Distributions that are both log-symmetric and R-symmetric

Abstract: Two concepts of symmetry for the distributions of positive random variables $Y$ are log-symmetry (symmetry of the distribution of $\log Y$) and R-symmetry [7]. In this paper, we characterise the distributions that have both properties, which we call doubly symmetric. It turns out that doubly symmetric distributions constitute a subset of those distributions that are moment-equivalent to the lognormal distribution. They include the lognormal, some members of the Berg/Askey class of distributions, and a number of others for which we give an explicit construction (based on work of A.J. Pakes) and note some properties; Stieltjes classes, however, are not doubly symmetric.


Degenerating curves and surfaces: first results

Abstract: Let $\mathcal S\to\mathbb A^1$ be a smooth family of surfaces whose general fibre is a smooth surface of $\mathbb P^3$ and whose special fibre has two smooth components, intersecting transversally along a smooth curve $R$. We consider the Universal Severi-Enriques variety $\mathcal V$ on $\mathcal S\to\mathbb A^1$. The general fibre of $\mathcal V$ is the variety of curves on $\mathcal S_t$ in the linear system $|\mathcal O_{\mathcal S_t}(n)|$ with $k$ cusps and $\delta$ nodes as singularities. Our problem is to find all irreducible components of the special fibre of $\mathcal V$. In this paper, we consider only the cases $(k,\delta)=(0,1)$ and $(k,\delta)=(1,0)$. In particular, we determine all singular curves on the special fibre of $\mathcal S$ which, counted with the right multiplicity, are a limit of 1-cuspidal curves on the general fibre of $\mathcal S$.


Principal noncommutative torus bundles

Abstract: In this paper we study continuous bundles of C*-algebras which are non-commutative analogues of principal torus bundles. We show that all such bundles, although in general being very far away from being locally trivial bundles, are at least locally trivial with respect to a suitable bundle version of bivariant K-theory (denoted RKK-theory) due to Kasparov. Using earlier results of Echterhoff and Williams, we shall give a complete classification of principal non-commutative torus bundles up to equivariant Morita equivalence. We then study these bundles as topological fibrations (forgetting the group action) and give necessary and sufficient conditions for any non-commutative principal torus bundle being RKK-equivalent to a commutative one. As an application of our methods we shall also give a K-theoretic characterization of those principal torus-bundles with H-flux, as studied by Mathai and Rosenberg which possess "classical" T-duals.


On doubling inequalities for elliptic systems

Abstract: We prove doubling inequalities for solutions of elliptic systems with an iterated Laplacian as diagonal principal part and for solutions of the Lame' system of isotropic linearized elasticity. These inequalities depend on global properties of the solutions.


Compactification de varietes de Siegel aux places de mauvaise reduction

Abstract: We construct arithmetic toroidal compactifications of the moduli stack of principally polarized abelian varieties with parahoric level structure. To this end, we extend the methods of Faltings and Chai to a case of bad reduction. ----- Nous construisons des compactifications toroidales arithm\'etiques du champ de modules des vari\'et\'es ab\'eliennes principalement polaris\'ees munies d'une structure de niveau parahorique. Pour ce faire, nous \'etendons la m\'ethode de Faltings et Chai \`a un cas de mauvaise r\'eduction.


Fibrations with noncommutative fibers

Abstract: We study an analogue of fibrations of topological spaces with the homotopy lifting property in the setting of C*-algebra bundles. We then derive an analogue of the Leray-Serre spectral sequence to compute the K-theory of the fibration in terms of the cohomology of the base and the K-theory of the fibres. We present many examples which show that fibrations with noncommutative fibres appear in abundance in nature.


Varieties swept out by grassmannians of lines

Abstract: We classify complex projective varieties of dimension $2r \geq 8$ swept out by a family of codimension two grassmannians of lines $\mathbb{G}(1,r)$. They are either fibrations onto normal surfaces such that the general fibers are isomorphic to $\G(1,r)$ or the grassmannian $\mathbb{G}(1,r+1)$. The cases $r=2$ and $r=3$ are also considered in the more general context of varieties swept out by codimension two linear spaces or quadrics.


Smoothing nodal Calabi-Yau n-folds

Abstract: Let X be an n-dimensional Calabi-Yau with ordinary double points, where n is odd. Friedman showed that for n=3 the existence of a smoothing of X implies a specific type of relation between homology classes on a resolution of X. (The converse is also true, due to work of Friedman, Kawamata and Tian.) We sketch a more topological proof of this result, and then extend it to higher dimensions. For n>3 the "Yukawa product" on the middle dimensional (co)homology plays an unexpected role. We also discuss a converse, proving it for nodal Calabi-Yau hypersurfaces in projective space.


Bounds on the volume entropy and simplicial volume in Ricci curvature $L^p$ bounded from below

Abstract: Let $(M,g)$ be a compact manifold with Ricci curvature almost bounded from below and $\pi:\bar{M}\to M$ be a normal, Riemannian cover. We show that, for any nonnegative function $f$ on $M$, the means of $f\o\pi$ on the geodesic balls of $\bar{M}$ are comparable to the mean of $f$ on $M$. Combined with logarithmic volume estimates, this implies bounds on several topological invariants (volume entropy, simplicial volume, first Betti number and presentations of the fundamental group) in Ricci curvature $L^p$-bounded from below.


Infinitesimal Variations of Hodge Structure at Infinity

Abstract: By analyzing the local and infinitesimal behavior of degenerating polarized variations of Hodge structure the notion of infinitesimal variation of Hodge structure at infinity is introduced. It is shown that all such structures can be integrated to polarized variations of Hodge structure and that, conversely, all are limits of infinitesimal variations of Hodge structure (IVHS) at finite points. As an illustration of the rich information encoded in this new structure, some instances of the maximal dimension problem for this type of infinitesimal variation are presented and contrasted with the "classical" case of IVHS at finite points.


The Hermitian Laplace Operator on Nearly K\"ahler Manifolds

Abstract: The moduli space NK of infinitesimal deformations of a nearly K\"ahler structure on a compact 6-dimensional manifold is described by a certain eigenspace of the Laplace operator acting on co-closed primitive (1,1) forms. Using the Hermitian Laplace operator and some representation theory, we compute the space NK on all 6-dimensional homogeneous nearly K\"ahler manifolds. It turns out that the nearly K\"ahler structure is rigid except for the flag manifold F(1,2)=SU_3/T^2, which carries an 8-dimensional moduli space of infinitesimal nearly K\"ahler deformations, modeled on the Lie algebra su_3 of the isometry group.


Maximal totally complex submanifolds of $\mathbb{H}\mathbb{P}^n$: homogeneity and normal holonomy

Abstract: We prove that a maximal totally complex submanifold $N^{2n}$ of the quaternionic projective space $\mathbb{H}\mathbb{P}^n$ ($n\geq 2$) is a parallel submanifold, provided one of the following conditions is satisfied: (1) $N$ is the orbit of a compact Lie group of isometries, (2) the restricted normal holonomy is a proper subgroup of ${\rm U}(n)$.


Euler characteristic and quadrilaterals of normal surfaces

Abstract: Let $M$ be a compact 3-manifold with a triangulation $\tau$. We give an inequality relating the Euler characteristic of a surface $F$ normally embedded in $M$ with the number of normal quadrilaterals in $F$. This gives a relation between a topological invariant of the surface and a quantity derived from its combinatorial description. Secondly, we obtain an inequality relating the number of normal triangles and normal quadrilaterals of $F$, that depends on the maximum number of tetrahedrons that share a vertex in $\tau$.


Simultaneous Stabilization in $A_\mathbb{R}(\mathbb{D})$

Abstract: In this note we study the problem of simultaneous stabilization for the algebra $A_\R(\D)$. Invertible pairs $(f_j,g_j)$, $j=1,..., n$, in a commutative unital algebra are called \textit{simultaneously stabilizable} if there exists a pair $(\alpha,\beta)$ of elements such that $\alpha f_j+\beta g_j$ is invertible in this algebra for $j=1,..., n$. For $n=2$, the simultaneous stabilization problem admits a positive solution for any data if and only if the Bass stable rank of the algebra is one. Since $A_\R(\D)$ has stable rank two, we are faced here with a different situation. When $n=2$, necessary and sufficient conditions are given so that we have simultaneous stability in $A_\R(\D)$. For $n\geq 3$ we show that under these conditions simultaneous stabilization is not possible and further connect this result to the question of which pairs $(f,g)$ in $A_\R(\D)^2$ are totally reducible; that is, for which pairs do there exist two units $u$ and $v$ in $A_\R(\D)$ such that $uf+vg=1$.


Periodic perturbations with delay of autonomous differential equations on manifolds

Abstract: We apply topological methods to the study of the set of harmonic solutions of periodically perturbed autonomous ordinary differential equations on differentiable manifolds, allowing the perturbing term to contain a fixed delay. In the crucial step, in order to cope with the delay, we define a suitable (infinite dimensional) notion of Poincar\'e $T$-translation operator and prove a formula that, in the unperturbed case, allows the computation of its fixed point index.


Finite groups of units and their composition factors in the integral group rings of the groups $\text{PSL}(2,q)$

Abstract: Let $G$ denote the projective special linear group $\text{PSL}(2,q)$, for a prime power $q$. It is shown that a finite 2-subgroup of the group $V(\mathbb{Z}G)$ of augmentation 1 units in the integral group ring $\mathbb{Z}G$ of $G$ is isomorphic to a subgroup of $G$. Furthermore, it is shown that a composition factor of a finite subgroup of $V(\mathbb{Z}G)$ is isomorphic to a subgroup of $G$.


Incompressibility and normal minimal surfaces

Abstract: In this paper we describe a procedure for refining the given triangulation of a 3-manifold that scales the PL-metric according to a given weight function while creating no new normal surfaces. It is known that an incompressible surface $F$ in a triangulated 3-manifold $M$ is isotopic to a normal surface that is of minimal PL-area in the isotopy class of $F$. Using the above scaling refinement we prove the converse. If $F$ is a surface in a closed 3-manifold $M$ such that for any triangulation $\tau$ of $M$, $F$ is isotopic to a $\tau$-normal surface $F(\tau)$ that is of minimal PL-area in its isotopy class, then we show that $F$ is incompressible.


Global attractors for doubly nonlinear evolution equations with non-monotone perturbations

Abstract: This paper proposes an abstract theory concerned with dynamical systems generated by doubly nonlinear evolution equations governed by subdifferential operators with non-monotone perturbations in a reflexive Banach space setting. In order to construct global attractors, an approach based on the notion of generalized semiflow is employed instead of the usual semi-group approach, since solutions of the Cauchy problem for the equation might not be unique. Moreover, the preceding abstract theory is applied to a generalized Allen-Cahn equation whose potential is divided into a convex part and a non-convex part as well as a semilinear parabolic equation with a nonlinear term involving gradients.


Skew Invariant Theory of Symplectic Groups, Pluri-Hodge Groups and 3-Manifold Invariants

Abstract: This article deals with a number of topics which are, somewhat surprisingly, related. Firstly, the fundamental theorem of skew invariant theory for the symplectic group giving the generators and relations of symplectic invariants is established. The relations are the so called P_n relations which appear in the study of certain 3-manifold invariants. Next a class of cohomology groups are introduced, called Pluri-Hodge groups (somewhat in keeping with the notion of pluri-canonical groups). These are Dolbeault groups on a complex manifold X with values in tensor powers of sheaves of holomorphic forms of various degrees. By Riemann-Roch one shows that knowledge of the Pluri-Hodge groups gives precise formulae for all Chern numbers of the manifold. When X is holomorphic symplectic the Pluri-Hodge groups form representations of Sp(g) where g counts the number of tensor products. Still with X holomorphic symplectic, the Pluri-Hodge groups are vectors in the vector space H_g(X) of states in the topological field theory of Rozansky and Witten for a 3-manifold with boundary a Riemann surface of genus g. A formulation of the Murakami-Ohtsuki invariants, due to Sawon, allows us to show that quotients of the spaces of trivalent graphs B_g carry representations of the symplectic group. There is a weight system W_X: B_g -> H_g(X) and we show that W_X preserves the various symplectic group actions.


On real analytic Banach manifolds

Abstract: Let $X$ be a real Banach space with an unconditional basis (e.g., $X=\ell_2$ Hilbert space), $\Omega\subset X$ open, $M\subset\Omega$ a closed split real analytic Banach submanifold of $\Omega$, $E\to M$ a real analytic Banach vector bundle, and ${\Cal A}^E\to M$ the sheaf of germs of real analytic sections of $E\to M$. We show that the sheaf cohomology groups $H^q(M,{\Cal A}^E)$ vanish for all $q\ge1$, and there is a real analytic retraction $r:U\to M$ from an open set $U$ with $M\subset U\subset\Omega$ such that $r(x)=x$ for all $x\in M$. Some applications are also given, e.g., we show that any infinite dimensional real analytic Hilbert submanifold of separable affine or projective Hilbert space is real analytically parallelizable.


$L^p$ spectral theory and heat dynamics of locally symmetric spaces

Abstract: In this paper we first derive several results concerning the $L^p$ spectrum of arithmetic locally symmetric spaces whose $\Q$-rank equals one. In particular, we show that there is an open subset of $\C$ consisting of eigenvalues of the $L^p$ Laplacian if $p <2$ and that corresponding eigenfunctions are given by certain Eisenstein series. On the other hand, if $p>2$ there is at most a discrete set of real eigenvalues of the $L^p$ Laplacian. These results are used in the second part of this paper in order to show that the dynamics of the $L^p$ heat semigroups for $p<2$ is very different from the dynamics of the $L^p$ heat semigroups if $p\geq 2$.


The root closure of a ring of mixed characteristic

Abstract: We define a closure operation for rings of mixed characteristic and verify that the closure is a ring. We then show that this closure produces a ring with good properties with respect to its Fontaine ring and give an example to show that rings that are not closed in this sense do not satisfy these properties.


On a continuous time game with incomplete information

Abstract: For zero-sum two-player continuous-time games with integral payoff and incomplete information on one side, one shows that the optimal strategy of the informed player can be computed through an auxiliary optimization problem over some martingale measures. One also characterizes the optimal martingale measures and compute it explicitely in several examples.


A note on Sierpi\'{n}ski problem related to triangular numbers

Abstract: In this note we show that the system of equations t_{x}+t_{y}=t_{p},\quad t_{y}+t_{z}=t_{q},\quad t_{x}+t_{z}=t_{r}, where $t_{x}=x(x+1)/2$ is a triangular number, has infinitely many solutions in integers. Moreover we show that this system has rational three-parametric solution. Using this result we show that the system t_{x}+t_{y}=t_{p},\quad t_{y}+t_{z}=t_{q},\quad t_{x}+t_{z}=t_{r},\quad t_{x}+t_{y}+t_{z}=t_{s} has infinitely many rational two-parametric solutions.


Differential operators on an affine curve: ideal classes and Picard groups

Abstract: Let X be a smooth complex affine curve, and let R be the space of right ideal classes in the ring D of differential operators on X. We introduce and study a fibration \gamma : R \to Pic(X). We relate this fibration to the corresponding one in the classical limit, and derive an integer invariant $ n $ which indexes the decomposition of the fibres of \gamma into Calogero-Moser spaces (see [BC]). We also study the action of the group Pic(D) on our fibration; and we explain how to define \gamma in the framework of the Grassmannian description of R due to Cannings and Holland.


Rational points on certain quintic hypersurfaces

Abstract: Let $f(x)=x^5+ax^3+bx^2+cx \in \Z[x]$ and consider the hypersurface of degree five given by the equation \cal{V}_{f}: f(p)+f(q)=f(r)+f(s). Under the assumption $b\neq 0$ we show that there exists $\Q$-unirational elliptic surface contained in $\cal{V}_{f}$. If $b=0, a<0$ and $-a\not\equiv 2,18,34 \pmod {48}$ then there exists $\Q$-rational surface contained in $\cal{V}_{f}$. Moreover, we prove that for each $f$ of degree five there exists $\Q(i)$-rational surface contained in $\cal{V}_{f}$.


Reciprocity laws \`a la Iwasawa-Wiles

Abstract: This paper is a brief survey on explicit reciprocity laws of Artin-Hasse-Iwasawa-Wiles type for the Kummer pairing on local fields.


Tensor products of recurrent hypercyclic semigroups

Abstract: We study tensor products of strongly continuous semigroups on Banach spaces that satisfy the hypercyclicity criterion, the recurrent hypercyclicity criterion or are chaotic.


Integer Knapsacks: Average Behavior of the Frobenius Numbers

Abstract: The main result of the paper shows that the asymptotic growth of the Frobenius number in average is significantly slower than the growth of the maximum Frobenius number.


Groupoid normalizers of tensor products

Abstract: We consider an inclusion $B\subseteq M$ of finite von Neumann algebras satisfying $B'\cap M\subseteq B$. A partial isometry $v\in M$ is called a groupoid normalizer if $vBv^*, v^*Bv\subseteq B$. Given two such inclusions $B_i\subseteq M_i$, $i=1,2$, we find approximations to the groupoid normalizers of $B_1 \vnotimes B_2$ in $M_1\vnotimes M_2$, from which we deduce that the von Neumann algebra generated by the groupoid normalizers of the tensor product is equal to the tensor product of the von Neumann algebras generated by the groupoid normalizers. Examples are given to show that this can fail without the hypothesis $B_i'\cap M_i\subseteq B_i$, $i=1,2$. We also prove a parallel result where the groupoid normalizers are replaced by the intertwiners, those partial isometries $v\in M$ satisfying $vBv^*\subseteq B$ and $v^*v, vv^*\in B$.


Hyperbolic conservation laws on spacetimes. A finite volume scheme based on differential forms

Abstract: We consider nonlinear hyperbolic conservation laws, posed on a differential (n+1)-manifold with boundary referred to as a spacetime, and in which the "flux" is defined as a flux field of n-forms depending on a parameter (the unknown variable). We introduce a formulation of the initial and boundary value problem which is geometric in nature and is more natural than the vector field approach recently developed for Riemannian manifolds. Our main assumption on the manifold and the flux field is a global hyperbolicity condition, which provides a global time-orientation as is standard in Lorentzian geometry and general relativity. Assuming that the manifold admits a foliation by compact slices, we establish the existence of a semi-group of entropy solutions. Moreover, given any two hypersurfaces with one lying in the future of the other, we establish a "contraction" property which compares two entropy solutions, in a (geometrically natural) distance equivalent to the L1 distance. To carry out the proofs, we rely on a new version of the finite volume method, which only requires the knowledge of the given n-volume form structure on the (n+1)-manifold and involves the {\sl total flux} across faces of the elements of the triangulations, only, rather than the product of a numerical flux times the measure of that face.


Limiting distributions and large deviations for random walks in random environments

Abstract: This thesis concerns the study of random walks in random environments (RWRE). Since there are two levels of randomness for random walks in random environments, there are two different distributions for the random walk that can be studied. The quenched distribution is the law of the random walk conditioned on a given environment. The annealed distribution is the quenched law averaged over all environments. The main results of the thesis fall into two categories: quenched limiting distributions for one-dimensional, transient RWRE and annealed large deviations for multidimensional RWRE. The analysis of the quenched distributions for transient, one-dimensional RWRE falls into two separate cases. First, when an annealed central limit theorem holds, we prove that a quenched central limit theorem also holds but with a random (depending on the environment) centering. In contrast, when the annealed limit distribution is not Gaussian, we prove that there is no quenched limiting distribution for the RWRE. Moreover, we show that for almost every environment, there exist two random (depending on the environment) sequences of times, along which random walk has different quenched limiting distributions. While an annealed large deviation principle for multidimensional RWRE was known previously, very little qualitative information was available about the annealed large deviation rate function. We prove that if the law on environments is non-nestling, then the annealed large deviation rate function is analytic in a neighborhood of its unique zero (which is the limiting velocity of the RWRE).


The asymptotic behavior of least pseudo-Anosov dilatations

Abstract: For a surface $S$ with $n$ marked points and fixed genus $g\geq2$, we prove that the logarithm of the minimal dilatation of a pseudo-Anosov homeomorphism of $S$ is on the order of $(\log n)/n$. This is in contrast with the cases of genus zero or one where the order is $1/n$.


Affine and toric hyperplane arrangements

Abstract: We extend the Billera-Ehrenborg-Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For arrangements on the torus, we also generalize Zaslavsky's fundamental results on the number of regions.


Stability of holonomicity over quasi-projective varieties

Abstract: Let $\V$ be a mixed characteristic complete discrete valuation ring with perfect residue field $k$. We solve Berthelot's conjectures on the stability of the holonomicity over smooth projective formal $\V$-schemes. Then we build a category of complexes of arithmetic $\D$-modules over quasi-projective $k$-varieties with bounded, $F$-holonomic cohomology. We get its stability under Grothendieck's six operations.


On representable graphs, semi-transitive orientations, and the representation numbers

Abstract: A graph $G=(V,E)$ is representable if there exists a word $W$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $W$ if and only if $(x,y)\in E$ for each $x\neq y$. If $W$ is $k$-uniform (each letter of $W$ occurs exactly $k$ times in it) then $G$ is called $k$-representable. It was shown that a graph is representable if and only if it is $k$-representable for some $k$. Minimum $k$ for which a representable graph $G$ is $k$-representable is called its representation number. In this paper we give a characterization of representable graphs in terms of orientations. Namely, we show that a graph is representable if and only if it admits an orientation into a so-called \emph{semi-transitive digraph}. This allows us to prove a number of results about representable graphs, not the least that 3-colorable graphs are representable. We also prove that the representation number of a graph on $n$ nodes is at most $n$, from which one concludes that the recognition problem for representable graphs is in NP. This bound is tight up to a constant factor, as we present a graph whose representation number is $n/2$. We also answer several questions, in particular, on representability of the Petersen graph and local permutation representability.


Vertex decomposable graphs and obstructions to shellability

Abstract: Inspired by several recent papers on the edge ideal of a graph G, we study the equivalent notion of the independence complex of G. Using the tool of vertex decomposability from geometric combinatorics, we show that 5-chordal graphs with no chordless 4-cycles are shellable and sequentially Cohen-Macaulay. We use this result to characterize the obstructions to shellability in flag complexes, extending work of Billera, Myers, and Wachs. We also show how vertex decomposability may be used to show that certain graph constructions preserve shellability.


On boundedness, existence and uniqueness of strong solutions of the Navier-Stokes Equations in 3 dimensions

Abstract: In this paper we consider the Navier-Stokes Equations in 3 dimensions in the vorticity formulation in the absence of the external forces. We derive upper bounds on L_{infinity} norm of omega and use them together with the Local Existence and Uniqueness results to show Global Existence and Uniqueness of the solution provided that at t=0, L_{infinity} norm of omega is finite, or L_4 norm of omega is finite.


Smooth varieties up to A^1-homotopy and algebraic h-cobordisms

Abstract: We start to study the problem of classifying smooth proper varieties over a field k from the standpoint of A^1-homotopy theory. Motivated by the topological theory of surgery, we discuss the problem of classifying up to isomorphism all smooth proper varieties having a specified A^1-homotopy type. Arithmetic considerations involving the sheaf of A^1-connected components lead us to introduce several different notions of connectedness in A^1-homotopy theory. We provide concrete links between these notions, connectedness of points by chains of affine lines, and various rationality properties of algebraic varieties (e.g., rational connectedness). We introduce the notion of an A^1-h-cobordism, an algebro-geometric analog of the topological notion of h-cobordism, and use it as a tool to produce non-trivial A^1-weak equivalences of smooth proper varieties. Also, we give explicit computations of refined A^1-homotopy invariants, such as the A^1-fundamental sheaf of groups, for some A^1-connected varieties. We observe that the A^1-fundamental sheaf of groups plays a central yet mysterious role in the structure of A^1-h-cobordisms. As a consequence of these observations, we completely solve the classification problem for rational smooth proper surfaces over an algebraically closed field: while there exist arbitrary dimensional moduli of such surfaces, there are only countably many A^1-homotopy types, each uniquely determined by the isomorphism class of its A^1-fundamental sheaf of groups.


On Banach Spaces containing $l_p$ or $c_0$

Abstract: We use the Gowers block Ramsey theorem to characterize Banach spaces containing isomorphs of $\ell_p$ (for some $1 \leq p < \infty$) or $c_0$.


An equivariant index formula for almost-CR manifolds

Abstract: We consider a consider the case of a compact manifold M, together with the following data: the action of a compact Lie group H and a smooth H-invariant distribution E, such that the H-orbits are transverse to E. These data determine a natural equivariant differential form with generalized coefficients J(E,X) whose properties we describe. When E is equipped with a complex structure, we define a class of symbol mappings in terms of the resulting almost-CR structure that are H-transversally elliptic whenever the action of H is transverse to E. We determine a formula for the H-equivariant index of such symbols that involves only J(E,X) and standard equivariant characteristic classes. This formula generalizes the formula given in arXiv:0712.2431 for the case of a contact manifold.


A chain complex and Quadrilaterals for normal surfaces

Abstract: We interpret a normal surface in a (singular) three-manifold in terms of the homology of a chain complex. This allows us to study the relation between normal surfaces and their quadrilateral co-ordinates. Specifically, we give a proof of an (unpublished) observation independently given by Casson and Rubinstein saying that quadrilaterals determine a normal surface up to vertex linking spheres. We also characterise the quadrilateral coordinates that correspond to a normal surface in a (possibly ideal) triangulation.


Non-cyclic graph associated with a group

Abstract: We associate a graph $\mathcal{C}_G$ to a non locally cyclic group $G$ (called the non-cyclic graph of $G$) as follows: take $G\backslash Cyc(G)$ as vertex set, where $Cyc(G)=\{x\in G | < x,y> \text{is cyclic for all} y\in G\}$ is called the cyclicizer of $G$, and join two vertices if they do not generate a cyclic subgroup. For a simple graph $\Gamma$, $w(\Gamma)$ denotes the clique number of $\Gamma$, which is the maximum size (if it exists) of a complete subgraph of $\Gamma$. In this paper we characterize groups whose non-cyclic graphs have clique numbers at most 4. We prove that a non-cyclic group $G$ is solvable whenever $w(\mathcal{C}_G)<31$ and the equality for a non-solvable group $G$ holds if and only if $G/Cyc(G)\cong A_5$ or $S_5$.


Interacting multi-class transmissions in large stochastic networks

Abstract: The mean-field limit of a Markovian model describing the interaction of several classes of permanent connections in a network is analyzed. Each of the connections has a self-adaptive behavior in that its transmission rate along its route depends on the level of congestion of the nodes of the route. Since several classes of connections going through the nodes of the network are considered, an original mean-field result in a multi-class context is established. It is shown that, as the number of connections goes to infinity, the behavior of the different classes of connections can be represented by the solution of an unusual nonlinear stochastic differential equation depending not only on the sample paths of the process, but also on its distribution. Existence and uniqueness results for the solutions of these equations are derived. Properties of their invariant distributions are investigated and it is shown that, under some natural assumptions, they are determined by the solutions of a fixed-point equation in a finite-dimensional space.


Finitely presented algebras and groups defined by permutation relations

Abstract: The class of finitely presented algebras over a field $K$ with a set of generators $a_{1},..., a_{n}$ and defined by homogeneous relations of the form $a_{1}a_{2}... a_{n} =a_{\sigma (a)} a_{\sigma (2)} ... a_{\sigma (n)}$, where $\sigma$ runs through a subset $H$ of the symmetric group $\Sym_{n}$ of degree $n$, is introduced. The emphasis is on the case of a cyclic subgroup $H$ of $\Sym_{n}$ of order $n$. A normal form of elements of the algebra is obtained. It is shown that the underlying monoid, defined by the same (monoid) presentation, has a group of fractions and this group is described. Properties of the algebra are derived. In particular, it follows that the algebra is a semiprimitive domain. Problems concerning the groups and algebras defined by arbitrary subgroups $H$ of $\Sym_{n}$ are proposed.


Arithmetic infinite Grassmannians and the induced central extensions

Abstract: The construction of families of Sato Grassmannians, their determinant line bundles and the extensions induced by them are given. The base scheme is an arbitrary scheme.


Commutative rings in which every finitely generated ideal is quasi-projective

Abstract: This paper studies the multiplicative ideal structure of commutative rings in which every finitely generated ideal is quasi-projective. Section 2 provides some preliminaries on quasi-projective modules over commutative rings. Section 3 investigates the correlation with well-known Prufer conditions; namely, we prove that this class of rings stands strictly between the two classes of arithmetical rings and Gaussian rings. Thereby, we generalize Osofsky's theorem on the weak global dimension of arithmetical rings and partially resolve Bazzoni-Glaz's related conjecture on Gaussian rings. We also establish an analogue of Bazzoni-Glaz results on the transfer of Prufer conditions between a ring and its total ring of quotients. Section 4 examines various contexts of trivial ring extensions in order to build new and original examples of rings where all finitely generated ideals are subject to quasi-projectivity, marking their distinction from related classes of Prufer rings.


Robust Asymptotic Stabilization of Nonlinear Systems with Non-Hyperbolic Zero Dynamics

Abstract: In this paper we present a general tool to handle the presence of zero dynamics which are asymptotically but not locally exponentially stable in problems of robust nonlinear stabilization by output feedback. We show how it is possible to design locally Lipschitz stabilizers under conditions which only rely upon a partial detectability assumption on the controlled plant, by obtaining a robust stabilizing paradigm which is not based on design of observers and separation principles. The main design idea comes from recent achievements in the field of output regulation and specifically in the design of nonlinear internal models.


On o-minimal homotopy groups

Abstract: We work over an o-minimal expansion of a real closed field. The o-minimal homotopy groups of a definable set are defined naturally using definable continuous maps. We prove that any two semialgebraic maps which are definably homotopic are also semialgebraically homotopic. This result together with the study of semialgebraic homotopy done by H. Delfs and M. Knebusch allows us to develop an o-minimal homotopy theory. In particular, we obtain o-minimal versions of the Hurewicz theorems and the Whitehead theorem.


Isometric action of SL(2,R) on homogeneous spaces

Abstract: We investigate the SL(2,R) invariant geodesic curves with the as- sociated invariant distance function in parabolic geometry. Parabolic geom- etry naturally occurs in the study of SL(2,R) and is placed in between the elliptic and the hyperbolic (also known as the Lobachevsky half-plane and 2- dimensional Minkowski half-plane space-time) geometries. Initially we attempt to use standard methods of finding geodesics but they lead to degeneracy in this setup. Instead, by studying closely the two related elliptic and hyperbolic geometries we discover a unified approach to a more exotic and less obvious parabolic case. With aid of common invariants we describe the possible dis- tance functions that turn out to have some unexpected, interesting properties.


On the field intersection problem of generic polynomials: a survey

Abstract: Let $k$ be a field of characteristic $\neq 2$. We survey a general method of the field intersection problem of generic polynomials via formal Tschirnhausen transformation. We announce some of our recent results of cubic, quartic and quintic cases the details of which are to appear elsewhere. In this note, we give an explicit answer to the problem in the cases of cubic and dihedral quintic by using multi-resolvent polynomials.


A Lindemann-Weierstrass theorem for semiabelian varieties over function fields

Abstract: We prove an analogue of the Lindemann-Weierstrass theorem (that the exponentials of Q-linearly independent algebraic numbers are algebraically independent) for commutative algebraic groups G without unipotent quotients, over function fields. We concentrate on solutions to the the differential algebraic relations satisfied by exp from LG to G.


Pointwise estimates for the Bergman kernel of the weighted Fock space

Abstract: We prove upper pointwise estimates for the Bergman kernel of the weighted Fock space of entire functions in $L^2(e^{-2\phi})$ where $\phi$ is a subharmonic function with $\Delta \phi$ a doubling measure. We derive estimates for the canonical solution operator to the inhomogeneous Cauchy-Riemann equation and we characterize the compactness of this operator in terms of $\Delta \phi$.


Decision problems and profinite completions of groups

Abstract: We consider pairs of finitely presented, residually finite groups $P\hookrightarrow\G$ for which the induced map of profinite completions $\hat P\to \hat\G$ is an isomorphism. We prove that there is no algorithm that, given an arbitrary such pair, can determine whether or not $P$ is isomorphic to $\G$. We construct pairs for which the conjugacy problem in $\G$ can be solved in quadratic time but the conjugacy problem in $P$ is unsolvable. Let $\mathcal J$ be the class of super-perfect groups that have a compact classifying space and no proper subgroups of finite index. We prove that there does not exist an algorithm that, given a finite presentation of a group $\G$ and a guarantee that $\G\in\mathcal J$, can determine whether or not $\G\cong\{1\}$. We construct a finitely presented acyclic group $\H$ and an integer $k$ such that there is no algorithm that can determine which $k$-generator subgroups of $\H$ are perfect.


Passage-time moments and hybrid zones for the exclusion-voter model

Abstract: We study the non-equilibrium dynamics of a one-dimensional interacting particle system that is a mixture of the voter model and the exclusion process. With the process started from a finite perturbation of the ground state Heaviside configuration consisting of 1's to the left of the origin and 0's elsewhere, we study the relaxation time $\tau$, that is, the first hitting time of the ground state configuration (up to translation). We give conditions for $\tau$ to be finite and for certain moments of $\tau$ to be finite or infinite, and prove a result that approaches a conjecture of Belitsky et al. (Bernoulli 7 (2001) 119--144). Ours are the first non-existence-of-moments results for $\tau$ for the mixture model. Moreover, we give almost sure asymptotics for the evolution of the size of the hybrid (disordered) region. Most of our results pertain to the discrete-time setting, but several transfer to continuous-time. As well as the mixture process, some of our results also cover pure exclusion. We state several significant open problems.


On a symmetric space attached to polyzeta values

Abstract: Quickly convergent series are given to compute polyzeta numbers. The formula involves an intricate combination of (generalized) polylogarithms at 1/2. However, the combinatorics has a very simple geometric interpretation: it corresponds with the square map on some symmetric space P.


Quantum $\mathrm{SO}(3)$ groups and quantum group actions on $M_2$

Abstract: Answering a question of Shuzhou Wang we give a description of quantum $\SO(3)$ groups of Podle\'s as universal objects. We use this result to give a complete classification of all continuous compact quantum group actions on $M_2$.


Direct factors of profinite completions and decidability

Abstract: We consider finitely presented,residually finite groups $G$ and finitely generated normal subgroups $A$ such that the inclusion $A\hookrightarrow G$ induces an isomorphism from the profinite completion of $A$ to a direct factor of the profinite completion of $G$. We explain why $A$ need not be a direct factor of a subgroup of finite index in $G$; indeed $G$ need not have a subgroup of finite index that splits as a non-trivial direct product. We prove that there is no algorithm that can determine whether $A$ is a direct factor of a subgroup of finite index in $G$.


On the relation between cluster and classical tilting

Abstract: Let D be a triangulated category with a cluster tilting subcategory U. The quotient category D/U is abelian; suppose that it has finite global dimension. We show that projection from D to D/U sends cluster tilting subcategories of D to support tilting subcategories of D/U, and that, in turn, support tilting subcategories of D/U can be lifted uniquely to maximal 1-orthogonal subcategories of D.


Fixed point loci of moduli spaces of sheaves on toric varieties

Abstract: Extending work of Klyachko and Perling, we develop a combinatorial description of pure equivariant sheaves of any dimension on an arbitrary nonsingular toric variety $X$. Using geometric invariant theory (GIT), this allows us to construct explicit moduli spaces of pure equivariant sheaves on $X$ corepresenting natural moduli functors (similar to work of Payne in the case of equivariant vector bundles). The action of the algebraic torus on $X$ lifts to the moduli space of all Gieseker stable sheaves on $X$ and we express its fixed point locus explicitly in terms of moduli spaces of pure equivariant sheaves on $X$. One of the problems arising is to find an equivariant line bundle on the side of the GIT problem, which precisely recovers Gieseker stability. In the case of torsion free equivariant sheaves, we can always construct such equivariant line bundles. As a by-product, we get a combinatorial description of the fixed point locus of the moduli space of $\mu$-stable reflexive sheaves on $X$. As an application, we show in a sequel how these methods can be used to compute generating functions of Euler characteristics of moduli spaces of $\mu$-stable torsion free sheaves on nonsingular complete toric surfaces.


Continuity of ring *-homomorphisms between C*-algebras

Abstract: The purpose of this short note is to prove that if $A$ and $B$ are unital C*-algebras and $\phi : A \to B$ is a unital *-preserving ring homomorphism, then $\phi$ is contractive; i.e., $\| \phi (a) \| \leq \| a \|$ for all $a \in A$. (Note that we do not assume $\phi$ is linear.) We use this result to deduce a number of corollaries as well as characterize the form of such unital *-preserving ring homomorphisms. (This note may be of interest to C*-algebraists as well as algebraists who study noncommutative rings and algebras. It is meant to be accessible to a general mathematician and does not require any prior knowledge of C*-algebras.)


A New Bijection Between Forests and Parking Functions

Abstract: In 1980, G. Kreweras gave a recursive bijection between forests and parking functions. In this paper we construct a nonrecursive bijection from forests onto parking functions, which answers a question raised by R. Stanley. As a by-product, we obtain a bijective proof of Gessel and Seo's formula for lucky statistic on parking functions.


Estimating the Parameters of Binomial and Poisson Distributions via Multistage Sampling

Abstract: In this paper, we have developed a new class of sampling schemes for estimating parameters of binomial and Poisson distributions. Without any information of the unknown parameters, our sampling schemes rigorously guarantee prescribed levels of precision and confidence.


Combinatorial Hopf algebras

Abstract: We define a "combinatorial Hopf algebra" as a Hopf algebra which is free (or cofree) and equipped with a given isomorphism to the free algebra over the indecomposables (resp. the cofree coalgebra over the primitives). The choice of such an isomorphism implies the existence a finer algebraic structure on the Hopf algebra and on the indecomposables (resp. the primitives). For instance a cofree-cocommutative right-sided combinatorial Hopf algebra is completely determined by its primitive part which is a pre-Lie algebra. The key example is the Connes-Kreimer Hopf algebra. The study of all these combinatorial Hopf algebra types gives rise to several good triples of operads. It involves the operads: dendriform, pre-Lie, brace, and variations of them.


Obstacle problem for SPDE with nonlinear Neumann boundary condition via reflected generalized backward doubly SDEs

Abstract: This paper is intended to give a probabilistic representation for stochastic viscosity solution of semi-linear reflected stochastic partial differential equations with nonlinear Neumann boundary condition. We use it connection with reflected generalized backward doubly stochastic differential equation.


Extended Zeilberger's Algorithm for Identities on Bernoulli and Euler Polynomials

Abstract: We present a computer algebra approach to proving identities on Bernoulli polynomials and Euler polynomials by using the extended Zeilberger's algorithm given by Chen, Hou and Mu. The key idea is to use the contour integral definitions of the Bernoulli and Euler numbers to establish recurrence relations on the integrands. Such recurrence relations have certain parameter free properties which lead to the required identities without computing the integrals.


Irreducible Lie-Yamaguti algebras

Abstract: Lie-Yamaguti algebras (or generalized Lie triple systems) are binary-ternary algebras intimately related to reductive homogeneous spaces. The Lie-Yamaguti algebras which are irreducible as modules over their Lie inner derivation algebra are the algebraic counterpart of the isotropy irreducible homogeneous spaces. These systems will be shown to split into three disjoint types: adjoint type, non-simple type and generic type. The systems of the first two types will be classified and most of them will be shown to be related to a Generalized Tits Construction of Lie algebras.


A map from Lawson homology to Deligne Cohomology

Abstract: A natural map from Lawson homology to Deligne cohomology groups for smooth complex projective varieties is constructed by using the Harvey-Lawson spark complexes. We also compare this to Abel-Jacobi type constructions by others.


Polynomial maps over $p$-adics and residual properties of mapping tori of group endomorphisms

Abstract: We continue our study of residual properties of mapping tori of free group endomorphisms. In this paper, we prove that each of these groups are virtually residually (finite $p$)-groups for all but finitely many primes$p$. The method involves further studies of polynomial maps over finite fields and $p$-adic completions of number fields.


Free holomorphic automorphisms of the unit ball of $B(H)^n$

Abstract: The theory of characteristic functions for row contractions is used to determine the group $Aut(B(H)^n_1)$ of all free holomorphic automorphisms of the unit ball of $B(H)^n$. We show that the noncommutative Poisson transform commutes with the action of the automorphism group $Aut(B(H)^n_1)$. This leads to a characterization of the unitarily implemented automorphisms of the Cuntz-Toeplitz algebra $C^*(S_1,..., S_n)$, which leave invariant the noncommutative disc algebra $\cA_n$. This result provides new insight into Voiculescu's group of automorphisms of the Cuntz-Toeplitz algebra and reveals new connections with noncommutative multivariable operator theory, especially, the theory of characteristic functions for row contractions and the noncommutative Poisson transforms. We study the isometric dilations and the characteristic functions of row contractions under the action of the automorphism group $Aut(B(H)^n_1)$. This enables us to obtain some results concerning the behavior of the curvature and the Euler characteristic of a row contraction under $Aut(B(H)^n_1)$. We prove a maximum principle for free holomorphic functions on the noncommutative ball $[B(H)^n]_1$ and provide some extensions of the classical Schwarz lemma to our noncommutative setting.


Linear extension of the Erdos-Heilbronn conjecture

Abstract: The famous Erdos-Heilbronn conjecture plays an important role in the development of additive combinatorics. In 2007 Z. W. Sun made the following further conjecture (which is the linear extension of the Erdos-Heilbronn conjecture): For any finite subset A of a field F and nonzero elements $a_1,...,a_n$ of F, the set {a_1x_1+...+a_nx_n: x_1,....,x_n are distinct elements of A} has cardinality at least min{p(F)-delta, n(|A|-n)+1}, where the additive order p(F) of the multiplicative identity of F is different from n+1, and delta=0,1 takes the value 1 if and only if n=2 and $a_1+a_2=0$. In this paper we prove this conjecture of Sun when $p(F)\geq n(3n-5)/2$. We also obtain a sharp lower bound for the cardinality of the restricted sumset {x_1+...+x_n: x_1\in A_1,...,x_n\in A_n, and P(x_1,...,x_n)\not=0}, where $A_1,...,A_n$ are finite subsets of a field F and $P(x_1,...,x_n)$ is a general polynomial over F.


The Algebra of Formal Twisted Pseudodifferential Symbols and a Noncommutative Residue

Abstract: We extend the Adler-Manin trace on the algebra of pseudodifferential symbols to a twisted setting.


Exact solutions to Waring's problem for finite fields

Abstract: The Waring function $g(k,q)$ measures the difficulty of Waring's problem for $k$th powers in the field of $q$ elements. Its calculation seems to be difficult, and many partial results have been published, notably upper bounds for certain regions of the $k$-$q$-plane. In this paper, we compute the exact value of $g(k,q)$ for two infinite families of exponent-field pairs. In these, $k$ is large compared to $q$. We use a new method of proof that is mainly combinatorial in nature.


On some Fano--Enriques threefolds

Abstract: We give a classification of Fano threefolds $X$ with canonical Gorenstein singularities such that $X$ possess a regular involution, which acts freely on some smooth surface in $|-K_X|$, and the linear system $|-K_X|$ gives a morphism which is not an embedding. From this classification one gets, in particular, a description of some natural class of Fano--Enriques threefolds.


Nonlinear stability of time-periodic viscous shocks

Abstract: In order to understand the nonlinear stability of many types of time-periodic travelling waves on unbounded domains, one must overcome two main difficulties: the presence of embedded neutral eigenvalues and the time-dependence of the associated linear operator. This problem is studied in the context of time-periodic Lax shocks in systems of viscous conservation laws. Using spatial dynamics and a decomposition into separate Floquet eigenmodes, it is shown that the linear evolution for the time-dependent operator can be represented using a contour integral similar to that of the standard time-independent case. By decomposing the resulting Green's distribution, the leading order behavior associated with the embedded eigenvalues is extracted. Sharp pointwise bounds are then obtained, which are used to prove that time-periodic Lax shocks are linearly and nonlinearly stable under the necessary conditions of spectral stability and minimal multiplicity of the translational eigenvalues. The latter conditions hold, for example, for small-oscillation time-periodic waves that emerge through a supercritical Hopf bifurcation from a family of time-independent Lax shocks of possibly large amplitude.


An overview of abelian varieties in homotopy theory

Abstract: This is an expository article on the theory of formal group laws in homotopy theory, with the goal of leading to the connection with higher-dimensional abelian varieties and automorphic forms. These are roughly based on a talk at the conference "New Topological Contexts for Galois Theory and Algebraic Geometry."


Drawing disconnected graphs on the Klein bottle

Abstract: We prove that two disjoint graphs must always be drawn separately on the Klein bottle, in order to minimize the crossing number of the whole drawing.


Coxeter group actions on 4F3(1) hypergeometric series

Abstract: We investigate a certain linear combination $K(\vec{x})=K(a;b,c,d;e,f,g)$ of two Saalschutzian hypergeometric series of type ${_4}F_3(1)$. We first show that $K(a;b,c,d;e,f,g)$ is invariant under the action of a certain matrix group $G_K$, isomorphic to the symmetric group $S_6$, acting on the affine hyperplane $V=\{(a,b,c,d,e,f,g)\in\Bbb C^7\colon e+f+g-a-b-c-d=1\}$. We further develop an algebra of three-term relations for $K(a;b,c,d;e,f,g)$. We show that, for any three elements $\mu_1,\mu_2,\mu_3$ of a certain matrix group $M_K$, isomorphic to the Coxeter group $W(D_6)$ (of order 23040), and containing the above group $G_K$, there is a relation among $K(\mu_1\vec{x})$, $K(\mu_2\vec{x})$, and $K(\mu_3\vec{x})$, provided no two of the $\mu_j$'s are in the same right coset of $G_K$ in $M_K$. The coefficients in these three-term relations are seen to be rational combinations of gamma and sine functions in $a,b,c,d,e,f,g$. The set of $({|M_K|/|G_K|\atop 3})=({32\atop 3})=4960$ resulting three-term relations may further be partitioned into five subsets, according to the Hamming type of the triple $(\mu_1,\mu_2,\mu_3) $ in question. This Hamming type is defined in terms of Hamming distance between the $\mu_j$'s, which in turn is defined in terms of the expression of the $\mu_j$'s as words in the Coxeter group generators. Each three-term relation of a given Hamming type may be transformed into any other of the same type by a change of variable. An explicit example of each of the five types of three-term relations is provided.


Boundary estimates for positive solutions to second order elliptic equations

Abstract: Consider positive solutions to second order elliptic equations with measurable coefficients in a bounded domain, which vanish on a portion of the boundary. We give simple necessary and sufficient geometric conditions on the domain, which guarantee the Hopf-Oleinik type estimates and the boundary Lipschitz estimates for solutions. These conditions are sharp even for harmonic functions.


Finite simple groups with narrow prime spectrum

Abstract: We find the nonabelian finite simple groups with order prime divisors not exceeding 1000. More generally, we determine the sets of nonabelian finite simple groups whose maximal order prime divisor is a fixed prime less than 1000. Our results are based on calculations in the computer algebra system GAP.


Intersection exponents for biased random walks on discrete cylinders

Abstract: We prove existence of intersection exponents xi(k,lambda) for biased random walks on d-dimensional half-infinite discrete cylinders, and show that, as functions of lambda, these exponents are real analytic. As part of the argument, we prove convergence to stationarity of a time-inhomogeneous Markov chain on half-infinite random paths. Furthermore, we show this convergence takes place at exponential rate, an estimate obtained via a coupling of weighted half-infinite paths.


A note on Kaehler-Ricci flow

Abstract: Let $g(t)$ with $t\in [0,T)$ be a complete solution to the Kaehler-Ricci flow: $\frac{d}{dt}g_{i\bar j}=-R_{i\bar j}$ where $T$ may be $\infty$. In this article, we show that the curvatures of $g(t)$ is uniformly bounded if the solution $g(t)$ is uniformly equivalet. This result is stronger than the main result in \v{S}e\v{s}um \cite{sesum} within the category of K\"ahler-Ricci flow.


A geometric proof of the classification of complex vector cross product

Abstract: In this article, we give a geometric proof of the classification of complex vector cross product due to Lee-Leung.


Negativity of Perelman's Li-Yau-Hamilton type expression

Abstract: Chau-Tam-Yu has proved the non-positivity of Perelman's new Li-Yau-Hamilton type expression on noncompact manifolds. In this article, we further prove that $v$ is negative if the Ricci flow is not end up with an Euclidean space.


Some Estimates of Fundamental Solution on noncompact manifolds with time-dependent metrics

Abstract: In this article, we obtain some further estimates of fundamental solutions comparing to the result of Chau-Tam-Yu and give some applications of the estimates on asymptotic behaviors of fundamental solutions.


Almost Sure Convergence of Extreme Order Statistics

Abstract: Let $M_n^{(k)}$ denote the $k$th largest maximum of a sample $(X_1,X_2,...,X_n)$ from parent $X$ with continuous distribution. Assume there exist normalizing constants $a_n>0$, $b_n\in \mathbb{R}$ and a nondegenerate distribution $G$ such that $a_n^{-1}(M_n^{(1)}-b_n)\stackrel{w}{\to}G$. Then for fixed $k\in \mathbb{N}$, the almost sure convergence of \[\frac{1}{D_N}\sum_{n=k}^Nd_n\mathbb{I}\{M_n^{(1)}\le a_nx_1+b_n,M_n^{(2)}\le a_nx_2+b_n,...,M_n^{(k)}\le a_nx_k+b_n\}\] is derived if the positive weight sequence $(d_n)$ with $D_N=\sum_{n=1}^Nd_n$ satisfies conditions provided by H\"{o}rmann.


Cobordism categories of manifolds with corners

Abstract: In this paper we study the topology of cobordism categories of manifolds with corners. Specifically, if {Cob}_{d,<k>} is the category whose objets are a fixed dimension d, with corners of codimension less than or equal to k, then we identify the homotopy type of the classifying space B{Cob}_{d,<k>} as the zero space of a homotopy colimit of certain diagram of Thom spectra. We also identify the homotopy type of the corresponding cobordism category when extra tangential structure is assumed on the manifolds. These results generalize the results of Galatius, Madsen, Tillmann and Weiss, and the proofs are an adaptation of the their methods. As an application we describe the homotopy type of the category of open and closed strings with a background space X, as well as its higher dimensional analogues. This generalizes work of Baas-Cohen-Ramirez and Hanbury.


Differentiability of Banach Spaces via Constructible Sets

Abstract: the main goal of this paper is to prove that any Banach space X, that every dual ball in X** is weak* -separable, or every weak* -closed convex subset in X** is weak* -separable, or every norm-closed convex set in X* is constructible, admits an equivalent Frechet differentiable norm.


Convexity of Chebyshev Sets Through Differentiability of Distance Function

Abstract: In this paper, we study a part of approximation theory that presents the conditions under which a \Ceby\sev set in a Banach space is convex. To do so, we use Gateaux differentiability of the distance function.


Expansive homoclinic classes

Abstract: We prove that for $C^1$ generic diffeomorphisms, every expansive homoclinic class is hyperbolic.


On certain permutation groups and sums of two squares

Abstract: We consider the question of existence of ramified covers over P_1 matching certain prescribed ramification conditions. This problem has already been faced in a number of papers, but we discuss alternative approaches for an existence proof, involving elliptic curves and universal ramified covers with signature. We also relate the geometric problem with finite permutation groups and with the Fermat-Euler Theorem on the representation of a prime as a sum of two squares.


Information inequalities and a dependent Central Limit Theorem

Abstract: We adapt arguments concerning information-theoretic convergence in the Central Limit Theorem to the case of dependent random variables under Rosenblatt mixing conditions. The key is to work with random variables perturbed by the addition of a normal random variable, giving us good control of the joint density and the mixing coefficient. We strengthen results of Takano and of Carlen and Soffer to provide entropy-theoretic, not weak convergence.


On the Combinatorics of the Boros-Moll Polynomials

Abstract: The Boros-Moll polynomials arise in the evaluation of a quartic integral. The original double summation formula does not imply the fact that the coefficients of these polynomials are positive. Boros and Moll proved the positivity by using Ramanujan's Master Theorem to reduce the double sum to a single sum. Based on the structure of reluctant functions introduced by Mullin and Rota along with an extension of Foata's bijection between Meixner endofunctions and bi-colored permutations, we find a combinatorial proof of the positivity. In fact, from our combinatorial argument one sees that it is essentially the binomial theorem that makes it possible to reduce the double sum to a single sum.


On quantum semigroup actions on finite quantum spaces

Abstract: We show that a continuous action of a quantum semigroup $\mathcal{S}$ on a finite quantum space (finite dimensional $\mathrm{C}^*$-algebra) preserving a faithful state comes from a continuous action of the quantum Bohr compactification $\mathfrak{b}\mathcal{S}$ of $\mathcal{S}$. Using the classification of continuous compact quantum group actions on $M_2$ we give a complete description of all continuous quantum semigroup actions on this quantum space preserving a faithful state.


Translated Poisson approximation for Markov chains

Abstract: The paper is concerned with approximating the distribution of a sum W of n integer valued random variables Y_i, whose distributions depend on the state of an underlying Markov chain X. The approximation is in terms of a translated Poisson distribution, with mean and variance chosen to be close to those of W, and the error is measured with respect to the total variation norm. Error bounds comparable to those found for normal approximation with respect to the weaker Kolmogorov distance are established, provided that the distribution of the sum of the Y_i's between the successive visits of X to a reference state is aperiodic. Without this assumption, approximation in total variation cannot be expected to be good.


On meromorphic extendibility

Abstract: Let D be a bounded domain in the complex plane whose boundary bD consists of finitely many pairwise disjoint real analytic simple closed curves. Let f be an integrable function on bD. In the paper we show how to compute the candidates for poles of a meromorphic extension of f through D and thus reduce the question of meromorphic extendibility to the question of holomorphic extendibility. Let A(D) be the algebra of all continuous functions on the closure of D which are holomorphic on D. For continuous functions f on bD we obtain a characterization of meromorphic extendibility in terms of the argument principle: f extends meromorphically through D if and only if there is a nonnegative integer N such that the winding number of Pf+Q along bD is bounded below by -N for all P, Q in A(D) such that Pf+Q has no zero on bD. If this is the case then the meromorphic extension of f has at most N poles in D, counting multiplicity.


Rough Sets Determined by Quasiorders

Abstract: In this paper, the ordered set of rough sets determined by a quasiorder relation $R$ is investigated. We prove that this ordered set is a complete, completely distributive lattice. We show that on this lattice can be defined three different kinds of complementation operations, and we describe its completely join-irreducible elements. We also characterize the case in which this lattice is a Stone lattice. Our results generalize some results of J. Pomykala and J. A. Pomykala (1988) and M. Gehrke and E. Walker (1992) in case $R$ is an equivalence.


Geometric structure in the principal series of the p-adic group G_2

Abstract: In the representation theory of reductive $p$-adic groups $G$, the issue of reducibility of induced representations is an issue of great intricacy. It is our contention, expressed as a conjecture in [3], that there exists a simple geometric structure underlying this intricate theory. We will illustrate here the conjecture with some detailed computations in the principal series of $G_2$. A feature of this article is the role played by cocharacters $h_c$ attached to two-sided cells $c$ in certain extended affine Weyl groups. The quotient varieties which occur in the Bernstein programme are replaced by extended quotients. We form the disjoint union $A(G)$ of all these extended quotient varieties. We conjecture that, after a simple algebraic deformation, the space $A(G)$ is a model of the smooth dual $Irr(G)$. In this respect, our programme is a conjectural refinement of the Bernstein programme. The algebraic deformation is controlled by cocharacters $h_c$, one for each two-sided cell $c$ in certain extended affine Weyl groups. The cocharacters themselves appear to be closely related to Langlands parameters.


Noncommutative hyperbolic geometry on the unit ball of $B(H)^n$

Abstract: In this paper we introduce a hyperbolic distance $\delta$ on the noncommutative open ball $[B(H)^n]_1$, where $B(H)$ is the algebra of all bounded linear operators on a Hilbert space $H$, which is a noncommutative extension of the Poincare-Bergman metric on the open unit ball of $C^n$. We prove that $\delta$ is invariant under the action of the group $Aut([B(H)^n]_1)$ of all free holomorphic automorphisms of $[B(\cH)^n]_1$, and show that the $\delta$-topology and the usual operator norm topology coincide on $[B(H)^n]_1$. Moreover, we prove that $[B(H)^n]_1$ is a complete metric space with respect to the hyperbolic metric and obtained an explicit formula for $\delta$ in terms of the reconstruction operator. A Schwarz-Pick lemma for bounded free holomorphic functions on $[B(H)^n]_1$, with respect to the hyperbolic metric, is also obtained.


Elliptic Equations Involving Meausres

Abstract: We present the moste recent results dealing with the theory of semilinear elliptic equations with measures data


From persistent random walks to the telegraph noise

Abstract: We study a family of memory-based persistent random walks and we prove weak convergences after space-time rescaling. The limit processes are not only Brownian motions with drift. We have obtained a continuous but non-Markov process $(Z_t)$ which can be easely expressed in terms of a counting process $(N_t)$. In a particular case the counting process is a Poisson process, and $(Z_t)$ permits to represent the solution of the telegraph equation. We study in detail the Markov process $((Z_t,N_t); t\ge 0)$.


Reducibility of quasiperiodic cocycles in linear Lie groups

Abstract: Let G be a linear Lie group. We define the G-reducibility of a continuous or discrete cocycle modulo N. We show that a G-valued continuous or discrete cocycle which is GL(n,C)-reducible is in fact G-reducible modulo 2 if G=GL(n,R),SL(n,R),Sp(n,R) or O(n) and modulo 1 if G=U(n).


Disques J-holomorphes contenus dans une hypersurface

Abstract: We study germs of J-Holomorphic curves contained in $M$, a real analytic hypersurface of an symplectic manifold of dimension 4. We show, under topological hypothesis on $M$, that if $M$ is compact then $M$ is of finite type and so there is no germs of $J$-holomorphic curves on $M$(with $J$ adapted with the symplectic form). In $\Bbb{C}^2$ with the standard complex structure, this is a classical result of Diederich-Fornaess.


The p-Laplace heat equation with a source term : self-similar solutions revisited

Abstract: We study the self-similar solutions of any sign of the equation u_{t}-div(|∇u|^{p-2}∇u)=|u|^{q-1}u, in R^{N}, where p,q>1. We extend the results of Haraux-Weissler obtained for p=2 to the case q>p-1>0. In particular we study the existence of slow or fast decaying solutions. For given t>0, the fast solutions u(t,.) have a compact support in R^{N} when p>2, and |x|^{p/(2-p)}u(t,x) is bounded at infinity when p<2. We describe the behaviour for large |x| of all the solutions. According to the position of q with respect to the first critical exponent p-1+p/N and the critical Sobolev exponent q^{∗}, we study the existence of positive solutions, or the number of the zeros of u(t,.). We prove that any solution u(t,.) is oscillatory when p<2 and q is closed to 1.


Hyperbolic geometry on the unit ball of $B(H)^n$ and dilation theory

Abstract: In this paper we continue our investigation concerning the hyperbolic geometry on the noncommutative ball $[B(H)^n]_1^-$, where $B(H)$ is the algebra of all bounded linear operators on a Hilbert space $H$, and its implications to noncommutative function theory. The central object is an intertwining operator $L_{B,A}$ of the minimal isometric dilations of $A, B\in [B(H)^n]_1^-$, which establishes a strong connection between noncommutative hyperbolic geometry on $[B(H)^n]_1^-$ and multivariable dilation theory. The goal of this paper is to study the operator $L_{B,A}$ and its connections to the hyperbolic metric $\delta$ on the Harnack parts of $[B(H)^n]_1^-$. We study the geometric structure of the operator $L_{B,A}$ and obtain new characterizations for the Harnack domination (resp. equivalence) in $[B(H)^n]_1^-$. We express $\|L_{B,A}\|$ in terms of the reconstruction operators $R_A$ and $R_B$, and obtain a Schwartz-Pick lemma for contractive free holomorphic functions on $[B(H)^n]_1$ with respect to the intertwining operator $L_{B,A}$. As a consequence, we deduce a Schwartz-Pick lemma for operator-valued multipliers of the Drury-Arveson space, with respect to the hyperbolic metric.


Quantum Isometry groups of the Podles Spheres

Abstract: For $\mu \in (0,1), c> 0,$ we identify the quantum group $SO_\mu(3)$ as the universal object in the category of compact quantum groups acting by `orientation and volume preserving isometries' in the sense of \cite{goswami2} on the natural spectral triple on the Podles sphere $S^2_{\mu, c}$ constructed by Dabrowski, D'Andrea, Landi and Wagner in \cite{{Dabrowski_et_al}}.


Brownian motion and the parabolicity of minimal graphs

Abstract: We prove that minimal graphs (other than planes) are parabolic in the sense that any bounded harmonic function is determined by its boundary values. The proof relies on using the coupling introduced in the author's earlier paper "A martingale approach to minimal surfaces" to show that Brownian motion on such a minimal graph almost surely strikes the boundary in finite time.


Bordered Heegaard Floer homology: Invariance and pairing

Abstract: We construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes in two different versions, one of which (type D) is a module over the algebra and the other of which (type A) is an A-infinity module. Both are well-defined up to chain homotopy equivalence. For a decomposition of a 3-manifold into two pieces, the A-infinity tensor product of the type D module of one piece and the type A module from the other piece is HF^ of the glued manifold. As a special case of the construction, we specialize to the case of three-manifolds with torus boundary. This case can be used to give another proof of the surgery exact triangle for HF^. We relate the bordered Floer homology of a three-manifold with torus boundary with the knot Floer homology of a filling.


Regular functions on covers of nilpotent coadjoint orbits

Abstract: In this paper I explore the relationship between regular functions associated to local systems on nilpotent orbits and unipotent representations in the complex groups.


A recursive presentation for Mihailova's subgroup

Abstract: We give an explicit recursive presentation for Mihailova's subgroup $M(H)$ of $F_n \times F_n$ corresponding to a finite, concise and Peiffer aspherical presentation $H=< x_1,..., x_n \mid R_1,..., R_m>$. This partially answers a question of R.I. Grigorchuk, [8, Problem 4.14]. As a corollary, we construct a finitely generated recursively presented orbit undecidable subgroup of $Aut(F_3)$.


Slicing planar grid diagrams: a gentle introduction to bordered Heegaard Floer homology

Abstract: We describe some of the algebra underlying the decomposition of planar grid diagrams. This provides a useful toy model for an extension of Heegaard Floer homology to 3-manifolds with parametrized boundary. This paper is meant to serve as a gentle introduction to the subject, and does not itself have immediate topological applications.


Birational aspects of the geometry of M_g

Abstract: We discuss topics on the geometry of the moduli space of curves. We present a short proof of the Harris-Mumford theorem on the Kodaira dimension of the moduli space which replaces the computations on the stack of admissible covers by a simple study of tautological Koszul bundles on M_g. We also present a streamlined self-contained account of Verra's recent work on the unirationality of M_g. Finally, we discuss a proof that the moduli space of curves of genus 22 is of general type. Written for Surveys in Differential Geometry.


On the fundamental group of II_1 factors and equivalence relations arising from group actions

Abstract: Given a countable group G, we consider the sets S_factor(G), S_eqrel(G), of subgroups F of the positive real line for which there exists a free ergodic probability measure preserving action G on X such that the fundamental group of the associated II_1 factor, respectively orbit equivalence relation, equals F. We prove that if G is the free product of Z and infinitely many copies of a non-trivial group \Gamma, then S_factor(G) and S_eqrel(G) contain R_+ itself, all of its countable subgroups, as well as uncountable subgroups whose log can have any Hausdorff dimension in the interval (0,1). We then prove that if G=\Gamma*\Lambda, with \Gamma, \Lambda finitely generated ICC groups, one of which has property (T), then S_factor(G)=S_eqrel(G)={1}. We also show that there exist II_1 factors M such that the fundamental group of M is R_+, but the associated II_\infty factor M tensor B(l^2) admits no continuous trace scaling action of R_+.


Statistics of incomplete quotients of continued fractions of quadratic irrationalities

Abstract: V.I. Arnold has experimentally established that the limit of the statistics of incomplete quotients of partial continued fractions of quadratic irrationalities coincides with the Gauss--Kuz'min statistics. Below we briefly prove this fact for roots of the equation $r x^2+p x=q$ with fixed $p$ and $r$ ($r>0$), and with random $q$, $q\le R$, $R\to \infty$. In Section 3 we estimate the sum of incomplete quotients of the period. According to the obtained bound, prior to the passage to the limit, incomplete quotients in average are logarithmically small. We also upper estimate the proportion of the "red" numbers among those representable as a sum of two squares.


A negative mass theorem for surfaces of positive genus

Abstract: We define the "sum of squares of the wavelengths" of a Riemannian surface (M,g) to be the regularized trace of the inverse of the Laplacian. We normalize by scaling and adding a constant, to obtain a "mass", which is scale invariant and vanishes at the round sphere. This is an anlaog for closed surfaces of the ADM mass from general relativity. We show that if M has positive genus then on each conformal class, the mass attains a negative minimum. For the minimizing metric, there is a sharp logarithmic Hardy-Littlewood-Sobolev inequality and a Moser-Trudinger-Onofri type inequality.


BCOV theory via Givental group action on cohomological field theories

Abstract: In a previous paper (arXiv:0704.1001), Losev, me, and Shneiberg constructed a full descendant potential associated to an arbitrary cyclic Hodge dGBV algebra. This contruction extended the construction of Barannikov and Kontsevich of solution of the WDVV equation, based on the earlier paper of Bershadsky, Cecotti, Ooguri, and Vafa. In the present paper, we give an interpretation of this full descendant potential in terms of Givental group action on cohomological field theories. In particular, the fact that it satisfies all tautological equations becomes a trivial observation.


A primer on Seshadri constants

Abstract: Seshadri constants express the so called local positivity of a line bundle on a projective variety. They were introduced by Demailly. The original idea of using them towards a proof of the Fujita conjecture failed but they quickly became a subject of intensive study quite in their own right. Lazarsfeld's book "Positivity in Algebraic Geometry" contains a whole chapter devoted to local positivity and serves as a very enjoyable introduction to Seshadri constants. Since this book has appeared, the subject witnessed quite a bit of development. It is the aim of these notes to give an account of recent progress as well as to discuss many open questions and provide some examples.


On the structure of Goulden-Jackson-Vakil formula

Abstract: We study the structure of the Goulden-Jackson-Vakil formula that relates Hurwitz numbers to some conjectural "intersection numbers" on a conjectural family of varieties $X_{g,n}$ of dimension $4g-3+n$. We give explicit formulas for the properly arranged generating function for these "intersection numbers", and prove that it satisfies Hirota equations. This generalizes and substantially simplifies our earlier results with Zvonkine in arXiv:math/0602457.


A naive parametrization for the vortex-sheet problem

Abstract: We consider the dynamics of a vortex sheet that evolves by the Birkhoff-Rott equations. The fluid evolution is understood as a weak solution of the incompressible Euler equations where the vorticity is given by a delta function on a curve multiplied by an amplitude. The solutions we study are with finite energy, which implies zero mean amplitude. In this context we choose a parametrization for the motion of the vortex sheet for which the equation is well-posed for analytic initial data. For the equation of the amplitude we show ill-posedness for non-analytic initial data.


A note on Elkin's improvement of Behrend's construction

Abstract: We provide a short proof of a recent result of Elkin in which large subsets of the integers 1 up to N free of 3-term progressions are constructed.


Morley sequences in dependent theories

Abstract: We characterize nonforking (Morley) sequences in dependent theories in terms of a generalization of Poizat's special sequences and show that average types of Morley sequences are stationary over their domains. We characterize generically stable types in terms of the structure of the "eventual" type. We then study basic properties of "strict Morley sequences", based on Shelah's notion of strict nonforking. In particular we prove "Kim's lemma" for such sequences, and a weak version of local character.


Unstable classes of metric structures

Abstract: We prove a strong non-structure theorem for a class of metric structures with an unstable pair of formulae. As a consequence, we show that weak categoricity (that is, categoricity up to isomorphisms and not isometries) implies several versions of stability. This is the first step in the direction of the investigation of weak categoricity and weak stability of metric classes.


Soliton dynamics for CNLS systems with potentials

Abstract: The soliton dynamics in the semiclassical limit for a weakly coupled nonlinear focusing Schr\"odinger systems in presence of a nonconstant potential is studied by taking as initial data some rescaled ground state solutions of an associate elliptic system.


A General Fredholm Theory III: Fredholm Functors and Polyfolds

Abstract: We describe a very general (nonlinear) Fredholm theory for a new class of ambient spaces, called polyfolds. The basic feature of these new spaces is that in general they may have locally varying dimensions. These new spaces are needed for a functional analytic treatment of nonlinear problems involving analytic limiting behavior. This theory is applicable to Gromov-Witten and Floer Theory as well as Symplectic Field Theory.


Variational and Geometric Structures of Discrete Dirac Mechanics

Abstract: In this paper, we develop the theoretical foundations of discrete Dirac mechanics, that is, discrete mechanics of degenerate Lagrangian/Hamiltonian systems with constraints. We first construct discrete analogues of Tulczyjew's triple and induced Dirac structures by considering the geometry of symplectic maps and their associated generating functions. We demonstrate that this framework provides a means of deriving discrete Lagrange-Dirac and nonholonomic Hamiltonian systems. In particular, this yields nonholonomic Lagrangian and Hamiltonian integrators. We also introduce discrete Lagrange-d'Alembert-Pontryagin and Hamilton-d'Alembert variational principles, which provide an alternative derivation of the same set of integration algorithms. The paper provides a unified treatment of discrete Lagrangian and Hamiltonian mechanics in the more general setting of discrete Dirac mechanics, as well as a generalization of symplectic and Poisson integrators to the broader category of Dirac integrators.


Observer design for invariant systems with homogeneous observations

Abstract: This paper considers the design of nonlinear observers for invariant systems posed on finite-dimensional connected Lie groups with measurements generated by a transitive group action on an associated homogeneous space. We consider the case where the group action has the opposite invariance to the system invariance and show that the group kinematics project to a minimal realisation of the systems observable dynamics on the homogeneous output space. The observer design problem is approached by designing an observer for the projected output dynamics and then lifting to the Lie-group. A structural decomposition theorem for observers of the projected system is provided along with characterisation of the invariance properties of the associated observer error dynamics. We propose an observer design based on a gradient-like construction that leads to strong (almost) global convergence properties of canonical error dynamics on the homogeneous output space. The observer dynamics are lifted to the group in a natural manner and the resulting gradient-like error dynamics of the observer on the Lie-group converge almost globally to the unobservable subgroup of the system, the stabiliser of the group action.


Class-Specific Tests of Spatial Segregation Based on Nearest Neighbor Contingency Tables

Abstract: The spatial interaction between two or more classes (or species) has important consequences in many fields and might cause multivariate clustering patterns such as segregation or association. The spatial pattern of segregation occurs when members of a class tend to be found near members of the same class (i.e., conspecifics), while association occurs when members of a class tend to be found near members of the other class or classes. These patterns can be tested using a nearest neighbor contingency table (NNCT). The null hypothesis is randomness in the nearest neighbor (NN) structure, which may result from -- among other patterns -- random labeling (RL) or complete spatial randomness (CSR) of points from two or more classes (which is called the CSR independence, henceforth). In this article, we consider Dixon's class-specific tests of segregation and introduce a new class-specific test, which is a new decomposition of Dixon's overall chi-square segregation statistic. We demonstrate that the tests we consider provide information on different aspects of the spatial interaction between the classes and they are conditional under the CSR independence pattern, but not under the RL pattern. We analyze the distributional properties and prove the consistency of these tests; compare the empirical significant levels (Type I error rates) and empirical power estimates of the tests using Monte Carlo simulations. We demonstrate that the new class-specific tests also have comparable performance with the currently available tests based on NNCTs in terms of Type I error and power estimates. For illustrative purposes, we use three example data sets. We also provide guidelines for using these tests.


Modeling the Dialectic

Abstract: Three formal first-order finite dialectical schemes are investigated. It is shown that schemes 1 and 2 have significantly different finite models. Further, an infinite natural number model for schemes 1, 2, 3 is constructed, and it is shown that scheme 3 has no finite model.


Generators and relations for wreath products

Abstract: Generators and defining relations for wreath products of groups are given. Under some condition (conormality of the generators) they are minimal. In particular, it is just the case for the Sylow subgroups of the symmetric groups.


Convexity of Chebyshev Sets in Hilbert Spaces

Abstract: The aim of this paper is state of conditions that ensure the convexity of a Chebyshev sets in Hilbert spaces .


A Review on Some Geometric Results of the Smulian s Theorem on Frechet Differentiability of Norms

Abstract: In this paper, we prove the Smulian s theorem on Frechet differentiability of norm,and present some of its geometric results concerning the Gateaux and Frechet differentiability of norm and properties of the allied space and its dual such as reflexivity and strict convexity.


Relaxed Lyapunov Criteria for Robust Global Stabilization of Nonlinear Systems

Abstract: The notion of the relaxed Robust Control Lyapunov Function (relaxed RCLF) is introduced and is exploited for the design of robust feedback stabilizers for nonlinear systems. Particularly, it is shown for systems with input constraints that relaxed RCLFs can be easily obtained, while RCLFs are not available. Moreover, it is shown that the use of relaxed RCLFs usually results to different feedback designs from the ones obtained by the use of the standard RCLF methodology. Using the relaxed RCLFs feedback design methodology, a simple controller that guarantees robust global stabilization of a perturbed chemostat model is provided.


Badly approximable numbers related to the Littlewood conjecture

Abstract: By means of Peres-Schlag's method we prove the existence of real numbers $\alpha, \beta$ such that $$ \liminf_{q\to \infty} (q\log^2 q)||\alpha q|| ||\beta q|| > 0.


Contributions to Khovanov Homology

Abstract: Khovanov homology ist a new link invariant, discovered by M. Khovanov, and used by J. Rasmussen to give a combinatorial proof of the Milnor conjecture. In this thesis, we give examples of mutant links with different Khovanov homology. We prove that Khovanov's chain complex retracts to a subcomplex, whose generators are related to spanning trees of the Tait graph, and we exploit this result to investigate the structure of Khovanov homology for alternating knots. Further, we extend Rasmussen's invariant to links. Finally, we generalize Khovanov's categorifications of the colored Jones polynomial, and study conditions under which our categorifications are functorial with respect to colored framed link cobordisms. In this context, we develop a theory of Carter--Saito movie moves for framed link cobordisms.


On the theory of $q$-complete spaces

Abstract: In this article, we show that if $X$ is a Stein space of dimension $n$ and, if $D$ is a locally $q$-complete open set in $X$, then $D$ is $q$-complete. This gives, in particular, a positive answer to the local Steinness problem, which is one of the most classical problem in complex analytic geometry, namely, if $D$ is a locally Stein open set in a Stein space $X$, then $D$ is Stein


Cuspidal representations of reductive groups

Abstract: This paper proves the existence of cuspidal automorphic forms for a reductive group, invariant under an automorphism of finite order. The techniques used are a local analysis of orbital integrals and the Arthur-Selberg trace formula.


On some representations of degenerate affine Hecke algebras of type $BC_{n}$

Abstract: Let G=U(p,q) and K=U(p)xU(q). In arXiv:0801.1530, the authors construct a functor from the category of Harish-Chandra modules for the pair (G,K) to the category of representations of the degenerate affine Hecke algebra of type B_n. In this paper, we study the image of the principal series modules of G under this functor. We describe the dAHA module structure of the image by constructing a family of common eigenvectors of {y_i}. In the rank 1 case, we write down an explicit formula for the generator y_{1} in terms of central elements of U(g) and calculate the eigenvalue of y_{1} by using the central characters.


Character sheaves on unipotent groups in positive characteristic: foundations

Abstract: In this article we formulate and prove the main theorems of the theory of character sheaves on unipotent groups over an algebraically closed field of characteristic p>0. In particular, we show that every admissible pair for such a group G gives rise to an L-packet of character sheaves on G, and that, conversely, every L-packet of character sheaves on G arises from a (non-unique) admissible pair. In the appendices we discuss two abstract category theory patterns related to the study of character sheaves. The first appendix sketches a theory of duality for monoidal categories, which generalizes the notion of a rigid monoidal category and is close in spirit to the Grothendieck-Verdier duality theory. In the second one we use a topological field theory approach to define the canonical braided monoidal structure and twist on the equivariant derived category of constructible sheaves on an algebraic group; moreover, we show that this category carries an action of the surface operad. The third appendix proves that the "naive" definition of the equivariant constructible derived category with respect to a unipotent algebraic group is equivalent to the "correct" one.


Asymptotic energy of graphs

Abstract: The energy of a simple graph $G$ arising in chemical physics, denoted by $\mathcal E(G)$, is defined as the sum of the absolute values of eigenvalues of $G$. We consider the asymptotic energy per vertex (say asymptotic energy) for lattice systems. In general for a type of lattice in statistical physics, to compute the asymptotic energy with toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions are different tasks with different hardness. In this paper, we show that if $\{G_n\}$ is a sequence of finite simple graphs with bounded average degree and $\{G_n'\}$ a sequence of spanning subgraphs of $\{G_n\}$ such that almost all vertices of $G_n$ and $G_n'$ have the same degrees, then $G_n$ and $G_n'$ have the same asymptotic energy. Thus, for each type of lattices with toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions, we have the same asymptotic energy. As applications, we obtain the asymptotic formulae of energies per vertex of the triangular, $3^3.4^2$, and hexagonal lattices with toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions simultaneously.


An Alternative Definition of the Completion of Metric Spaces

Abstract: In this article, the author proposes another way to define the completion of a metric space, which is different from the classical one via the dense property, and prove the equivalence between two definitions. This definition is based on considerations from category theory, and can be generalized to arbitrary categories.


Robust Stabilization of Nonlinear Systems by Quantized and Ternary Control

Abstract: Results on the problem of stabilizing a nonlinear continuous-time system by a finite number of control or measurement values are presented. The basic tool is a discontinuous version of the so-called semi-global backstepping lemma. We derive robust practical stabilizability results by quantized and ternary controllers and apply them to some significant control problems.


Rational homotopy theory and differential graded category

Abstract: We propose a generalization of Sullivan's de Rham homotopy theory to non-simply connected spaces. The formulation is such that the real homotopy type of a manifold should be the closed tensor dg-category of flat bundles on it much the same as the real homotopy type of a simply connected manifold is the de Rham algebra in original Sullivan's theory. We prove the existence of a model category structure on the category of small closed tensor dg-categories and as a most simple case, confirm an equivalence between the homotopy category of spaces whose fundamental groups are finite and whose higher homotopy groups are finite dimensional rational vector spaces and the homotopy category of small closed tensor dg-categories satisfying certain conditions.


Dynamics in several complex variables: endomorphisms of projective spaces and polynomial-like mapping

Abstract: The emphasis of this course is on pluripotential methods in complex dynamics in higher dimension. They are based on the compactness properties of plurisubharmonic functions and on the theory of positive closed currents. Applications of these methods are not limited to the dynamical systems that we consider here. We choose to show their effectiveness and to describe the theory for two large families of maps. The first chapter deals with holomorphic endomorphisms of the projective space P^k. We establish the first properties and give several constructions for the Green currents and the equilibrium measure \mu. The emphasis is on quantitative properties and speed of convergence. We then treat equidistribution problems and establish ergodic properties of \mu: K-mixing, exponential decay of correlations for various classes of observables, central limit theorem and large deviations theorem. Finally, we study the entropy, the Lyapounov exponents and the dimension of \mu. The second chapter develops the theory of polynomial-like maps in higher dimension. We introduce the dynamical degrees and construct the equilibrium measure \mu of maximal entropy. Then, under a natural assumption, we prove equidistribution properties of points and various statistical properties of the measure \mu. The assumption is stable under small pertubations on the map. We also study the dimension of \mu, the Lyapounov exponents and their variation. Our aim is to get a self-contained text that requires only a minimal background. In order to help the reader, an appendix gives the basics on p.s.h. functions, positive closed currents and super-potentials on projective spaces. Some exercises are proposed and an extensive bibliography is given.


Superconnections and Index Theory

Abstract: We investigate index theory in the context of Dirac operators coupled to superconnections. In particular, we prove a local index theorem for such operators, and for families of such operators. We investigate eta-invariants and prove an APS-theorem, and construct a geometric determinant line bundle for families of such operators, computing its curvature and holonomy in terms of familiar index theoretic quantities.


The Newton stratification on deformations of local G-shtukas

Abstract: Bounded local G-shtukas are function field analogs for p-divisible groups with extra structure. We describe their deformations and moduli spaces. The latter are analogous to Rapoport-Zink spaces for p-divisible groups. The underlying schemes of these moduli spaces are affine Deligne-Lusztig varieties. For basic Newton polygons the closed Newton stratum in the universal deformation of a local G-shtuka is isomorphic to the completion of a corresponding affine Deligne-Lusztig variety in that point. This yields bounds on the dimension and proves equidimensionality of the basic affine Deligne-Lusztig varieties.


Some properties of (C,E,P)-algebras : Overgneration and 0-order estimates

Abstract: We give a new definition of the so-called overgenerated rings, which are the usual tool used to define the asymptotic structure of a (C,E,P)-algebra, written as a factor space M_{(A,E,P)}/N_{(I_{A},E,P)}. With this new definition and in the particular case of E=C^{∞}, we show that a moderate element i.e. in M_{(A,E,P)} is negligible if and only if it satisfies the C⁰-order estimate for the ideal N_{(I_{A},E,P)}.


Multivariate Determinateness

Abstract: The uniqueness question of the multivariate moment problem is studied by different methods: Hilbert space operators, complex function theory, polynomial approximation, disintegration, integral geometry. Most of the known results in the multi-dimensional case are reviewed and reproved, and a number of new determinacy criteria are developed.


Structural properties of acyclic heaps of pieces with Kazhdan--Lusztig theory

Abstract: We introduce the notions of boundary vertex, linear equivalence and effective boundary vertex in the context of Viennot's heaps of pieces. We prove that in the heap of a fully commutative element in a star reducible Coxeter group, every boundary vertex is linearly equivalent to an effective boundary vertex. Using this result, we establish Property W (in the sense of math.QA/0509363) for star reducible Coxeter groups; this corrects a mistake in the latter paper.


Permanence criteria for semi-free profinite groups

Abstract: We introduce the condition of a profinite group being semi-free, which is more general than being free and more restrictive than being quasi-free. In particular, every projective semi-free profinite group is free. We prove that the usual permanence properties of free groups carry over to semi-free groups. Using this, we conclude that if k is a separably closed field, then many field extensions of k((x,y)) have free absolute Galois groups.


A Basic Elementary Extension of the Duchet-Meyniel Theorem

Abstract: The Conjecture of Hadwiger implies that the Hadwiger number $h$ times the independence number $\alpha$ of a graph is at least the number of vertices $n$ of the graph. In 1982 Duchet and Meyniel proved a weak version of the inequality, replacing the independence number $\alpha$ by $2\alpha-1$, that is, $$(2\alpha-1)\cdot h \geq n.$$ In 2005 Kawarabayashi, Plummer and the second author published an improvement of the theorem, replacing $2\alpha - 1$ by $2\alpha - 3/2$ when $\alpha$ is at least 3. Since then a further improvement by Kawarabayashi and Song has been obtained, replacing $2\alpha - 1$ by $2\alpha - 2$ when $\alpha$ is at least 3. In this paper a basic elementary extension of the Theorem of Duchet and Meyniel is presented. This may be of help to avoid dealing with basic cases when looking for more substantial improvements. The main unsolved problem (due to Seymour) is to improve, even just slightly, the theorem of Duchet and Meyniel in the case when the independence number $\alpha$ is equal to 2. The case $\alpha = 2$ of Hadwiger's Conjecture was first pointed out by Mader as an interesting special case.


Representations of finite special linear groups in non-defining characteristic

Abstract: We determine precisely the number of irreducible summands of an irreducible cross characteristic representation of $GL_{n}(q)$ on restriction to $SL_{n}(q)$. Combined with a recent result of C. Bonnafe, this yields a canonical labeling for irreducible $\ell$-modular representations of $SL_{n}(q)$, where $(\ell,q)=1$. As an application, we classify for the first time complex representations of $SL_{n}(q)$ whose reductions modulo $\ell$ are irreducible.