, associated graded Lie algebras, higher homotopy groups, cohomology rings, twisted homology with rank 1 complex coefficients, and Milnor fibers.


Regularizations of residue currents

Abstract: Under assumptions about complete intersection, we prove that Coleff-Herrera type currents satisfy a robust calculus in the sense that natural regularizations of such currents can be multiplied to yield regularizations of the Coleff-Herrera product of the currents.


Higher order energy decay rates for damped wave equations with variable coefficients

Abstract: Under appropriate assumptions the energy of wave equations with damping and variable coefficients $c(x)u_{tt}-\hbox{div}(b(x)\nabla u)+a(x)u_t =h(x)$ has been shown to decay. Determining the rate of decay for the higher order energies involving the $k$th order spatial and time derivatives has been an open problem with the exception of some sparse results obtained for $k=1,2,3$. We establish estimates that optimally relate the higher order energies with the first order energy by carefully analyzing the effects of linear damping. The results concern weighted (in time) and also pointwise (in time) energy decay estimates. We also obtain $L^\infty$ estimates for the solution $u$. As an application we compute explicit decay rates for all energies which involve the dimension $n$ and the bounds for the coefficients $a(x)$ and $b(x)$ in the case $c (x)=1$ and $h(x)=0.$


An overview of arithmetic motivic integration

Abstract: This is an attempt at an elementary exposition, with examples, of the theory of motivic integration developed by R. Cluckers and F. Loeser, with the view towards applications in representation theory of p-adic groups.


P-values for high-dimensional regression

Abstract: Assigning significance in high-dimensional regression is challenging. Most computationally efficient selection algorithms cannot guard against inclusion of noise variables. Asymptotically valid p-values are not available. An exception is a recent proposal by Wasserman and Roeder (2008) which splits the data into two parts. The number of variables is then reduced to a manageable size using the first split, while classical variable selection techniques can be applied to the remaining variables, using the data from the second split. This yields asymptotic error control under minimal conditions. It involves, however, a one-time random split of the data. Results are sensitive to this arbitrary choice: it amounts to a `p-value lottery' and makes it difficult to reproduce results. Here, we show that inference across multiple random splits can be aggregated, while keeping asymptotic control over the inclusion of noise variables. We show that the resulting p-values can be used for control of both family-wise error (FWER) and false discovery rate (FDR). In addition, the proposed aggregation is shown to improve power while reducing the number of falsely selected variables substantially.


Variation and Rough Path Properties of Local Times of L\'evy Processes

Abstract: In this paper, we will prove that the local time of a L\'evy process is of finite $p$-variation in the space variable in the classical sense, a.s. for any $p>2$, $t\geq 0$, if the L\'evy measure satisfies $\int_{R\setminus \{0\}}(|y|^{3\over 2}\wedge 1)n(dy)<\infty$, and is a rough path of roughness $p$ a.s. for any $2<p<3$ under a slightly stronger condition for the L\'evy measure. Then for any function $g$ of finite $q$-variation ($1\leq q <3$), we establish the integral $\int_{-\infty}^{\infty}g(x)dL_t^x$ as a Young integral when $1\leq q<2$ and a Lyons' rough path integral when $2\leq q<3$. We therefore apply these path integrals to extend the Tanaka-Meyer formula for a continuous function $f$ if $\nabla ^-f$ exists and is of finite $q$-variation when $1\leq q<3$, for both continuous semi-martingales and a class of L\'evy processes.


Master Equation and Perturbative Chern-Simons theory

Abstract: We extend the Chern-Simons perturbative invariant of Axelrod and Singer to non-acyclic connections. We construct a solution of the quantum master equation on the space of functions on the cohomology of the connection. We prove that this solution is well defined up to master homotopy. We discuss also invariants of links.


Geometry of canonical self-similar tilings

Abstract: We give several different geometric characterizations of the situation in which the parallel set $F_\epsilon$ of a self-similar set $F$ can be described by the inner $\epsilon$-parallel set $T_{-\epsilon}$ of the associated canonical tiling $\mathcal T$, in the sense of \cite{SST}. For example, $F_\epsilon=T_{-\epsilon} \cup C_\epsilon$ if and only if the boundary of the convex hull $C$ of $F$ is a subset of $F$, or if the boundary of $E$, the unbounded portion of the complement of $F$, is the boundary of a convex set. In the characterized situation, the tiling allows one to obtain a tube formula for $F$, i.e., an expression for the volume of $F_\epsilon$ as a function of $\epsilon$. On the way, we clarify some geometric properties of canonical tilings. Motivated by the search for tube formulas, we give a generalization of the tiling construction which applies to all self-affine sets $F$ having empty interior and satisfying the open set condition. We also characterize the relation between the parallel sets of $F$ and these tilings.


Test Functions in Constrained Interpolation

Abstract: We give a set of test functions for the interpolation problem on $H_1^\infty$ , the constrained interpolation problem studied by Davidson, Paulsen, Raghupathi and Singh. We show that this set of test functions is minimal.


Splice diagram determining singularity links and universal abelian covers

Abstract: To a rational homology sphere graph manifold one can associate a weighted tree invariant called splice diagram. In this article we prove a sufficient numerical condition on the splice diagram for a graph manifold to be a singularity link. We also show that if two manifolds have the same splice diagram, then their universal abelian covers are homeomorphic. To prove the last theorem we have to generalize our notions to orbifolds.


Plumbers' knots

Abstract: We construct a new type of geometric knot theory, plumbers' knots, and solve the problems of distinguishing and enumerating such knots at a fixed level of complexity. (v2) Minor edits, added theorem 3.18. (v3) Substantial revisions, essentially completely rewritten in places.


Asymptotics of solutions of the wave equation on de Sitter-Schwarzschild space

Abstract: Solutions to the wave equation on de Sitter-Schwarzschild space with smooth initial data on a Cauchy surface are shown to decay exponentially to a constant at temporal infinity, with corresponding uniform decay on the appropriately compactified space.


Complete intersection points on general surfaces in $\PP^3$

Abstract: In this paper we consider the existence of complete intersection points of type $(a,b,c)$, on the generic degree $d$ surface of $\PP^3$. For any choice of $a, b, c$ we resolve the existence question asymptotically, i.e. for all $d \gg 0$. For small values of $a, b, c$ we resolve the existence problem completely.


Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds

Abstract: We obtain an asymptotic formula for the number of circles of curvature at most T in any given bounded Apollonian circle packing. For an integral packing, we obtain the upper bounds for the number of circles with prime curvature as well as of pairs of circles with prime curvatures, which are sharp up constant multiples. The main ingredient of our proof is the effective equidistribution of expanding horospheres on geometrically finite hyperbolic 3-manifolds under the assumption that the critical exponent of its fundamental group exceeds one.


The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells

Abstract: Using the notion of Gamma-convergence, we discuss the limiting behavior of the 3d nonlinear elastic energy for thin elliptic shells, as their thickness h converges to zero, under the assumption that the elastic energy of deformations scales like $h^\beta$ with $2<\beta<4$. We establish that, for the given scaling regime, the limiting theory reduces to the linear pure bending. Two major ingredients of the proofs are: the density of smooth infinitesimal isometries in the space of $W^{2,2}$ first order infinitesimal isometries, and a result on matching smooth infinitesimal isometries with exact isometric immersions on smooth elliptic surfaces.


Asymptotic analysis of a family of polynomials associated with the inverse error function

Abstract: We analyze the sequence of polynomials defined by the differential-difference equation $P_{n+1}(x)=P_{n}^{\prime}(x)+x(n+1)P_{n}(x)$ asymptotically as $n\to\infty$. The polynomials $P_{n}(x)$ arise in the computation of higher derivatives of the inverse error function $\operatorname{inverf}(x)$. We use singularity analysis and discrete versions of the WKB and ray methods and give numerical results showing the accuracy of our formulas.


On the cohomology of a simple normal crossings divisor

Abstract: We establish a formula which decomposes the cohomologies of various sheaves on a simple normal crossings divisor (SNC) $D$ in terms of the simplicial cohomologies of the dual complex $\Delta(D)$ with coefficients in a presheaf of vector spaces. This presheaf consists precisely of the corresponding cohomology data on the components of $D$ and on their intersections. We use this formula to give a Hodge decomposition for SNC divisors and investigate the toric setting. We also conjecture the existence of such a formula for effective non-reduced divisors with SNC support, and show that this would imply the vanishing of the higher simplicial cohomologies of the dual complex associated to a resolution of an isolated rational singularity.


Solutions of Polynomial Systems Derived from the Steady Cavity Flow Problem

Abstract: We propose a general algorithm to enumerate all solutions of a zero-dimensional polynomial system with respect to a given cost function. The algorithm is developed and is used to study a polynomial system obtained by discretizing the steady cavity flow problem in two dimensions. The key technique on which our algorithm is based is to solve polynomial optimization problems via sparse semidefinite programming relaxations (SDPR), which has been adopted successfully to solve reaction-diffusion boundary value problems recently. The cost function to be minimized is derived from discretizing the fluid's kinetic energy. The enumeration algorithm's solutions are shown to converge to the minimal kinetic energy solutions for SDPR of increasing order. We demonstrate the algorithm with SDPR of first and second order on polynomial systems for different scenarios of the cavity flow problem and succeed in deriving the $k$ smallest kinetic energy solutions. The question whether these solutions converge to solutions of the steady cavity flow problem is discussed, and we pose a conjecture for the minimal energy solution for increasing Reynolds number.


The endpoint case of the Bennett-Carbery-Tao multilinear Kakeya conjecture

Abstract: Bennett, Carbery, and Tao formulated an n-linear analogue of the Kakeya conjecture in R^n. They proved the conjecture except for the endpoint case. We prove the endpoint case.


On the tunnel number and the Morse-Novikov number of knots

Abstract: We prove that the Morse-Novikov number of a link L in a 3-sphere is less than or equal to twice the tunnel number of L.


Solving a characteristic Cauchy problem

Abstract: In this paper we give a meaning to the nonlinear characteristic Cauchy problem for the Wave Equation in base form by replacing it by a family of non-characteristic problems in an appropriate algebra of generalized functions. We prove existence of a solution and we precise how it depends on the choice made. We also check that in the classical case (non-characteristic) our new solution coincides with the classical one.


Minoration du rang des courbes elliptiques sur les corps de classes de Hilbert

Abstract: Soit $E/\BmQ$ une courbe elliptique. Soit $D<0$ un discriminant fondamental suffisamment grand. Si $E(\bar{\BmQ})$ contient des points de Heegner de discriminant $D$, ces points engendrent un sous-groupe dont le rang est sup\'erieur \`a $\pabs{D}^{0.0009}$. Ce r\'esultat est en accord avec la conjecture de Birch et Swinnerton-Dyer. --- Let $E/\BmQ$ be an elliptic curve. Let $D<0$ be a sufficiently large fundamental discriminant. If $E(\bar{\BmQ})$ contains Heegner points of discriminant $D$, these points generate a subgroup of rank greater than $\pabs{D}^{0.0009}$. This result is in agreement with the conjecture of Birch and Swinnerton-Dyer.


Oriented bivariant theories, I

Abstract: In 1981 W. Fulton and R. MacPherson introduced the notion of bivariant theory (BT), which is a sophisticated unification of covariant theories and contravariant theories. This is for the study of singular spaces. In 2001 M. Levine and F. Morel introduced the notion of algebraic cobordism, which is a universal oriented Borel-Moore functor with products (OBMF) of geometric type, in an attempt to understand better V. Voevodsky's (higher) algebraic cobordism. In this paper we introduce a notion of oriented bivariant theory (OBT), a special case of which is nothing but the oriented Borel-Moore functor with products. The present paper is a first one of the series to try to understand Levine-Morel's algebraic cobordism from a bivariant-theoretical viewpoint, and its first step is to introduce OBT as a unification of BT and OBMF.


Supersymmetric field theories and cohomology

Abstract: This is the Ph.D. dissertation of the author. The project has been motivated by the conjecture that the Hopkins-Miller tmf spectrum can be described in terms of `spaces' of conformal field theories. In this dissertation, spaces of field theories are constructed as classifying spaces of categories whose objects are certain types of field theories. If such a category has a symmetric monoidal structure and its components form a group, by work of Segal, its classifying space is an infinite loop space and defines a cohomology theory. This has been carried out for two classes of field theories: (i) For each integer n, there is a category SEFT_n whose objects are the Stolz-Teichner (1|1)-dimensional super Euclidean field theories of degree n. It is proved that the classifying space |SEFT_n| represents degree-n K or KO cohomology, depending on the coefficients of the field theories. (ii) For each integer n, there is a category AFT_n whose objects are a kind of (2|1)-dimensional field theories called `annular field theories,' defined using supergeometric versions of circles and annuli only. It is proved that the classifying space |AFT_n| represents the degree-n elliptic cohomology associated with the Tate curve. To the author's knowledge, this is the first time the definitions of low-dimensional supersymmetric field theories are given in full detail.


Products of longitudinal pseudodifferential operators on flag varieties

Abstract: Associated to each set $S$ of simple roots for $SL(n,\mathbb{C})$ is an equivariant fibration $X\to X_S$ of the space $X$ of complete flags of $\mathbb{C}^n$. To each such fibration we associate an algebra $J_S$ of operators on $L^2(X)$ which contains, in particular, the longitudinal pseudodifferential operators of negative order tangent to the fibres. These form a lattice of operator ideals whose common intersection is the compact operators. As a consequence, the product of fibrewise smoothing operators (for instance) along the fibres of two such fibrations, $X\to X_S$ and $X\to X_T$, is a compact operator if $S\cup T$ is the full set of simple roots. The construction uses noncommutative harmonic analysis, and hinges upon a representation theoretic property of subgroups of SU(n), which may be described as `essential orthogonality of subrepresentations'.


Homological Stability among Moduli Spaces of Holomorphic Curves in Complex Projective Space

Abstract: The primary goal of this paper is to find a homotopy theoretic approximation to moduli spaces of holomorphic maps Riemann surfaces into complex projective space. There is a similar treatment of a partial compactification of these moduli spaces of consisting of irreducible stable maps in the sense of Gromov-Witten theory. The arguments follow those from a paper of G. Segal on the topology of the space of rational functions.


Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison

Abstract: It is now established that under quite general circumstances, including in models with jumps, the existence of a solution to a reflected BSDE is guaranteed under mild conditions, whereas the existence of a solution to a doubly reflected BSDE is essentially equivalent to the so-called Mokobodski condition. As for uniqueness of solutions, this holds under mild integrability conditions. However, for practical purposes, existence and uniqueness are not enough. In order to further develop these results in Markovian set-ups, one also needs a (simply or doubly) reflected BSDE to be well posed, in the sense that the solution satisfies suitable bound and error estimates, and one further needs a suitable comparison theorem. In this paper, we derive such estimates and comparison results. In the last section, applicability of the results is illustrated with a pricing problem in finance.


Horizontal normal map on the Heisenberg group

Abstract: We investigate the notion of H-subdifferential and H-normal map of a function on the Heisenberg group, based on its sub-Riemannian structure. In particular, a characterization of the convexity of a function is given via the nonemptiness of the H-subdifferential at every point.


Geometric Cobordism Categories

Abstract: In this paper we study cobordism categories consisting of manifolds which are endowed with geometric structure. Examples of such geometric structures include symplectic structures, flat connections on principal bundles, and complex structures along with a holomorphic map to a target complex manifold. A general notion of "geometric structure" is defined using sheaf theoretic constructions. Our main theorem is the identification of the homotopy type of such cobordism categories in terms of certain Thom spectra. This extends work of Galatius-Madsen-Tillmann-Weiss who identify the homotopy type of cobordism categories of manifolds with fiberwise structures on their tangent bundles. Interpretations of the main theorem are discussed which have relevance to topological field theories, moduli spaces of geometric structures, and h-principles. Applications of the main theorem to various examples of interest in geometry, particularly holomorphic curves, are elaborated upon.


High-dimensional stochastic optimization with the generalized Dantzig estimator

Abstract: We propose a generalized version of the Dantzig selector. We show that it satisfies sparsity oracle inequalities in prediction and estimation. We consider then the particular case of high-dimensional linear regression model selection with the Huber loss function. In this case we derive the sup-norm convergence rate and the sign concentration property of the Dantzig estimators under a mutual coherence assumption on the dictionary.


Spaces of operator-valued functions measurable with respect to the strong operator topology

Abstract: Let $X$ and $Y$ be Banach spaces and $(\Omega,\Sigma,\mu)$ a finite measure space. In this note we introduce the space $L^p[\mu;L(X,Y)]$ consisting of all (equivalence classes of) functions $\Phi:\Omega \mapsto L(X,Y)$ such that $\omega \mapsto \Phi(\omega)x$ is strongly $\mu$-measurable for all $x\in X$ and $\omega \mapsto \Phi(\omega)f(\omega)$ belongs to $L^1(\mu;Y)$ for all $f\in L^{p'}(\mu;X)$, $1/p+1/p'=1$. We show that functions in $L^p[\mu;\L(X,Y)]$ define operator-valued measures with bounded $p$-variation and use these spaces to obtain an isometric characterization of the space of all $L(X,Y)$-valued multipliers acting boundedly from $L^p(\mu;X)$ into $L^q(\mu;Y)$, $1\le q< p<\infty$.


On the SU(2,1) representation space of the Brieskorn homology spheres

Abstract: In this paper, we give a parameterization of the SU(2,1) representation space of the Brieskorn homology spheres using the trace coordinates. As applications, we give an example which shows that the orbifold Toledo invariant in \cite{krebs} does not distinguish the connected components of the PU(2,1) representation space.


Configuration of nilpotent groups and isomorphism

Abstract: The concept of configuration was first introduced by Rosenblatt and Willis to give a condition for amenability of groups. We show that if $G_1$ and $G_2$ have the same configuration sets and $H_1$ is a normal subgroup of $G_1$ with abelian quotient, then there is a normal subgroup $H_2$ of $G_2$ such that $\frac{G_1}{H_1}\cong\frac{G_2}{H_2}.$ Also configuration of FC-groups and isomorphism is studied.


General branching processes in discrete time as random trees

Abstract: The simple Galton--Watson process describes populations where individuals live one season and are then replaced by a random number of children. It can also be viewed as a way of generating random trees, each vertex being an individual of the family tree. This viewpoint has led to new insights and a revival of classical theory. We show how a similar reinterpretation can shed new light on the more interesting forms of branching processes that allow repeated bearings and, thus, overlapping generations. In particular, we use the stable pedigree law to give a transparent description of a size-biased version of general branching processes in discrete time. This allows us to analyze the $x\log x$ condition for exponential growth of supercritical general processes as well as relation between simple Galton--Watson and more general branching processes.


Lower Order Terms for the One-Level Density of Elliptic Curve L-Functions

Abstract: It is believed that, in the limit as the conductor tends to infinity, correlations between the zeros of elliptic curve $L$-functions averaged within families follow the distribution laws of the eigenvalues of random matrices drawn from the orthogonal group. For test functions with restricted support, this is known to be the true for the one- and two-level densities of zeros within the families studied to date. However, for finite conductor Miller's experimental data reveal an interesting discrepancy from these limiting results. Here we use the L-functions ratios conjectures to calculate the 1-level density for the family of even quadratic twists of an elliptic curve L-function for large but finite conductor. This gives a formula for the leading and lower order terms up to an error term that is conjectured to be significantly smaller. The lower order terms explain many of the features of the zero statistics for relatively small conductor and model the very slow convergence to the infinite conductor limit. However, our main observation is that they do not capture the behaviour of zeros in the important region very close to the critical point and so do not explain Miller's discrepancy. This therefore implies that a more accurate model for statistics near to this point needs to be developed.


Alexander-equivalent Zariski pairs of irreducible sextics

Abstract: The existence of Alexander-equivalent Zariski pairs dealing with irreducible curves of degree 6 was proved by A. Degtyarev. However, up to now, no explicit example of such a pair was available (only the existence was known). In this paper, we construct the first concrete example.


Bifurcation of critical periods from Pleshkan's isochrones

Abstract: Pleshkan proved in 1969 that, up to a linear transformation and a constant rescaling of time, there are four isochrones in the family of cubic centers with homogeneous nonlinearities $\mathscr C_3.$ In this paper we prove that if we perturb any of these isochrones inside $\mathscr C_3,$ then at most two critical periods bifurcate from its period annulus. Moreover we show that, for each $k=0,1,2,$ there are perturbations giving rise to exactly $k$ critical periods. As a byproduct, we obtain a partial result for the analogous problem in the family of quadratic centers $\mathscr C_2.$ Loud proved in 1964 that, up to a linear transformation and a constant rescaling of time, there are four isochrones in $\mathscr C_2.$ We prove that if we perturb three of them inside $\mathscr C_2,$ then at most one critical period bifurcates from its period annulus. In addition, for each $k=0,1,$ we show that there are perturbations giving rise to exactly $k$ critical periods. The quadratic isochronous center that we do not consider displays some peculiarities that are discussed at the end of the paper.


Coxeter Elements and Root Bases

Abstract: Let g be a Lie algebra of type A,D,E with fixed Cartan subalgebra h, root system R and Weyl group W. We show that a choice of Coxeter element C gives a root basis for g. Moreover we show that this root basis gives a purely combinatorial construction of g, where root vectors correspond to vertices of a certain quiver $Gammahat$, and show that with respect to this basis the structure constants of the Lie bracket are given by paths in $Gammahat$. This construction is then related to the constructions of Ringel and Peng and Xiao.


Transformations birationnelles de petit degr\'e

Abstract: Since the end of the XIXth century, we know that each birational map of the complex projective plane is the product of a finite number of quadratic birational maps of the projective plane; this motivates our work which essentially deals with these quadratic maps. We establish algebraic properties such as the classification of one parameter groups of quadratic birational maps or the smoothness of the set of quadratic birational maps in the set of rational maps. We prove that a finite number of generic quadratic birational maps generates a free group. We show that if f is a quadratic birational map or an automorphism of the projective plane, the normal subgroup generated by f is the full group of birational maps of the projective plane, which implies that this group is perfect. We study some dynamical properties: following an idea of Guillot, we translate some invariants for foliations in our context, in particular we obtain that if two generic quadratic birational maps are birationally conjugated, then they are conjugated by an automorphism of the projective plane. We are also interested in the presence of "invariant objects": curves, foliations, fibrations. Then follows a more experimental part: we draw orbits of quadratic birational maps with real coefficients and sets analogous to Julia sets for polynomials of one variable. We study birational maps of degree 3 and, by considering the different possible configurations of the exceptional curves, we give the "classification" of these maps. We can deduce from this that the set of the birational maps of degree 3 exactly is irreducible, in fact rationally connected.


A continuous semigroup of notions of independence between the classical and the free one

Abstract: In this paper, we investigate a continuous family of notions of independence which interpolates between the classical and free ones for non-commutative random variables. These notions are related to the liberation process introduced by D. Voiculescu. To each notion of independence correspond new convolutions of probability measures, for which we establish formulae and of which we compute simple examples. We prove that there exists no reasonable analogue of classical and free cumulants associated to these notions of independence.


On the Riesz Basis Property of the Eigen- and Associated Functions of Periodic and Antiperiodic Sturm-Liouville Problems

Abstract: The paper deals with the Sturm-Liouville operator $$ Ly=-y^{\prime\prime}+q(x)y,\qquad x\in\lbrack0,1], $$ generated in the space $L_{2}=L_{2}[0,1]$ by periodic or antiperiodic boundary conditions. Several theorems on Riesz basis property of the root functions of the operator $L$ are proved. One of the main results is the following. \textsl{Let $q$ belong to Sobolev space $W_{1}^{p}[0,1]$ with some integer $p\geq0$ and satisfy the conditions $q^{(k)}(0)=q^{(k)}(1)=0$ for $0\leq k\leq s-1$, where s}$\leq p.$ \textsl{Let the functions $Q$ and $S$ be defined by the equalities $Q(x)=\int_{0}^{x}q(t) dt, S(x)=Q^{2}(x)$ and let $q_{n}%, Q_{n},S_{n}$ be the Fourier coefficients of $q,Q,S$ with respect to the trigonometric system $\{e^{2\pi inx}\}_{-\infty}^{\infty}$. Assume that the sequence $q_{2n}-S_{2n}+2Q_{0}Q_{2n}$ decreases not faster than the powers $n^{-s-2}$. Then the system of eigen and associated functions of the operator $L$ generated by periodic boundary conditions forms a Riesz basis in the space $L_{2}[0,1]$ (provided that the eigenfunctions are normalized) if and only if the condition $$ q_{2n}-S_{2n}+Q_{0}Q_{2n}\asymp q_{-2n}-S_{-2n}+2Q_{0}Q_{-2n},\quad n>1, $$ holds.


F_p-repr\'esentations semi-stables

Abstract: Torsion semi-stable representations can be constructed and studied using Breuil modules. In this paper, we define the notion of pylonet and prove that some categories of Breuil modules naturally define pylonets. As a consequence, we are able to define full subcategories of Breuil's categories with very nice properties (in particular, they are abelian). In a second part of this work, we try to make very explicit some abstract constructions coming from the general theory of pylonets (developped earlier in the paper). These explicitations should be very useful to make computations with torsion semi-stable Galois representations.


Effective algebraic degeneracy

Abstract: We prove that any nonconstant entire holomorphic curve from the complex line C into a projective algebraic hypersurface X = X^n in P^{n+1}(C) of arbitrary dimension n (at least 2) must be algebraically degenerate provided X is generic if its degree d = deg(X) satisfies the effective lower bound: d larger than or equal to n^{{(n+1)}^{n+5}}.


A supplement to a theorem of Merker and Porten: a short proof of Hartogs' extension theorem for $(n-1)$-complete complex spaces

Abstract: We give a short proof for the Hartogs's extension theorem on (n-1)-complete complex spaces.


Counting closed geodesics in Moduli space

Abstract: We compute the asymptotics, as R tends to infinity, of the number of closed geodesics in Moduli space of length at most R, or equivalently the number of pseudo-Anosov elements of the mapping class group of translation length at most R.


Sturm and Sylvester algorithms revisited via tridiagonal determinantal representations

Abstract: First, we show that Sturm algorithm and Sylvester algorithm, which compute the number of real roots of a given univariate polynomial, lead to two dual tridiagonal determinantal representations of the polynomial. Next, we show that the number of real roots of a polynomial given by a tridiagonal determinantal representation is greater than the signature of this representation.


Mather invariants in groups of piecewise-linear homeomorphisms

Abstract: We describe the relation between two characterizations of conjugacy in groups of piecewise-linear homeomorphisms, discovered by Brin and Squier in [2] and Kassabov and Matucci in [5]. Thanks to the interplay between the techniques, we produce a simplified point of view of conjugacy that allows us to easily recover centralizers and lends itself to generalization.


Trees of cylinders and canonical splittings

Abstract: Let T be a tree with an action of a finitely generated group G. Given a suitable equivalence relation on the set of edge stabilizers of T (such as commensurability, co-elementarity in a relatively hyperbolic group, or commutation in a commutative transitive group), we define a tree of cylinders T_c. This tree only depends on the deformation space of T; in particular, it is invariant under automorphisms of G if T is a JSJ splitting. We thus obtain Out(G)-invariant cyclic or abelian JSJ splittings. Furthermore, T_c has very strong compatibility properties (two trees are compatible if they have a common refinement).


On Quasiconvexity and Relative Hyperbolic Structures

Abstract: Let G be a group which is hyperbolic relative to a collection of subgroups A, and it is also hyperbolic relative to a collection of subgroups B. Suppose that the collection A contains B. We characterize, for subgroups of G, when quasicovexity relative to A implies quasiconvexity relative to B. We also show that quasiconvexity relative to B implies quasiconvexity relative to A. Some applications are discussed.


Scott and Swarup's regular neighbourhood as a tree of cylinders

Abstract: Let G be a finitely presented group. Scott and Swarup have constructed a canonical splitting of G which encloses all almost invariant sets over virtually polycyclic subgroups of a given length. We give an alternative construction of this regular neighbourhood, by showing that it is the tree of cylinders of a JSJ splitting.


Flag Paraproducts

Abstract: We describe the theory of "flag paraproducts" and their relationship with the field of differential equations.


On the Safe Use of Inconsistent Mathematics

Abstract: A method is presented for using the consistent part of inconsistent axiomatic systems.


Regularity of smooth curves in biprojective spaces

Abstract: Maclagan and Smith \cite{MaclaganSmith} developed a multigraded version of Castelnuovo-Mumford regularity. Based on their definition we will prove in this paper that for a smooth curve $C\subseteq \P^a\times\P^b$ $(a, b\geq 2)$ of bidegree $(d_1,d_2)$ with nondegenerate birational projections the ideal sheaf $\mathcal{I}_{C|\P^a\times\P^b}$ is $(d_2-b+1,d_1-a+1)$-regular. We also give an example showing that in some cases this bound is the best possible.


Exceptional divisors which are not uniruled belong to the image of the Nash map

Abstract: We prove that, if X is a variety over an uncountable algebraically closed field k of characteristic zero, then any irreducible exceptional divisor E on a resolution of singularities of X which is not uniruled, belongs to the image of the Nash map, i.e. corresponds to an irreducible component of the space of arcs on X centered in Sing X. This reduces the Nash problem of arcs to understanding which uniruled essential divisors are in the image of the Nash map, more generally, how to determine the uniruled essential divisors from the space of arcs.


Asymptotic stability of oscillations of two-bodies vibrating screen with one-sided obstacle without clearances

Abstract: We prove asymptotic stability of periodic oscillations in two-bodies vibrating screen model under assumption that the frequencies w1 and w2 of the generating system (without obstacle and periodic driving) satisfy the assumption w1:w2=1:2. We also assume that the frequency of the external periodic driving equals to w1. These settings correspond to nonlinear resonance which is a well known phenomenon in industrial implementation of the vibrating screen. The justification is performed over nonsmooth analog of the second Bogolyubov's theorem proposed by the author in his previous papers. It is rigorously proven that the periodic oscillations obtained have two frequencies, in contrast with the case when the obtacle is absent.


Finite Square Lattice Vertex Cover by a Baseline Set Defined With a Minimum Sublattice

Abstract: Each straight infinite line defined by two vertices of a finite square point lattice contains (covers) these two points and a - possibly empty - subset of points that happen to be collinear to these. This work documents vertex subsets of minimum order such that the sum of the infinite straight lines associated with the edges of their complete subgraph covers the entire set of vertices (nodes). This is an abstraction to the problem of sending a light signal to all stations (receivers) in a square array with a minimum number of stations also equipped with transmitters to redirect the light to other transmitters.


Points in a triangle forcing small triangles

Abstract: An old theorem of Alexander Soifer's is the following: Given five points in a triangle of unit area, there must exist some three of them which form a triangle of area 1/4 or less. It is easy to check that this is not true if "five" is replaced by "four", but can the theorem be improved in any other way? We discuss in this article two different extensions of the original result. First, we allow the value of "small", 1/4, to vary. In particular, our main result is to show that given five points in a triangle of unit area, then there must exist some three of them determining a triangle of area 6/25 or less. Second, we put bounds on the minimum number of small triangles determined by n points in a triangle, and make a conjecture about the asymptotic right answer as n tends to infinity.


A Family of Runge-Kutta Methods with Zero Phase-Lag and Derivatives for the Numerical Solution of the Schr\"odinger Equation and Related Problems

Abstract: We construct a family of two new optimized explicit Runge-Kutta methods with zero phase-lag and derivatives for the numerical solution of the time-independent radial Schr\"odinger equation and related ordinary differential equations with oscillating solutions. The numerical results show the superiority of the new technique of nullifying both the phase-lag and its derivatives.


Two Optimized Symmetric Eight-Step Implicit Methods for Initial-Value Problems with Oscillating Solutions

Abstract: In this paper we present two optimized eight-step symmetric implicit methods with phase-lag order ten and infinite (phase-fitted). The methods are constructed to solve numerically the radial time-independent Schr\"odinger equation with the use of the Woods-Saxon potential. They can also be used to integrate related IVPs with oscillating solutions such as orbital problems. We compare the two new methods to some recently constructed optimized methods from the literature. We measure the efficiency of the methods and conclude that the new method with infinite order of phase-lag is the most efficient of all the compared methods and for all the problems solved.


A New Methodology for the Development of Numerical Methods for the Numerical Solution of the Schr\"odinger Equation

Abstract: In the present paper we introduce a new methodology for the construction of numerical methods for the approximate solution of the one-dimensional Schr\"odinger equation. The new methodology is based on the requirement of vanishing the phase-lag and its derivatives. The efficiency of the new methodology is proved via error analysis and numerical applications.


Commutative association schemes

Abstract: Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the "commutative case," has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, cometric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger algebra in Delsarte Theory as a central topic for future work.


High Order Multistep Methods with Improved Phase-Lag Characteristics for the Integration of the Schr\"odinger Equation

Abstract: In this work we introduce a new family of twelve-step linear multistep methods for the integration of the Schr\"odinger equation. The new methods are constructed by adopting a new methodology which improves the phase lag characteristics by vanishing both the phase lag function and its first derivatives at a specific frequency. This results in decreasing the sensitivity of the integration method on the estimated frequency of the problem. The efficiency of the new family of methods is proved via error analysis and numerical applications.


On certain diophantine equations related to triangular and tetrahedral numbers

Abstract: In this paper we give solutions of certain diophantine equations related to triangular and tetrahedral numbers and propose several problems connected with these numbers. The material of this paper was presented in part at the 11th International Workshop for Young Mathematicians - NUMBER THEORY, Krak\'{o}w, 14th-20th september 2008.


High Order Phase Fitted Multistep Integrators for the Schr\"odinger Equation with Improved Frequency Tolerance

Abstract: In this work we introduce a new family of 14-steps linear multistep methods for the integration of the Schr\"odinger equation. The new methods are phase fitted but they are designed in order to improve the frequency tolerance. This is achieved by eliminating the first derivatives of the phase lag function at the fitted frequency forcing the phase lag function to be '\textit{flat}' enough in the neighbor of the fitted frequency. The efficiency of the new family of methods is proved via error analysis and numerical applications.


Zero Dispersion and Zero Dissipation Implicit Runge-Kutta Methods for the Numerical Solution of Oscillating IVPs

Abstract: In this paper we present two new methods based on an implicit Runge-Kutta method Gauss which is of algebraic order fourth and has two stages: the first one has zero dispersion and the second one has zero dispersion and zero dissipation. The efficiency of these methods is measured while integrating the radial Schr\"odinger equation and other well known initial value problems.


A Phase-Fitted Runge-Kutta-Nystr\"om method for the Numerical Solution of Initial Value Problems with Oscillating Solutions

Abstract: A new Runge-Kutta-Nystr\"om method, with phase-lag of order infinity, for the integration of second-order periodic initial-value problems is developed in this paper. The new method is based on the Dormand and Prince Runge-Kutta-Nystr\"om method of algebraic order four\cite{pa}. Numerical illustrations indicate that the new method is much more efficient than the classical one.


Counting arithmetic lattices and surfaces

Abstract: We give estimates on the number $AL_H(x)$ of arithmetic lattices $\Gamma$ of covolume at most $x$ in a simple Lie group $H$. In particular, we obtain a first concrete estimate on the number of arithmetic 3-manifolds of volume at most $x$. Our main result is for the classical case $H=PSL(2,R)$ where we compute the limit of $\log AL_H(x) / x\log x$ when $x\to\infty$. The proofs use several different techniques: geometric (bounding the number of generators of $\Gamma$ as a function of its covolume), number theoretic (bounding the number of maximal such $\Gamma$) and sharp estimates on the character values of the symmetric groups (to bound the subgroup growth of $\Gamma$).


A mixed problem for a Boussinesq hyperbolic equation with integral condition

Abstract: A hyperbolic problem wich combines a classical(Dirichlet) and a non-local contraint is considered.The existence and uniqueness of strong solutions are proved,we use a functionnal analysis method based on a priori estimate and on the density of the range of the operator generated by the considered problem.


A statistical framework for differential privacy

Abstract: One goal of statistical privacy research is to construct a data release mechanism that protects individual privacy while preserving information content. An example is a {\em random mechanism} that takes an input database $X$ and outputs a random database $Z$ according to a distribution $Q_n(\cdot|X)$. {\em Differential privacy} is a particular privacy requirement developed by computer scientists in which $Q_n(\cdot |X)$ is required to be insensitive to changes in one data point in $X$. This makes it difficult to infer from $Z$ whether a given individual is in the original database $X$. We consider differential privacy from a statistical perspective. We consider several data release mechanisms that satisfy the differential privacy requirement. We show that it is useful to compare these schemes by computing the rate of convergence of distributions and densities constructed from the released data. We study a general privacy method, called the exponential mechanism, introduced by McSherry and Talwar (2007). We show that the accuracy of this method is intimately linked to the rate at which the probability that the empirical distribution concentrates in a small ball around the true distribution.


The Cauchy problem for a short-wave equation

Abstract: We prove an existence and uniqueness of solution for the Cauchy problem of the simplest nonlinear short-wave equation, $u_{tx}=u-3u^{2}$, with periodic boundary condition.


Mackey-functor structure on the Brauer groups of a finite Galois covering of schemes

Abstract: Past studies of the Brauer group of a scheme tells us the importance of the interrelationship among Brauer groups of its finite \'etale coverings. In this paper, we consider these groups simultaneously, and construct an integrated object "Brauer-Mackey functor". We realize this as a {\it cohomological Mackey functor} on the Galois category of finite \'etale coverings. For any finite \'etale covering of schemes, we can associate two homomorphisms for Brauer groups, namely the pull-back and the norm map. These homomorphisms make Brauer groups into a bivariant functor ($=$ Mackey functor) on the Galois category. As a corollary, Restricting to a finite Galois covering of schemes, we obtain a cohomological Mackey functor on its Galois group. This is a generalization of the result for rings by Ford. Moreover, applying Bley and Boltje's theorem, we can derive certain isomorphisms for the Brauer groups of intermediate coverings.


Cohomology of Substitution Tiling Spaces

Abstract: Anderson and Putnam showed that the cohomology of a substitution tiling space may be computed by collaring tiles to obtain a substitution which "forces its border." One can then represent the tiling space as an inverse limit of an inflation and substitution map on a cellular complex formed from the collared tiles; the cohomology of the tiling space is computed as the direct limit of the homomorphism induced by inflation and substitution on the cohomology of the complex. In earlier work, Barge and Diamond described a modification of the Anderson-Putnam complex on collared tiles for one-dimensional substitution tiling spaces that allows for easier computation and provides a means of identifying certain special features of the tiling space with particular elements of the cohomology. In this paper, we extend this modified construction to higher dimensions. We also examine the action of the rotation group on cohomology and compute the cohomology of the pinwheel tiling space.


Some consequences of the Karpenko-Merkurjev theorem

Abstract: We use a recent theorem of N. A. Karpenko and A. S. Merkurjev to settle several questions in the theory of essential dimension.


Conformal Deformation on Manifolds with Boundary

Abstract: We consider natural conformal invariants arising from the Gauss-Bonnet formulas on manifolds with boundary, and study conformal deformation problems associated to them. The key technique we used is to derive boundary C^2 estimates directly from C^0 estimates for fully nonlinear equations. The main result has appeared in the author's thesis.


On the classification of toric singularities

Abstract: For a toric log variety with standard coefficients, we show that the minimal log discrepancy at a closed invariant point bounds the Cartier index of a neighbourhood.


Yet another solution to the Burnside problem for matrix semigroups

Abstract: We use the kernel category to give a finiteness condition for semigroups. As a consequence we provide yet another proof that finitely generated periodic semigroups of matrices are finite.


On a result of Gelfand, Kapranov, and Zelevinsky

Abstract: In this paper I give new elementary proofs of basic results of Gelfand, Kapranov and Zelevinskywhich express discriminants and resultants in terms of determinants of direct images of Cayley-Koszul complexes of sheaves.


Projective duality and K-energy asymptotics

Abstract: Let X be a smooth, linearly normal n dimensional complex projective variety. Assume that the projective dual of X has codimension one with defining polynomial D(X). In this paper the log of the norm of D(X) is expressed as the restriction to the Bergman metrics of an energy functional on X. We show how, for smooth plane curves, this energy functional reduces to the standard action functionals of Kahler geometry.


Expanding Measures

Abstract: We prove that any C^{1+} transformation, possibly with a (non-flat) critical or singular region, admits an invariant probability measure absolutely continuous with respect to any expanding measure whose Jacobian satisfies a mild distortion condition. This is an extension to arbitrary dimension of a famous theorem of Keller for maps of the interval with negative Schwarzian derivative. We also show how to construct an induced Markov map F such that every expanding probability of the initial transformation lifts to an invariant probability of F. The induced time is bounded at each point by the corresponding first hyperbolic time (the first time the dynamics exhibits hyperbolic behavior). In particular, F may be used to study decay of correlations and others statistical properties of the initial map, relative to any expanding probability.


Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics

Abstract: Let X be a smooth, linearly normal algebraic variety. It is shown that the Mabuchi energy of X restricted to the Bergman metrics is completely determined by the X-hyperdiscriminant of format (n-1) and the Chow form of X. As a corollary it is shown that the Mabuchi energy is bounded from below for all degenerations in G if and only if the hyperdiscriminant polytope dominates the Chow polytope for all maximal algebraic tori H of G .


Solution of Partial Differential Equations by Method of Hyperholomorphic functions

Abstract: It is well known that the real and imaginary parts of any holomorphic function are harmonic functions of two variables. In this paper we generalize this property to finite-dimensional commutative algebras. We prove that if some basis of a subspace of a commutative algebra satisfies a polynomial equation then components of a hyperholomorphic function on the subspace are solutions of the respective partial differential equation.


Application of graph combinatorics to rational identities of type A

Abstract: To a word $w$, we associate the rational function $\Psi_w = \prod (x_{w_i} - x_{w_{i+1}})^{-1}$. The main object, introduced by C. Greene to generalize identities linked to Murnaghan-Nakayama rule, is a sum of its images by certain permutations of the variables. The sets of permutations that we consider are the linear extensions of oriented graphs. We explain how to compute this rational function, using the combinatorics of the graph $G$. We also establish a link between an algebraic property of the rational function (the factorization of the numerator) and a combinatorial property of the graph (the existence of a disconnecting chain).


The fundamental category of a stratified space

Abstract: The fundamental groupoid of a locally 0 and 1-connected space classifies covering spaces, or equivalently local systems. When the space is topologically stratified Treumann, based on unpublished ideas of MacPherson, constructed an `exit category' (in the terminology of this paper, the `fundamental category') which classifies constructible sheaves, equivalently stratified etale covers. This paper generalises this construction to homotopically stratified sets, in addition showing that the fundamental category dually classifies constructible cosheaves, equivalently stratified branched covers. The more general setting has several advantages. It allows us to remove a technical `tameness' condition which appears in Treumann's work; to show that the fundamental groupoid can be recovered by inverting all morphisms and, perhaps most importantly, to reduce computations to the two stratum case. This provides an approach to computing the fundamental category in terms of homotopy groups of strata and homotopy links. We apply these techniques to compute the fundamental category of symmetric products of R^2, stratified by collisions. Two appendices explain the close relations respectively between filtered and pre-ordered spaces and between cosheaves and branched covers (technically locally-connected uniquely-complete spreads).


On the Structure of Complex Homogeneous Supermanifolds

Abstract: For a Lie group $G$ and a closed Lie subgroup $H\subset G$, it is well known that the coset space $G/H$ can be equipped with the structure of a manifold homogeneous under $G$ and that any $G$-homogeneous manifold is isomorphic to one of this kind. An interesting problem is to find an analogue of this result in the case of supermanifolds. In the classical setting, $G$ is a real or a complex Lie group and $G/H$ is a real and, respectively, a complex manifold. Now, if $G$ is a real Lie supergroup and $H\subset G$ is a closed Lie subsupergroup, there is a natural way to consider $G/H$ as a supermanfold. Furthermore, any $G$-homogeneous real supermanifold can be obtained in this way, see \cite{Kostant}. The goal of this paper is to give a proof of this result in the complex case.


Shifted small deviations and Chung LIL for symmetric alpha-stable processes

Abstract: Consider a symmetric $\alpha$-stable L\'evy process with $\alpha\in (1,2)$. We study shifted small ball probabilities for these processes in the uniform topology, when the shift function is an arbitrary continuous function which starts at 0. We obtain the exact rate of decrease for these probabilities including constants. Using these small ball estimates, we obtain a functional LIL for $\alpha$-stable L\'evy process with attracting functions that are continuous. It occurs that the limit set for the family of renormalized $\alpha$-stable L\'evy processes is equal to the set of all continuous functions on $[0,1]$ which start at 0, under certain choice of normalizing functions.


Soliton dynamics for the nonlinear Schr\"odinger equation with magnetic field

Abstract: The semiclassical limit of a nonlinear focusing Schr\"odinger equation in presence of nonconstant electric and magnetic potentials V,A is studied by taking as initial datum the ground state solution of an associated autonomous elliptic equation. The concentration curve of the solutions is a parameterization of the solutions of a Newton ODE involving the electric force as well as the magnetic force via the Lorenz law of electrodynamics.


Period Doubling in Area-Preserving Maps: An Associated One Dimensional Problem

Abstract: It has been observed that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of $\field{R}^2$. A renormalization approach has been used in a computer-assisted proof of existence of an area-preserving map with orbits of all binary periods by J.-P. Eckmann, H. Koch and P. Wittwer (1982 and 1984). As it is the case with all non-trivial universality problems in non-dissipative systems in dimensions more than one, no analytic proof of this period doubling universality exists to date. We argue that the period doubling renormalization fixed point for area-preserving maps is almost one dimensional, in the sense that it is close to the following Henon-like map: $$H^*(x,u)=(\phi(x)-u,x-\phi(\phi(x)-u)),$$ where $\phi$ solves $$\phi(x)={2 \over \lambda} \phi(\phi(\lambda x))-x.$$ We then give a ``proof'' of existence of solutions of small analytic perturbations of this one dimensional problem, and describe some of the properties of this solution. The ``proof'' consists of an analytic argument for factorized inverse branches of $\phi$ together with verification of several inequalities and inclusions of subsets of $\field{C}$ numerically. Finally, we suggest an analytic approach to the full period doubling problem for area-preserving maps based on its proximity to the one dimensional. In this respect, the paper is an exploration of a possible analytic machinery for a non-trivial renormalization problem in a conservative two-dimensional system.


The Center of the Nilcoxeter and 0-Hecke Algebras

Abstract: We compute the dimension of the center of the 0-Hecke algebra $\mathcal{H}_n$ and of the Nilcoxeter algebra $\mathcal{NC}_n$ using a calculus of diagrams on the M\"{o}bius band. In the case of the Nilcoxeter algebra, this calculus is shown to produce a basis for $Z(\mathcal{NC}_n)$ and the table of multiplication in this basis is shown to be trivial. We conjecture that a basis for $Z(\mathcal{H}_n)$ can also be obtained in a specific way from this topological calculus.


Polynomial operators and local smoothness classes on the unit interval, II

Abstract: We prove the existence of quadrature formulas exact for integrating high degree polynomials with respect to Jacobi weights based on scattered data on the unit interval. We also obtain a characterization of local Besov spaces using the coefficients of a tight frame expansion.


Distributed Stochastic Subgradient Projection Algorithms for Convex Optimization

Abstract: We consider a distributed multi-agent network system where the goal is to minimize a sum of convex objective functions of the agents subject to a common convex constraint set. Each agent maintains an iterate sequence and communicates the iterates to its neighbors. Then, each agent combines weighted averages of the received iterates with its own iterate, and adjusts the iterate by using subgradient information (known with stochastic errors) of its own function and by projecting onto the constraint set. The goal of this paper is to explore the effects of stochastic subgradient errors on the convergence of the algorithm. We first consider the behavior of the algorithm in mean, and then the convergence with probability 1 and in mean square. We consider general stochastic errors that have uniformly bounded second moments and obtain bounds on the limiting performance of the algorithm in mean for diminishing and non-diminishing stepsizes. When the means of the errors diminish, we prove that there is mean consensus between the agents and mean convergence to the optimum function value for diminishing stepsizes. When the mean errors diminish sufficiently fast, we strengthen the results to consensus and convergence of the iterates to an optimal solution with probability 1 and in mean square.


Beltrami forms, affine surfaces and the Schwarz-Christoffel formula: a worked out example of straightening

Abstract: Consider the straightening phi of a Beltrami form that is constant on a square, with the corresponding ellipses having a vertical major axis, and null outside. A generalized Schwarz-Christoffel formula is used to express the inverse of phi. The formula is found by introducing an affine Riemann surface. This formula is used to draw on a computer the image of the square by phi, and practical aspects are discussed. The resulting shapes are shown for different values of the constant dilatation ratio of the ellipses (=major axis/minor axis). The limit when this ratio tends to infinity is surprising. A model of this limit is proposed, produced by an affine surface uniformization.


Reconstructing quasimorphisms from associated partial orders and a question of Polterovich

Abstract: We show that every continuous homogeneous quasimorphism on a finite-dimensional 1-connected simple Lie group arises as the relative growth of any continuous bi-invariant partial order on that group. More generally we show, that an arbitrary homogeneous quasimorphism can be reconstructed as the relative growth of a partial order subject to a certain sandwich condition. This provides a link between invariant orders and bounded cohomology and allows the concrete computation of relative growth for finite dimensional simple Lie groups as well as certain infinite-dimensional Lie groups arising from symplectic geometry.


Variations of Independence in Boolean Algebras

Abstract: We give a definition of some classes of boolean algebras generalizing free boolean algebras; they satisfy a universal property that certain functions extend to homomorphisms. We give a combinatorial property of generating sets of these algebras, which we call n -independent. The properties of these classes (n-free and omega-free boolean algebras) are investigated. These include connections to hypergraph theory and cardinal invariants on these algebras. Related cardinal functions, n Ind, which is the supremum of the cardinalities of n-independent subsets; i_n, the minimum size of a maximal n -independent subset; and i_omega, the minimum size of an omega-independent subset, are introduced and investigated. The values of i_n and i_omega on P(omega)/fin are shown to be independent of ZFC. Ideal-independence is also considered, and it is shown that the cardinal function p <= s_mm for infinite boolean algebras. We also define and consider moderately generated boolean algebras; that is, those boolean algebras that have a generating set consisting of elements that split finitely many elements of the boolean algebra.


Shape optimization for low Neumann and Steklov eigenvalues

Abstract: We give an overview of results on shape optimization for low eigenvalues of the Laplacian on bounded planar domains with Neumann and Steklov boundary conditions. These results share a common feature: they are proved using methods of complex analysis. In particular, we present modernized proofs of the classical inequalities due to Szego and Weinstock for the first nonzero Neumann and Steklov eigenvalues. We also extend the inequality for the second nonzero Neumann eigenvalue, obtained recently by Nadirashvili and the authors, to non-homogeneous membranes with log-subharmonic densities. In the homogeneous case, we show that this inequality is strict, which implies that the maximum of the second nonzero Neumann eigenvalue is not attained in the class of simply-connected membranes of a given mass. The same is true for the second nonzero Steklov eigenvalue, as follows from our results on the Hersch-Payne-Schiffer inequalities.


Geometric Langlands duality and forms of reductive groups

Abstract: The category of perverse sheaves on the affine Grassmannian of a complex reductive group $G$ gives a canonical geometric construction of the split form of the Langlands dual group $\check G_\bZ$ over the integers. Given a field $k$, we give a Tannakian construction of the quasi-split forms of $\check G_k$, as well as a construction of the gerbe associated to an inner form of $\check G_k$.


Predictability on finite horizon for processes with exponential decrease of energy on higher frequencies

Abstract: The paper presents sufficient conditions of predictability for continuous time processes in deterministic setting. We found that processes with exponential decay on energy for higher frequencies are predictable in some weak sense on some finite time horizon defined by the rate of decay. Moreover, this predictability can be achieved uniformly over classes of processes. Some explicit formulas for predictors are suggested.


Uniform Asymptotics of the Meixner Polynomials

Abstract: Using the steepest descent method of Deift-Zhou, we derive locally uniform asymptotic formulas for the Meixner polynomials. These include an asymptotic formula in a neighborhood of the origin, a result which as far as we are aware has not yet been obtained previously. This particular formula involves a special function, which is the uniformly bounded solution to a scalar Riemann-Hilbert problem, and which is asymptotically (as the polynomial degree $n$ tends to infinity) equal to the constant $"1"$ except at the origin. Numerical computation by using our formulas, and comparison with earlier results, are also given.


Maximizing the number of q-colorings

Abstract: Let P_G(q) denote the number of proper q-colorings of a graph G. This function, called the chromatic polynomial of G, was introduced by Birkhoff in 1912, who sought to attack the famous four-color problem by minimizing P_G(4) over all planar graphs G. Since then, motivated by a variety of applications, much research was done on minimizing or maximizing P_G(q) over various families of graphs. In this paper, we study an old problem of Linial and Wilf, to find the graphs with n vertices and m edges which maximize the number of q-colorings. We provide the first approach which enables one to solve this problem for many nontrivial ranges of parameters. Using our machinery, we show that for each q >= 4 and sufficiently large m < \kappa_q n^2 where \kappa_q is approximately 1/(q log q), the extremal graphs are complete bipartite graphs minus the edges of a star, plus isolated vertices. Moreover, for q = 3, we establish the structure of optimal graphs for all large m <= n^2/4, confirming (in a stronger form) a conjecture of Lazebnik from 1989.


Cohomology theory in 2-categories

Abstract: Recently, symmetric categorical groups are used for the study of the Brauer groups of symmetric monoidal categories. As a part of these efforts, some algebraic structures of the 2-category of symmetric categorical groups $\mathrm{SCG}$ are being investigated. In this paper, we consider a 2-categorical analogue of an abelian category, in such a way that it contains $\mathrm{SCG}$ as an example. As a main theorem, we construct a long cohomology 2-exact sequence from any extension of complexes in such a 2-category. Our axiomatic and self-dual definition will enable us to simplify the proofs, by analogy with abelian categories.


On Boundary Crossing Probabilities for Diffusion Processes

Abstract: In this paper, we establish a relationship between the asymptotic form of conditional boundary crossing probabilities and first passage time densities for diffusion processes. Namely, we show that, under broad assumptions, the first crossing time density of a general curvilinear boundary by a general time-homogeneous diffusion process has a product-form, the factors being the transition density of the process and the coefficient of the leading term in the asymptotic representation of the non-crossing probability of the boundary by the respective diffusion bridge (as the end-point of the bridge approaches the boundary). Using a similar technique, we also demonstrate that the boundary crossing probability is a Gateaux differentiable function of the boundary and give an explicit representation of its derivative.


A classification of spherical conjugacy classes in good characteristic

Abstract: We classify spherical conjugacy classes in a simple algebraic group over an algebraically closed field of good, odd characteristic.


Universal pointwise selection rule in multivariate function estimation

Abstract: In this paper, we study the problem of pointwise estimation of a multivariate function. We develop a general pointwise estimation procedure that is based on selection of estimators from a large parameterized collection. An upper bound on the pointwise risk is established and it is shown that the proposed selection procedure specialized for different collections of estimators leads to minimax and adaptive minimax estimators in various settings.


A Note on Gorenstein Flat Dimension

Abstract: Unlike the Gorenstein projective and injective dimensions, the majority of results on the Gorenstein flat dimension have been established only over Noetherian (or coherent) rings. Naturally, one would like to generalize these results to any associative ring. In this direction, we show that the Gorenstein flat dimension is a refinement of the classical flat dimension over any ring; and we investigate the relations between the Gorenstein projective dimension and the Gorenstein flat dimension.


Spherical designs from norm-3 shell of integral lattices

Abstract: A set of vectors all of which have a constant (non-zero) norm value in an Euclidean lattice is called a shell of the lattice. Venkov classified strongly perfect lattices of minimum 3 (R\'{e}seaux et "designs" sph\'{e}rique, 2001), whose minimal shell is a spherical 5-design. This note considers the classification of integral lattices whose shells of norm 3 are 5-designs.


One group of inequalities with altitudes and medians in triangle

Abstract: In the article we prove some inequalities that contain relations between altitudes and medians in triangle. At least one of these inequalities has not been considered in the literature before and the main theorem has also not been proved elsewhere in that form. Some immediate corollaries have been presented as well.


A wavelet analysis of the Rosenblatt process: chaos expansion and estimation of the self-similarity parameter

Abstract: By using chaos expansion into multiple stochastic integrals, we make a wavelet analysis of two self-similar stochastic processes: the fractional Brownian motion and the Rosenblatt process. We study the asymptotic behavior of the statistic based on the wavelet coefficients of these processes. Basically, when applied to a non-Gaussian process (such as the Rosenblatt process) this statistic satisfies a non-central limit theorem even when we increase the number of vanishing moments of the wavelet function. We apply our limit theorems to construct estimators for the self-similarity index and we illustrate our results by simulations.


Simple finite group schemes and their infinitesimal deformations

Abstract: We show that the classification of simple finite group schemes over an algebraically closed field reduces to the classification of abstract simple finite groups and of simple restricted Lie algebras in positive characteristic. Both these two simple objects have been classified. We review this classification. Finally, we address the problem of determining the infinitesimal deformations of simple finite group schemes.


The "north pole problem" and random orthogonal matrices

Abstract: This paper is motivated by the following observation. Take a 3 x 3 random (Haar distributed) orthogonal matrix $\Gamma$, and use it to "rotate" the north pole, $x_0$ say, on the unit sphere in $R^3$. This then gives a point $u=\Gamma x_0$ that is uniformly distributed on the unit sphere. Now use the same orthogonal matrix to transform u, giving $v=\Gamma u=\Gamma^2 x_0$. Simulations reported in Marzetta et al (2002) suggest that v is more likely to be in the northern hemisphere than in the southern hemisphere, and, morever, that $w=\Gamma^3 x_0$ has higher probability of being closer to the poles $\pm x_0$ than the uniformly distributed point u. In this paper we prove these results, in the general setting of dimension $p\ge 3$, by deriving the exact distributions of the relevant components of u and v. The essential questions answered are the following. Let x be any fixed point on the unit sphere in $R^p$, where $p\ge 3$. What are the distributions of $U_2=x'\Gamma^2 x$ and $U_3=x'\Gamma^3 x$? It is clear by orthogonal invariance that these distribution do not depend on x, so that we can, without loss of generality, take x to be $x_0=(1,0,...,0)'\in R^p$. Call this the "north pole". Then $x_0'\Gamma^ k x_0$ is the first component of the vector $\Gamma^k x_0$. We derive stochastic representations for the exact distributions of $U_2$ and $U_3$ in terms of random variables with known distributions.


On the vanishing of cohomology in triangulated categories

Abstract: We study the vanishing of cohomology in triangulated categories admitting a central ring action. In particular, we study vanishing gaps and symmetry.


C-Ideals of Lie Algebras

Abstract: A subalgebra B of a Lie algebra L is called a c-ideal of L if there is an ideal C of L such that L = B + C and B \cap C \leq B_L, where B_L is the largest ideal of $L$ contained in B. This is analogous to the concept of c-normal subgroup, which has been studied by a number of authors. We obtain some properties of c-ideals and use them to give some characterisations of solvable and supersolvable Lie algebras. We also classify those Lie algebras in which every one-dimensional subalgebra is a c-ideal.


Value-at-Risk Computation by Fourier Inversion with Explicit Error Bounds

Abstract: The value-at-risk of a delta-gamma approximated derivatives portfolio can be computed by numerical integration of the characteristic function. However, while the choice of parameters in any numerical integration scheme is paramount, in practice it often relies on ad hoc procedures of trial and error. For normal and multivariate $t$-distributed risk factors, we show how to calculate the necessary parameters for one particular integration scheme as a function of the data (the distribution of risk factors, and delta and gamma) \emph{in order to satisfy a given error tolerance}. This allows for implementation in a fully automated risk management system. We also demonstrate in simulations that the method is significantly faster than the Monte Carlo method, for a given error tolerance.


Orbitally but not asymptotically stable ground states for the discrete NLS

Abstract: We consider examples of discrete nonlinear Schroedinger equations in Z admitting ground states which are orbitally but not asymptotically stable in l ^2(Z). The ground states contain internal modes which decouple from the continuous modes. The absence of leaking of energy from discrete to continues modes leads to an almost conservation and perpetual oscillation of the discrete modes. This is quite different from what is known for nonlinear Schroedinger equations in eucliedean spaces. We do not investigate connections with work on quasi periodic solutions as in M.Johansson & S.Aubry and of Bambusi & Vella


Coloring plane graphs with independent crossings

Abstract: We show that every plane graph with maximum face size four whose all faces of size four are vertex-disjoint is cyclically 5-colorable. This answers a question of Albertson whether graphs drawn in the plane with all crossings independent are 5-colorable.


Cyclic colorings of plane graphs with independent faces

Abstract: Let G be a plane graph with maximum face size D. If all faces of G with size four or more are vertex disjoint, then G has a cyclic coloring with D+1 colors, i.e., a coloring such that all vertices incident with the same face receive distinct colors.


Trasferring $L^p$ eigenfunction bounds from $S^{2n+1}$ to $h^n$

Abstract: By using the notion of contraction of Lie groups, we transfer $L^p-L^2$ estimates for joint spectral projectors from the unit complex sphere $\sfera$ in ${{\mathbb{C}}}^{n+1}$ to the reduced Heisenberg group $h^{n}$. In particular, we deduce some estimates recently obtained by H. Koch and F. Ricci on $h^n$. As a consequence, we prove, in the spirit of Sogge's work, a discrete restriction theorem for the sub-Laplacian $L$ on $h^n$.


Irreducible complex skew-Berger algebras

Abstract: Irreducible skew-Berger algebras $\g\subset\gl(n,\Co)$, i.e. algebras spanned by the images of the linear maps $R:\odot^2\Co^n\to\g$ satisfying the Bianchi identity, are classified. These Lie algebras can be interpreted as irreducible complex Berger superalgebras contained in $\gl(0|n,\Co)$.


On the integral of $\log x\frac{dy}{y}-\log y\frac{dx}{x}$ over the A-polynomial curves

Abstract: In this note, we study the integral of the 1-form $\log x\frac{dy}{y}-\log y\frac{dx}{x}$ over certain plane curves defined by A-polynomials of knots. It is quite surprising that a Chern-Simons type invariant of 3-manifolds, which can be geometrically computed, may be used to get the exact values of those integrals. The arithmetic nature of these integrals is still unknown at the moment and deserved further investigation.


Uniformites et Continuity Spaces

Abstract: A semigroup A is an abelian semigroup with identity 0. A set of positives in A is an ordered down-directed set P containing with every r an element r/2 with r/2 + r/2 = r. A continuity space is an abstract set X equipped with a map d : XxX to A satisfying d(x, x) = 0 and d(x, z) d(x, y) + d(y, z). A quasi-uniform space is an abstract set X equipped with a filterbase of binary relations {U} such that each U contains the diagonal as well as for some V{U}. For each rP, the set } is seen to be a quasi-uniform filterbase on X . Indeed, the down-directedness of P ensures that U(r) is a filterbase of oversets of the diagonal and U(r) contains U(r/2)U(r/2). One obtains a uniform filterbase by symmetrization, i.e. by intersecting the U(r) with the U(s) = {(y, x)|d(y, x) <s}.


On peeling procedure applied to a Poisson point process

Abstract: In the focus of our attention is the asymptotic properties of the sequence of convex hulls which arise as a result of a peeling procedure applied to the convex hull generated by a Poisson point process. Processes of the considered type are tightly connected with empirical point processes and stable random vectors. Results are given about the limit shape of the convex hulls in the case of a discrete spectral measure. We give some numerical experiments to illustrate the peeling procedure for a more large class of Poisson point processes.


Classification of knotted tori in the 2-metastable dimension

Abstract: This paper is on the classical Knotting Problem: for a given manifold N and a number m describe the set of isotopy classes of embeddings $N\to S^m$. We study the specific case of knotted tori, i. e. the embeddings $S^p \times S^q \to S^m$. The classification of knotted tori up to isotopy in the metastable dimension range $m>p+\frac{3}{2}q+2$, $p\le q$, was given by A. Haefliger, E. Zeeman and A. Skopenkov. We consider the dimensions below the metastable range, and give an explicit criterion for the finiteness of this set of isotopy classes in the 2-metastable dimension: Theorem. Assume that $p+\frac{4}{3}q+2<m<p+\frac{3}{2}q+2$ and $m>2p+q+2$. Then the set of smooth embeddings $S^p \times S^q \to S^m$ up to isotopy is infinite if and only if either $q+1$ or $p+q+1$ is divisible by 4. Our approach to the classification is based on an analogue of the Koschorke exact sequence from the theory of link maps. This sequence involves a new $\beta$-invariant of knotted tori. The exactness is proved using embedded surgery and the Habegger-Kaiser techniques of studying the complement.


A semantics for obligations

Abstract: We analyze a number of properties obligations have or should have.


A view of mathematics research productivity at U.S. regional public universities

Abstract: Statistical summaries of certain kinds of mathematics research output are given for a large sample of U.S. regional public universities. These statistical summaries are reported using a variety of metrics that distinguish between single-authored and collaborative work and account for publication length.


Representations of the general linear groups which are irreducible over subgroups

Abstract: We classify all triples $(G,V,H)$ such that $SL_n(q)\leq G\leq GL_n(q)$, $V$ is a representation of $G$ of dimension greater than one over an algebraically closed field $\FF$ of characteristic coprime to $q$, and $H$ is a proper subgroup of $G$ such that the restriction $V\dar_{H}$ is irreducible. This problem is a natural part of the Aschbacher-Scott program on maximal subgroups of finite classical groups.


A partial analog of integrability theorem for distributions on p-adic spaces and applications

Abstract: Let X be a smooth real algebraic variety. Let $\xi$ be a distribution on it. One can define the singular support of $\xi$ to be the singular support of the $D_X$-module generated by $\xi$ (some times it is also called the characteristic variety). A powerful property of the singular support is that it is a coisotropic subvariety of $T^*X$. This is the integrability theorem (see [KKS, Mal, Gab]). This theorem turned out to be useful in representation theory of real reductive groups (see e.g. [AG_AMOT, AS, Say]). The aim of this paper is to give an analog of this theorem to the non-Archimedean case. The theory of D-modules is not available to us so we need a different definition of the singular support. We use the notion wave front set from [Hef] and define the singular support to be its Zariski closure. Then we prove that the singular support satisfies some property that we call weakly coisotropic, which is weaker than being coisotropic but is enough for some applications. We also prove some other properties of the singular support that were trivial in the Archimedean case (using the algebraic definition) but not obvious in the non-Archimedean case.


Quantitative asymptotics of graphical projection pursuit

Abstract: There is a result of Diaconis and Freedman which says that, in a limiting sense, for large collections of high-dimensional data most one-dimensional projections of the data are approximately Gaussian. This paper gives quantitative versions of that result. For a set of deterministic vectors $\{x_i\}_{i=1}^n$ in $\R^d$ with $n$ and $d$ fixed, let $\theta\in\s^{d-1}$ be a random point of the sphere and let $\mu_n^\theta$ denote the random measure which puts mass $\frac{1}{n}$ at each of the points $\inprod{x_1}{\theta},...,\inprod{x_n}{\theta}$. For a fixed bounded Lipschitz test function $f$, $Z$ a standard Gaussian random variable and $\sigma^2$ a suitable constant, an explicit bound is derived for the quantity $\ds\P[|\int f d\mu_n^\theta-\E f(\sigma Z)|>\epsilon]$. A bound is also given for $\ds\P[d_{BL}(\mu_n^\theta, N(0,\sigma^2))>\epsilon]$, where $d_{BL}$ denotes the bounded-Lipschitz distance, which yields a lower bound on the waiting time to finding a non-Gaussian projection of the $\{x_i\}$ if directions are tried independently and uniformly on $\s^{d-1}$.


A minimalist two-level foundation for constructive mathematics

Abstract: We present a two-level theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin. One level is given by an intensional type theory, called Minimal type theory. This theory extends the set-theoretic version introduced in the mentioned paper with collections. The other level is given by an extensional set theory which is interpreted in the first one by means of a quotient model. This two-level theory has two main features: it is minimal among the most relevant foundations for constructive mathematics; it is constructive thanks to the way the extensional level is linked to the intensional one which fulfills the "proofs-as-programs" paradigm and acts as a programming language.


The Julia set of a post-critically finite endomorphism of PC^2

Abstract: We construct a combinatorial model of the Julia set of the endomorphism $f(z, w)=((1-2z/w)^2, (1-2/w)^2)$ of $PC^2$.


Concrete Constructions of Real Equiangular Line Sets

Abstract: We give some concrete constructions of real equiangular line sets. The emphasis here is on {\em building blocks} for certain angles which are then used to build up larger equiangular line sets. We concentrate on angles greater than or equal to 1/7.


A Giambelli formula for isotropic Grassmannians

Abstract: Let X be a symplectic or odd orthogonal Grassmannian parametrizing isotropic subspaces in a vector space equipped with a nondegenerate (skew) symmetric form. We prove a Giambelli formula which expresses an arbitrary Schubert class in H^*(X,Z) as a polynomial in certain special Schubert classes. We study theta polynomials, a family of polynomials defined using raising operators whose algebra agrees with the Schubert calculus on X. Furthermore, we prove that theta polynomials are special cases of Billey-Haiman Schubert polynomials and use this connection to express the former as positive linear combinations of products of Schur Q-functions and S-polynomials.


Brunet-Derrida behavior of branching-selection particle systems on the line

Abstract: We consider a class of branching-selection particle systems on $\R$ similar to the one considered by E. Brunet and B. Derrida in their 1997 paper "Shift in the velocity of a front due to a cutoff". Based on numerical simulations and heuristic arguments, Brunet and Derrida showed that, as the population size $N$ of the particle system goes to infinity, the asymptotic velocity of the system converges to a limiting value at the unexpectedly slow rate $(\log N)^{-2}$. In this paper, we give a rigorous mathematical proof of this fact, for the class of particle systems we consider. The proof makes use of ideas and results by R. Pemantle, and by N. Gantert, Y. Hu and Z. Shi, and relies on a comparison of the particle system with a family of $N$ independent branching random walks killed below a linear space-time barrier.


Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions

Abstract: In this paper we consider a multi-dimensional wave equation with dynamic boundary conditions, related to the Kelvin-Voigt damping. Global existence and asymptotic stability of solutions starting in a stable set are proved. Blow up for solutions of the problem with linear dynamic boundary conditions with initial data in the unstable set is also obtained.


Center stable manifolds for quasilinear parabolic pde and conditional stability of nonclassical viscous shock waves

Abstract: Motivated by the study of conditional stability of traveling waves, we give an elementary $H^2$ center stable manifold construction for quasilinear parabolic PDE, sidestepping apparently delicate regularity issues by the combination of a carefully chosen implicit fixed-point scheme and "damping-type" $H^s$ energy estimates of a type familiar from the study of hyperbolic--parabolic and relaxation systems. An important feature of these methods is that they generalize to situations such as the hyperbolic--parabolic or relaxation case for which parabolic-type smoothing estimates are unavailable. As an application, we show conditional stability of Lax- or undercompressive shock waves of general quasilinear parabolic systems of conservation laws by a pointwise stability analysis on the center stable manifold.


Logarithm laws for unipotent flows, I

Abstract: We prove analogues of the logarithm laws of Sullivan and Kleinbock-Margulis in the context of unipotent flows. In particular, we obtain results for one-parameter actions on the space of lattices $SL(n, \R)/SL(n, \Z)$. The key lemma for our results says the measure of the set of unimodular lattices in $\R^n$ that does not intersect a `large' volume subset of $\R^n$ is `small'. This can be considered as a `random' analogue of the classical Minkowski theorem in the geometry of numbers.


A Nonseparably Connected Metric Space as a Dense Connected Graph

Abstract: We present a connected metric space that does not contain any nontrivial separable connected subspace. Our space is a dense connected graph of a function from the real line satisfying Cauchy's equation.