Comparison of relative cohomology theories with respect to semidualizing modules

Abstract: We compare and contrast various relative cohomology theories that arise from resolutions involving semidualizing modules. We prove a general balance result for relative cohomology over a Cohen-Macaulay ring with a dualizing module, and we demonstrate the failure of the naive version of balance one might expect for these functors. We prove that the natural comparison morphisms between relative cohomology modules are isomorphisms in several cases, and we provide a Yoneda-type description of the first relative Ext functor. Finally, we show by example that each distinct relative cohomology construction does in fact result in a different functor.


Singularity categories, Schur Functors and Triangular Matrix Rings

Abstract: We study certain Schur functors which preserve singularity categories of rings and we apply them to study the singularity category of triangular matrix rings. In particular, combining these results with Buchweitz-Happel's theorem, we can describe singularity categories of certain non-Gorenstein rings via the stable category of maximal Cohen-Macaulay modules. Three concrete examples of finite-dimensional algebras with the same singularity category are discussed.


Nearly Tight Frames and Space-Frequency Analysis on Compact Manifolds

Abstract: Let $\bf M$ be a smooth compact oriented Riemannian manifold, and let $\Delta$ be the Laplace-Beltrami operator on ${\bf M}$. Say $0 \neq f \in \mathcal{S}(\RR^+)$, and that $f(0) = 0$. For $t > 0$, let $K_t(x,y)$ denote the kernel of $f(t^2 \Delta)$. Suppose $f$ satisfies Daubechies' criterion, and $b > 0$. For each $j$, write ${\bf M}$ as a disjoint union of measurable sets $E_{j,k}$ with diameter at most $ba^j$, and comparable to $ba^j$ if $ba^j$ is sufficiently small. Take $x_{j,k} \in E_{j,k}$. We then show that the functions $\phi_{j,k}(x)=[\mu(E_{j,k})]^{1/2} \bar{K_{a^j}}(x_{j,k},x)$ form a frame for $(I-P)L^2({\bf M})$, for $b$ sufficiently small (here $P$ is the projection onto the constant functions). Moreover, we show that the ratio of the frame bounds approaches 1 nearly quadratically as the dilation parameter approaches 1, so that the frame quickly becomes nearly tight (for $b$ sufficiently small). Moreover, based upon how well-localized a function $F \in (I-P)L^2$ is in space and in frequency, we can describe which terms in the summation $F \sim SF = \sum_j \sum_k < F,\phi_{j,k} > \phi_{j,k}$ are so small that they can be neglected. If $n=2$ and $\bf M$ is the torus or the sphere, and $f(s)=se^{-s}$ (the "Mexican hat" situation), we obtain two explicit approximate formulas for the $\phi_{j,k}$, one to be used when $t$ is large, and one to be used when $t$ is small. Finally we explain in what sense the kernel $K_t(x,y)$ should itself be regarded as a continuous wavelet on ${\bf M}$, and characterize the H\"older continuous functions on ${\bf M}$ by the size of their continuous wavelet transforms, for H\"older exponents strictly between 0 and 1.


Dilatation structures with the Radon-Nikodym property

Abstract: In this paper I explain what is a pair of dilatation structures, one looking down to another. Such a pair of dilatation structures leads to the intrinsic definition of a distribution as a field of topological filters. To any pair of dilatation structures there is an associated notion of differentiability which generalizes the Pansu differentiability. This allows the introduction of the Radon-Nikodym property for dilatation structures, which is the straightforward generalization of the Radon-Nikodym property for Banach spaces. After an introducting section about length metric spaces and metric derivatives, is proved that for a dilatation structure with the Radon-Nikodym property the length of absolutely continuous curves expresses as an integral of the norms of the tangents to the curve, as in Riemannian geometry. Further it is shown that Radon-Nikodym property transfers from any "upper" dilatation structure looking down to a "lower" dilatation structure, theorem \ref{ttransfer}. Im my opinion this result explains intrinsically the fact that absolutely continuous curves in regular sub-Riemannian manifolds are derivable almost everywhere, as proved by Margulis-Mostow, Pansu (for Carnot groups) or Vodopyanov.


Betti numbers of Springer fibers in type A

Abstract: We determine the Betti numbers of the Springer fibers in type A. To do this, we construct a cell decomposition of the Springer fibers. The codimension of the cells is given by an analogue of the Coxeter length. This makes our cell decomposition well suited for the calculation of Betti numbers.


Coloring complexes and arrangements

Abstract: Steingrimsson's coloring complex and Jonsson's unipolar complex are interpreted in terms of hyperplane arrangements. This viewpoint leads to short proofs that all coloring complexes and a large class of unipolar complexes have convex ear decompositions. These convex ear decompositions impose strong new restrictions on the chromatic polynomials of all finite graphs. Similar results are obtained for characteristic polynomials of submatroids of type B_n arrangements.


The wave equation on asymptotically de Sitter-like spaces

Abstract: In this paper we obtain the asymptotic behavior of solutions of the Klein-Gordon equation on Lorentzian manifolds $(X^\circ,g)$ which are de Sitter-like at infinity. Such manifolds are Lorentzian analogues of the so-called Riemannian conformally compact (or asymptotically hyperbolic) spaces. Under global assumptions on the (null)bicharacteristic flow, namely that the boundary of the compactification X is a union of two disjoint manifolds, Y+ and Y-, and each bicharacteristic converges to one of these two manifolds as the parameter along the bicharacteristic goes to plus infinity, and to the other manifold as the parameter goes to minus infinity, we also define the scattering operator, and show that it is a Fourier integral operator associated to the bicharacteristic flow from Y+ to Y-.


Construction of eigenvarieties in small cohomological dimensions for semi-simple, simply connected groups

Abstract: We study low order terms of Emerton's spectral sequence for simply connected, simple groups. As a result, for real rank 1 groups, we show that Emerton's method for constructing eigenvarieties is successful in cohomological dimension 1. For real rank 2 groups, we show that a slight modification of Emerton's method allows one to construct eigenvarieties in cohomological dimension 2.


Conjugate varieties with distinct real cohomology algebras

Abstract: Using constructions of Voisin, we exhibit a smooth projective variety defined over a number field k and two complex embeddings of k, such that the two complex manifolds induced by these embeddings have non isomorphic cohomology algebras with real coefficients. This contrasts with the fact that the cohomology algebras with l-adic coefficients are canonically isomorphic for any prime number l, and answers a question of Grothendieck.


The Universal sl_2 Link Homology Theory

Abstract: We explore the complex associated to a link in the geometric formalism of Khovanov's (n=2) link homology theory, determine its exact underlying algebraic structure and find its precise universality properties for link homology functors. We present new methods of extracting all known link homology theories directly from this universal complex, and determine its relative strength as a link invariant by specifying the amount of information held within the complex. We achieve these goals by finding a complex isomorphism which reduces the complex into one in a simpler category. We introduce few tools and methods, including surface classification modulo the 4TU/S/T relations and genus generating operators, and use them to explore the relation between the geometric complex and its underlying algebraic structure. We identify the universal topological quantum field theory (TQFT) that can be used to create link homology and find that it is ``smaller'' than what was previously reported by Khovanov. We find new homology theories that hold a controlled amount of information relative to the known ones. The universal complex is computable efficiently using our reduction theorem. This allows us to explore the phenomenological aspects of link homology theory through the eyes of the universal complex in order to explain and unify various phenomena (such as torsion and thickness). The universal theory also enables us to state results regarding specific link homology theories derived from it. The methods developed in this thesis can be combined with other known techniques (such as link homology spectral sequences) or used in the various extensions of Khovanov link homology (such as sl_3 link homology).


The Mellin transform and spectral properties of toric varieties

Abstract: In this article we apply results of \cite{W} on the twisted Mellin transform to problems in toric geometry. In particular we use these results to describe the asymptotics of probability densities associated with the monomial eigenstates, $z^k$, $k \in \ZZ^d$, in Bargmann space and prove an "upstairs" version of the spectral density theorem of \cite{BGU}. We also obtain for the $z^k$'s, "upstairs" versions of the results of \cite{STZ} on distribution laws for eigenstates on toric varieties.


Comparing powers and symbolic powers of ideals

Abstract: We develop tools to study the problem of containment of symbolic powers $I^{(m)}$ in powers $I^r$ for a homogeneous ideal $I$ in a polynomial ring $k[{\bf P}^N]$ in $N+1$ variables over an algebraically closed field $k$. We obtain results on the structure of the set of pairs $(r,m)$ such that $I^{(m)}\subseteq I^r$. As corollaries, we show that $I^2$ contains $I^{(3)}$ whenever $S$ is a finite generic set of points in ${\bf P}^2$ (thereby giving a partial answer to a question of Huneke), and we show that the containment theorems of Ein-Lazarsfeld-Smith and Hochster-Huneke are optimal for every fixed dimension and codimension.


The Bergman kernel and projection on non-smooth worm domains

Abstract: This paper provides a precise asymptotic expansion for the Bergman kernel on the non-smooth worm domains of Christer Kiselman in complex 2-space. Applications are given to the failure of Condition R, to deviant boundary behavior of the kernel, and to L^p mapping properties of the kernel.


Point counting on reductions of CM elliptic curves

Abstract: We give explicit formulas for the number of points on reductions of elliptic curves with complex multiplication by any imaginary quadratic field. We also find models for CM $\mathbf{Q}$-curves in certain cases. This generalizes earlier results of Gross, Stark, and others.


Weyl modules and opers without monodromy

Abstract: We prove that the algebra of endomorphisms of a Weyl module of critical level is isomorphic to the algebra of functions on the space of monodromy-free opers on the disc with regular singularity and residue determined by the highest weight of the Weyl module. This result may be used to test the local geometric Langlands correspondence proposed in our earlier work arXiv:math/0508382.


Combinatorial and hybrid principles for sigma-directed families of countable sets modulo finite

Abstract: We consider strong combinatorial principles for sigma-directed families of countable sets in the ordering by inclusion modulo finite, e.g. P-ideals of countable sets. We try for principles as strong as possible while remaining compatible with CH, and we also consider principles compatible with the existence of nonspecial Aronszajn trees. The main thrust is towards abstract principles with game theoretic formulations. Some of these principles are purely combinatorial, while the ultimate principles are primarily combinatorial but also have aspects of forcing axioms.


Higher Rank TQFT Representations of SL(2,Z) are Reducible

Abstract: In this article we give examples which show that the TQFT representations of the mapping class groups derived from quantum SU(N) for N>2 are generically decomposable. One general decomposition of the representations is induced by the symmetry which exchanges SU(N) representation labels by their conjugates. The respective summands of a given parity are typically still reducible into many further components. Specifically, we give an explicit basis for an irreducible direct summand in the SL(2,Z) representation obtained from quantum PSU(3) when the order of the root of unity is a prime r=2 mod 3. We show that this summand is isomorphic to the respective PSU(2) representation.


Equivariant Littlewood-Richardson Skew Tableaux

Abstract: We give a positive equivariant Littlewood-Richardson rule also discovered independently by Molev. Our proof generalizes a proof by Stembridge of the ordinary Littlewood-Richardson rule. We describe a weight-preserving bijection between our indexing tableaux and the Knutson-Tao puzzles.


Gorenstein cohomology in abelian categories

Abstract: We investigate relative cohomology functors on subcategories of abelian categories via Auslander-Buchweitz approximations and the resulting strict resolutions. We verify that certain comparison maps between these functors are isomorphisms and introduce a notion of perfection for this context. Our main theorem is a balance result for relative cohomology that simultaneously recovers theorems of Holm and the current authors as special cases.


Randomized series and Geometry of Banach spaces

Abstract: We study some properties of the randomized series and their applications to the geometric structure of Banach spaces. For $n\ge 2$ and $1<p<\infty$, it is shown that $\ell_\infty^n$ is representable in a Banach space $X$ if and only if it is representable in the Lebesgue-Bochner $L_p(X)$. New criteria for various convexity properties in Banach spaces are also studied. It is proved that a Banach lattice $E$ is uniformly monotone if and only if its $p$-convexification $E^{(p)}$ is uniformly convex and that a K\"othe function space $E$ is upper locally uniformly monotone if and only if its $p$-convexification $E^{(p)}$ is midpoint locally uniformly convex.


The Bloch-Okounkov correlation functions of negative levels

Abstract: Bloch and Okounkov introduced an $n$-point correlation function on the fermionic Fock space and found a closed formula in terms of theta functions. This function affords several distinguished interpretations and in particular can be formulated as correlation functions on irreducible $\hat{gl}_\infty$-modules of level one. These correlation functions have been generalized for irreducible integrable modules of $\hat{gl}_\infty$ and its classical Lie subalgebras of positive levels by the authors. In this paper we extend further these results and compute the correlation functions as well as the $q$-dimensions for modules of $\hat{gl}_\infty$ and its classical subalgebras at negative levels.


Gale duality for complete intersections

Abstract: We show that every complete intersection of Laurent polynomials in an algebraic torus is isomorphic to a complete intersection of master functions in the complement of a hyperplane arrangement, and vice versa. We call this association Gale duality because the exponents of the monomials in the polynomials annihilate the weights of the master functions. We use Gale duality to give a Kouchnirenko theorem for the number of solutions to a system of master functions and to compute some topological invariants of generic master function complete intersections.


On Monge-Ampere equations with homogeneous right hand side

Abstract: We study the regularity and behavior at the origin of solutions to the two-dimensional degenerate Monge-Ampere equation with homogeneous right hand side of degree alpha, alpha>-2. We show that when alpha > 0, solutions admit only two possible behaviors near the origin, radial and non-radial. We also show that the radial behavior is unstable. For alpha<0 we prove that solutions admit only the radial behavior near the origin.


Sharp linear and bilinear restriction estimates for paraboloids in the cylindrically symmetric case

Abstract: For cylindrically symmetric functions dyadically supported on the paraboloid, we obtain a family of sharp linear and bilinear adjoint restriction estimates. As corollaries, we first extend the ranges of exponents for the classical \textit{linear or bilinear adjoint restriction conjectures} for such functions and verify the \textit{linear adjoint restriction conjecture} for the paraboloid. We also interpret the restriction estimates in terms of solutions to the Schr\"odinger equation and establish the analogous results when the paraboloid is replaced by the lower third of the sphere.


Uppers to zero and semistar operations in polynomial rings

Abstract: Given a stable semistar operation of finite type $\star$ on an integral domain $D$, we show that it is possible to define in a canonical way a stable semistar operation of finite type $[\star]$ on the polynomial ring $D[X]$, such that $D$ is a $\star$-quasi-Pr\"ufer domain if and only if each upper to zero in $D[X]$ is a quasi-$[\star]$-maximal ideal. This result completes the investigation initiated by Houston-Malik-Mott \cite[Section 2]{hmm} in the star operation setting. Moreover, we show that $D$ is a Pr\"ufer $\star$-multiplication (resp., a $\star$-Noetherian; a $\star$-Dedekind) domain if and only if $D[X]$ is a Pr\"ufer $[\star]$-multiplication (resp., a $[\star]$-Noetherian; a $[\star]$-Dedekind) domain. As an application of the techniques introduced here, we obtain a new interpretation of the Gabriel-Popescu localizing systems of finite type on an integral domain $D$ (Problem 45 of \cite{cg}), in terms of multiplicatively closed sets of the polynomial ring $D[X]$.


Valdivia compact Abelian groups

Abstract: Let R denote the smallest class of compact spaces containing all metric compacta and closed under limits of continuous inverse sequences of retractions. Class R is striclty larger than the class of Valdivia compact spaces. We show that every compact connected Abelian group which is a topological retract of a space from class R is necessarily isomorphic to a product of metric groups. This completes the result of V. Uspenskij and the author, where a compact connected Abelian group outside class R has been described.


Order from Randomness

Abstract: We consider an elementary discrete process which starts from purely random configuration and leads to well-ordered and stable state. Complete analytical solution to this problem is presented.


On systematic scan for sampling H-colourings of the path

Abstract: This paper is concerned with sampling from the uniform distribution on H-colourings of the n-vertex path using systematic scan Markov chains. An H-colouring of the n-vertex path is a homomorphism from the n-vertex path to some fixed graph H. We show that systematic scan for H-colourings of the n-vertex path mixes in O(log n) scans for any fixed H. This is a significant improvement over the previous bound on the mixing time which was O(n^5) scans. Furthermore we show that for a slightly more restricted family of H (where any two vertices are connected by a 2-edge path) systematic scan also mixes in O(log n) scans for any scan order. Finally, for completeness, we show that a random update Markov chain mixes in O(n log n) updates for any fixed H, improving the previous bound on the mixing time from O(n^5) updates.


Inverse Systems and I-Favorable Spaces

Abstract: A compact space X is I-favorable if, and only if X can be representing as a limit of $\sigma$-complete inverse system of compact metrizable spaces with skeletal bonding maps.


Geometric generalizations in Kresin-Maz'ya Sharp Real-Part Theorems

Abstract: In the present article we give geometric generalizations of the estimates from Chapters 5,6,7 from \cite{krem:gnus}, while extending their sharpness to new cases.


Random Sampling of Entire Functions of Exponential Type in Several Variables

Abstract: We consider the problem of random sampling for band-limited functions. When can a band-limited function $f$ be recovered from randomly chosen samples $f(x_j), j\in \mathbb{N}$? We estimate the probability that a sampling inequality of the form A\|f\|_2^2 \leq \sum_{j\in \mathbb{N}} |f(x_j)|^2 \leq B \|f\|_2^2 hold uniformly all functions $f\in L^2(\mathbb{R}^d)$ with supp $\hat{f} \subseteq [-1/2,1/2]^d$ or some subset of \bdl functions. In contrast to discrete models, the space of band-limited functions is infinite-dimensional and its functions "live" on the unbounded set $\mathbb{R}^d$. This fact raises new problems and leads to both negative and positive results. (a) With probability one, the sampling inequality fails for any reasonable definition of a random set on $\mathbb{R}^d$, e.g., for spatial Poisson processes or uniform distribution over disjoint cubes. (b) With overwhelming probability, the sampling inequality holds for certain compact subsets of the space of band-limited functions and for sufficiently large sampling size.


Invariant deformations of orbit closures in $\mathfrak{sl}_n$

Abstract: We study deformations of orbit closures for the action of a connected semisimple group $G$ on its Lie algebra $\mathfrak{g}$, especially when $G$ is the special linear group. The tools we use are on the one hand the invariant Hilbert scheme and on the other hand the sheets of $\mathfrak{g}$. We show that when $G$ is the special linear group, the connected components of the invariant Hilbert schemes we get are the geometric quotients of the sheets of $\mathfrak{g}$. These quotients were constructed by Katsylo for a general semisimple Lie algebra $\mathfrak{g}$; in our case, they happen to be affine spaces.


Mok-Siu-Yeung type formulas on contact locally sub-symmetric spaces

Abstract: We derive Mok-Siu-Yeung type formulas for horizontal maps from compact contact locally sub-symmetric spaces into strictly pseudoconvex CR manifolds and we obtain some rigidity theorems for the horizontal pseudoharmonic maps.


Arithmetic lattices and weak spectral geometry

Abstract: This note is an expansion of three lectures given at the workshop "Topology, Complex Analysis and Arithmetic of Hyperbolic Spaces" held at Kyoto University in December of 2006 and will appear in the proceedings for this workshop.


D-modules over rings with finite F-representation type

Abstract: Smith and Van den Bergh introduced the notion of finite F-representation type as a characteristic $p$ analogue of the notion of finite representation type. In this paper, we prove two finiteness properties of rings with finite F-representation type. The first property states that if $R=\bigoplus_{n \ge 0}R_n$ is a Noetherian graded ring with finite (graded) F-representation type, then for every non-zerodivisor $x \in R$, $R_x$ is generated by $1/x$ as a $D_{R}$-module. The second one states that if $R$ is a Gorenstein ring with finite F-representation type, then $H_I^n(R)$ has only finitely many associated primes for any ideal $I$ of $R$ and any integer $n$. We also include a result on the discreteness of F-jumping exponents of ideals of rings with finite (graded) F-representation type as an appendix.


Graphical Representation of some Duality Relations in Stochastic Population Models

Abstract: We derive a unified stochastic picture for the duality of a resampling-selection model with a branching-coalescing particle process (cf. http://www.ams.org/mathscinet-getitem?mr=MR2123250) and for the self-duality of Feller's branching diffusion with logistic growth (cf. math/0509612). The two dual processes are approximated by particle processes which are forward and backward processes in a graphical representation. We identify duality relations between the basic building blocks of the particle processes which lead to the two dualities mentioned above.


Countable groups of isometries on Banach spaces

Abstract: A group G is representable in a Banach space X if G is isomorphic to the group of isometries on X in some equivalent norm. We prove that a countable group G is representable in a separable real Banach space X in several general cases, including when $G=\{-1,1\} \times H$, H finite and $\dim X \geq |H|$, or when G contains a normal subgroup with two elements and X is of the form c_0(Y) or $\ell_p(Y)$, $1 \leq p <+\infty$. This is a consequence of a result inspired by methods of S. Bellenot and stating that under rather general conditions on a separable real Banach space X and a countable bounded group G of isomorphisms on X containing -Id, there exists an equivalent norm on X for which G is equal to the group of isometries on X. We also extend methods of K. Jarosz to prove that any complex Banach space of dimension at least 2 may be renormed to admit only trivial real isometries, and that any real Banach space which is a cartesian square may be renormed to admit only trivial and conjugation real isometries. It follows that every real space of dimension at least 4 and with a complex structure up to isomorphism may be renormed to admit exactly two complex structures up to isometry, and that every real cartesian square may be renormed to admit a unique complex structure up to isometry.


Understanding preservation theorems: Chapter VI of Proper and Improper Forcing

Abstract: This expository paper covers the first two sections of chapter VI of Shelah's book "Proper and Improper Forcing," including the preservationn (using CS iterations of proper forcings) of omega-omega bounding, Sacks property, Lavewr property, P-point property, and others.


The classification question for Leavitt path algebras

Abstract: We prove an algebraic version of the Gauge-Invariant Uniqueness Theorem, a result which gives information about the injectivity of certain homomorphisms between ${\mathbb Z}$-graded algebras. As our main application of this theorem, we obtain isomorphisms between the Leavitt path algebras of specified graphs. From these isomorphisms we are able to achieve two ends. First, we show that the $K_0$ groups of various sets of purely infinite simple Leavitt path algebras, together with the position of the identity element in $K_0$, classifies the algebras in these sets up to isomorphism. Second, we show that the isomorphism between matrix rings over the classical Leavitt algebras, established previously using number-theoretic methods, can be reobtained via appropriate isomorphisms between Leavitt path algebras.


Simplicial volume of Hilbert modular varieties

Abstract: The simplicial volume introduced by Gromov provides a topologically accessible lower bound for the minimal volume. Lafont and Schmidt proved that the simplicial volume of closed, locally symmetric spaces of non-compact type is positive. In this paper, we present a generalization of this result to certain non-compact locally symmetric spaces of finite volume, to so-called Hilbert modular varieties. The key idea is to reduce the problem to the compact case by first relating the simplicial volume of these manifolds to the Lipschitz simplicial volume and then taking advantage of a proportionality principle for the Lipschitz simplicial volume. Moreover, using computations of Bucher-Karlsson for the simplicial volume of products of closed surfaces, we obtain the exact value of the simplicial volume of Hilbert modular surfaces.


The Implicit Function Theorem for continuous functions

Abstract: In the present paper we obtain a new homological version of the implicit function theorem and some versions of the Darboux theorem. Such results are proved for continuous maps on topological manifolds. As a consequence, some versions of these classic theorems are proved when we consider differenciable (not necessarily C^1) maps.


On the rank of a Coxeter group

Abstract: Let W be a Coxeter group with Coxeter generators S. The rank of the Coxeter system (W,S) is the cardinality |S| of S. The Coxeter system (W,S) has finite rank if and only if W is finitely generated. If (W,S) has infinite rank, then |S| = |W|, since every element of W is represented by a finite product of elements of S. Thus if W is not finitely generated, the rank of (W,S) is uniquely determined by W. If W is finitely generated, then W may have sets of Coxeter generators S and S' of different ranks. In this paper, we determine the set of all possible ranks for an arbitrary finitely generated Coxeter group W.


Triviality of vector bundles on sufficiently twisted ind-Grassmannians

Abstract: Twisted ind-Grassmannians are ind-varieties $\GG$ obtained as direct limits of Grassmannians $G(r_m,V^{r_m})$, for $m\in\ZZ_{>0}$, under embeddings $\phi_m:G(r_m,V^{r_m})\to G(r_{m+1}, V^{r_{m+1}})$ of degree greater than one. It has been conjectured in \cite{PT} and \cite{DP} that any vector bundle of finite rank on a twisted ind-Grassmannian is trivial. We prove this conjecture under the assumption that the ind-Grassmannian $\GG$ is sufficiently twisted, i.e. that $\lim_{m\to\infty}\frac{r_m}{\deg \phi_1...\deg\phi_m}=0$.


Holomorphic curves in Exploded Torus Fibrations: Compactness

Abstract: The category of exploded torus fibrations is an extension of the category of smooth manifolds in which some adiabatic limits look smooth. (For example, the limits considered in tropical geometry appear smooth, also degenerations corresponding to an algebraic family with normal crossing singularities are smooth.) In this paper we prove a compactness theorem for (pseudo)-holomorphic curves in exploded torus fibrations. In the case of smooth manifolds, this is just a version of Gromov's compactness theorem in a topology strong enough for gluing analysis.


A Linearization of Connes' Embedding Problem

Abstract: We show that Connes' embedding problem for II_1-factors is equivalent to a statement about distributions of sums of self-adjoint operators with matrix coefficients. This is an application of a linearization result for finite von Neumann algebras, which is proved using asymptotic second order freeness of Gaussian random matrices.


Nonparametric estimation for dependent data with an application to panel time series

Abstract: In this paper we consider nonparametric estimation for dependent data, where the observations do not necessarily come from a linear process. We study density estimation and also discuss associated problems in nonparametric regression using the 2-mixing dependence measure. We compare the results under 2-mixing with those derived under the assumption that the process is linear. In the context of panel time series where one observes data from several individuals, it is often too strong to assume the joint linearity of processes. Instead the methods developed in this paper enable us to quantify the dependence through 2-mixing which allows for nonlinearity. We propose an estimator of the panel mean function and obtain its rate of convergence. We show that under certain conditions the rate of convergence can be improved by allowing the number of individuals in the panel to increase with time.


Rips complexes and covers in the uniform category

Abstract: James \cite{Jam} introduced uniform covering maps as an analog of covering maps in the topological category. Subsequently Berestovskii and Plaut \cite{BP3} introduced a theory of covers for uniform spaces generalizing their results for topological groups \cite{BP1}-\cite{BP2}. Their main concepts are discrete actions and pro-discrete actions, respectively. In case of pro-discrete actions Berestovskii and Plaut provided an analog of the universal covering space and their theory works well for the so-called coverable spaces. As will be seen in Section \ref{SECTION-Comparison}, \cite{BP3} generalizes only regular covering maps in topology and pro-discrete actions may not be preserved by compositions. In this paper we redefine the uniform covering maps and we generalize pro-discrete actions using Rips complexes and the chain lifting property. We expand the concept of generalized paths of Krasinkiewicz and Minc \cite{KraMin}.


Thorn independence in the field of real numbers with a small multiplicative group

Abstract: We characterize thorn-independence in a variety of structures, focusing on the field of real numbers expanded by predicate defining a dense multiplicative subgroup, G, satisfying the Mann property and whose pth powers are of finite index in G. We also show such structures are super-rosy and eliminate imaginaries up to codes for small sets.


Maps between moduli spaces of vector bundles and the base locus of the theta divisor

Abstract: Given a vector bundle $E$ of rank $r$ and degree $d$ on a curve $C$ of genus $g$, one can associate to $E$ in a natural way several other vector bundles. For example, one can take wedge powers of $E$. If $E$ is generated by global sections, the kernel of the evaluation map of sections is again a vector bundle. Also, new vector bundles can be produced by taking elementary transformations centered at a fixed point. Under suitable conditions on degree and rank, these constructions can be carried out globally. While all this processes seem quite elementary, very little is known about the resulting maps. The purpose of this paper is to fill in this gap.


Notes on $\pi_1$ of Smooth Loci of Log Del Pezzo Surfaces

Abstract: It is known that the fundamental groups of smooth loci of Log del Pezzo Surfaces are finite groups. The aim of this note is to study these finite groups. A short table containing these groups is given. And lots of groups on the table are proved to be fundamental groups.


Abstract: We consider compact hyperbolic Coxeter polytopes whose Coxeter diagram contains a unique dotted edge. We prove that such a polytope in d-dimensional hyperbolic space has at most d+3 facets. In view of results of Lann\'er, Kaplinskaja, Esselmann, and the second author, this implies that compact hyperbolic Coxeter polytopes with a unique pair of non-intersecting facets are completely classified. They do exist only up to dimension 6 and in dimension 8.


Multiscale inference about a density

Abstract: We introduce a multiscale test statistic based on local order statistics and spacings that provides simultaneous confidence statements for the existence and location of local increases and decreases of a density or a failure rate. The procedure provides guaranteed finite-sample significance levels, is easy to implement and possesses certain asymptotic optimality and adaptivity properties.


Raynaud vector bundles

Abstract: We construct vector bundles $R^r_\mu$ on a smooth projective curve $X$ having the property that for all sheaves $E$ of slope $\mu$ and rank $r$ on $X$ we have an equivalence: $E$ is a semistable vector bundle $\iff$ $Hom(R^r_\mu,E)=0$. As a byproduct of our construction we obtain effective bounds on $r$ such that the linear system $|R \cdot \Theta|$ has base points on the moduli space $U_X(r,r(g-1))$.


On the L^p-distorsion of finite quotients of amenable groups

Abstract: We study the L^p-distortion of finite quotients of amenable groups. In particular, for every number p larger or equal than 2, we prove that the l^p-distortion of the finite lamplighter group grows like (\log n)^{1/p}. We also give the asymptotic behavior of the l^p-distortion of finite quotients of certain metabelian polycyclic groups and of the solvable Baumslag-Solitar groups BS(m,1). The proofs are short and elementary.


Whitehead-Torsion und Faserungen

Abstract: This work treats on the question whether a given map f: M -> B of smooth closed manifolds is homotopic to a smooth fiber bundle. We define a first obstruction in H^1(B;Wh(\pi_1(E))) and, provided that this obstruction vanishes and one additional condition is verified, a second obstruction in Wh(\pi_1(E)) >. Both elements vanish if the answer to the above question is positive. In the case where B is the 1-sphere and the dimension of M exceeds five, we show that the converse is also true, using a relationship with two obstructions defined by Farrell in this particular situation.


On the Greenfield-Wallach and Katok conjectures

Abstract: We survey recent progress on the Greenfield-Wallach and Katok conjectures on globally hypoelliptic and cohomology free vector fields and derive a proof of the conjectures in dimension three. The argument is primarily based on recent work of F. and J. Rodriguez Hertz which allows us to reduce the question to the case of a Reeb flow for a contact form. The contact case is settled by invoking the Weinstein conjecture (which has been recently announced by C. Taubes).


Distributions associated with general runs and patterns in hidden Markov models

Abstract: This paper gives a method for computing distributions associated with patterns in the state sequence of a hidden Markov model, conditional on observing all or part of the observation sequence. Probabilities are computed for very general classes of patterns (competing patterns and generalized later patterns), and thus, the theory includes as special cases results for a large class of problems that have wide application. The unobserved state sequence is assumed to be Markovian with a general order of dependence. An auxiliary Markov chain is associated with the state sequence and is used to simplify the computations. Two examples are given to illustrate the use of the methodology. Whereas the first application is more to illustrate the basic steps in applying the theory, the second is a more detailed application to DNA sequences, and shows that the methods can be adapted to include restrictions related to biological knowledge.


The order completion method for systems of nonlinear PDEs: Pseudo-topological perspectives

Abstract: By setting up appropriate uniform convergence structures, we are able to reformulate the Order Completion Method of Oberguggenberger and Rosinger in a setting that more closely resembles the usual topological constructions for solving PDEs. As an application, we obtain existence and uniqueness results for the solutions of arbitrary continuous, nonlinear PDEs.


Local results for flows whose speed or height satisfies a bound of the form $\frac c t$

Abstract: In this paper we prove local results for solutions to the Ricci flow (heat flow) whose speed (height) is bounded by $\frac c t$ for some time interval $ t \in (0,T)$. These results are contained in chapter 7 of the author's habilitation thesis, University of Freiburg, Germany, 2006.


Stochastic control problems for systems driven by normal martingales

Abstract: In this paper we study a class of stochastic control problems in which the control of the jump size is essential. Such a model is a generalized version for various applied problems ranging from optimal reinsurance selections for general insurance models to queueing theory. The main novel point of such a control problem is that by changing the jump size of the system, one essentially changes the type of the driving martingale. Such a feature does not seem to have been investigated in any existing stochastic control literature. We shall first provide a rigorous theoretical foundation for the control problem by establishing an existence result for the multidimensional structure equation on a Wiener--Poisson space, given an arbitrary bounded jump size control process; and by providing an auxiliary counterexample showing the nonuniqueness for such solutions. Based on these theoretical results, we then formulate the control problem and prove the Bellman principle, and derive the corresponding Hamilton--Jacobi--Bellman (HJB) equation, which in this case is a mixed second-order partial differential/difference equation. Finally, we prove a uniqueness result for the viscosity solution of such an HJB equation.


Induction and computation of Bass Nil Groups for finite groups

Abstract: Let G be a finite group. We show that the Bass Nil-groups $NK_n(RG)$, $n \in Z$, are generated from the p-subgroups of G by induction maps, certain twisting maps depending on elements in the centralizers of the p-subgroups, and the Verschiebung homomorphisms. As a consequence, the groups $NK_n(RG)$ are generated by induction from elementary subgroups. For $NK_0(ZG)$ we get an improved estimate of the torsion exponent.


How much information about a dynamical system do its recurrences contain?

Abstract: We show that, under suitable assumptions, Poincare recurrences of a dynamical system determine its topology in phase space. Therefore, dynamical systems with the same recurrences are topologically equivalent.


Stopped diffusion processes: boundary corrections and overshoot

Abstract: For a stopped diffusion process in a multidimensional time-dependent domain $\D$, we propose and analyse a new procedure consisting in simulating the process with an Euler scheme with step size $\Delta$ and stopping it at discrete times $(i\Delta)_{i\in\N^*}$ in a modified domain, whose boundary has been appropriately shifted. The shift is locally in the direction of the inward normal $n(t,x)$ at any point $(t,x)$ on the parabolic boundary of $\D$, and its amplitude is equal to $0.5826 (...) |n^*\sigma|(t,x)\sqrt \Delta$ where $\sigma$ stands for the diffusion coefficient of the process. The procedure is thus extremely easy to use. In addition, we prove that the rate of convergence w.r.t. $\Delta$ for the associated weak error is higher than without shifting, generalizin g previous results by \cite{broa:glas:kou:97} obtained for the one dimensional Brownian motion. For this, we establish in full generality the asymptotics of the triplet exit time/exit position/overshoot for the discretely stopped Euler scheme. Here, the overshoot means the distance to the boundary of the process when it exits the domain. Numerical experiments support these results.


Continuous spectrum for a class of nonhomogeneous differential operators

Abstract: We study the boundary value problem $-{\rm div}((|\nabla u|^{p_1(x)-2}+|\nabla u|^{p_2(x)-2})\nabla u)=\lambda|u|^{q(x)-2}u$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded domain in $\RR^N$ with smooth boundary, $\lambda$ is a positive real number, and the continuous functions $p_1$, $p_2$, and $q$ satisfy $1<p_2(x)<q(x)<p_1(x)<N$ and $\max_{y\in\bar\Omega}q(y)<\frac{N p_2(x)}{N-p_2(x)}$ for any $x\in\bar\Omega$. The main result of this paper establishes the existence of two positive constants $\lambda_0$ and $\lambda_1$ with $\lambda_0\leq\lambda_1$ such that any $\lambda\in[\lambda_1,\infty)$ is an eigenvalue, while any $\lambda\in(0,\lambda_0)$ is not an eigenvalue of the above problem.


Toward the classification of cohomology-free vector fields

Abstract: In 1984, Anatole Katok conjectured that the only closed orientable manifolds that support cohomology-free vector fields are tori and these vector fields are smoothly conjugated to Diophantine (constant) ones. In this work we present a proof of Katok conjecture for 3-manifolds.


The pentagon relation for the quantum dilogarithm and quantized M_{0,5}

Abstract: We introduce and study a Schwarz space S in the space of functions on the real line. It is a module over the algebra L of regular functions on the (modular double of the) non-commutative q-deformation of the moduli space of configurations of 5 cyclically ordered points on the projective line. The algebra L has an order five automorphism corresponding to the cyclic shift of the points. The quantum dilogarithm gives rise to an automorphism of the space Schwarz S intertwining the automorphism of L. This easily implies the pentagon relation for the quantum dilogarithm function. The triple (L, S, the automorphism) is the quantized moduli space of configurations of 5 points on the projective line. It is the simplest example of a quantized cluster X-variety.


Weighted Low-Regularity Solutions of the KP-I Initial Value Problem

Abstract: In this paper we establish local well-posedness of the KP-I problem, with initial data small in the intersection of the natural energy space with the space of functions which are square integrable when multiplied by the weight y. The result is proved by the contraction mapping principle. A similar (but slightly weaker) result was the main Theorem in the paper " Low regularity solutions for the Kadomstev-Petviashvili I equation " by Colliander, Kenig and Staffilani (GAFA 13 (2003),737-794 and math.AP/0204244). Ionescu found a counterexample (included in the present paper) to the main estimate used in the GAFA paper, which renders incorrect the proof there. The present paper thus provides a correct proof of a strengthened version of the main result in the GAFA paper.


Minimal $\gamma$--sheaves

Abstract: In this note we show that finitely generated unit $O_X[\sigma]$--modules for $X$ regular and $F$--finite have a minimal root (in the sense of [Lyubeznik, F-modules] Definition~3.6). This problem was posed by Lyubeznik and answered by himself in the case that $X=\Spec R$ is a complete local ring. One immediate consequence of this result is that the parameter test module of tight closure theory commutes with localization. As a further application of the methods in this paper we give new proofs of the results on discreteness and rationality of $F$--thresholds [arXiv:0705.1210] and on $D$-module generation [arXiv:math/0502405v1]. The new proofs are valid in a slightly more general setting such that they also party cover the generalizations recently obtained in [arXiv:0706.3028].


Equisingularite reelle : invariants locaux et conditions de regularite

Abstract: For germs of subanalytic sets, we define two finite sequences of new numerical invariants. The first one is obtained by localizing the classical Lipschitz-Killing curvatures, the second one is the real analogue of the evanescent characteristics introduced by M. Kashiwara. We show that each invariant of one sequence is a linear combination of the invariants of the other sequence. We then connect our invariants to the geometry of the discriminants of all dimension. Finally we prove that these invariants are continuous along Verdier strata of a closed subanalytic set.


Universal abelian covers of rational surface singularities and multi-index filtrations

Abstract: In previous papers, there were computed the Poincare series of some (multi-index) filtrations on the ring of germs of functions on a rational surface singularity. These Poincare series were written as the integer parts of certain fractional power series, an interpretation of whom was not given. Here we show that, up to a simple change of variables, these fractional power series are specializations of the equivariant Poincare series for filtrations on the ring of germs of functions on the universal abelian cover of the surface singularity. We compute these equivariant Poincare series.


An effective criterion and a new example for ballistic diffusions in random environment

Abstract: In the setting of multidimensional diffusions in random environment, we carry on the investigation of condition $(T')$, introduced by Sznitman [Ann. Probab. 29 (2001) 723--764] and by Schmitz [Ann. Inst. H. Poincar\'{e} Probab. Statist. 42 (2006) 683--714] respectively in the discrete and continuous setting, and which implies a law of large numbers with nonvanishing limiting velocity (ballistic behavior) as well as a central limit theorem. Specifically, we show that when $d\geq2$, $(T')$ is equivalent to an effective condition that can be checked by local inspection of the environment. When $d=1$, we prove that condition $(T')$ is merely equivalent to almost sure transience. As an application of the effective criterion, we show that when $d\geq4$ a perturbation of Brownian motion by a random drift of size at most $\epsilon>0$ whose projection on some direction has expectation bigger than $\epsilon^{2-\eta},\eta>0$, satisfies condition $(T')$ when $\epsilon$ is small and hence exhibits ballistic behavior. This class of diffusions contains new examples of ballistic behavior which in particular do not fulfill the condition in [Ann. Inst. H. Poincar\'{e} Probab. Statist. 42 (2006) 683--714], (5.4) therein, related to Kalikow's condition.


Abstract: A generalized version of Chernoff's theorem has been obtained. Namely, the version of Chernoff's theorem for semigroups obtained in a paper by Smolyanov, Weizsaecker, and Wittich is generalized for a time-inhomogeneous case. The main theorem obtained in the current paper, Chernoff's theorem for evolution families, deals with a family of time-dependent generators of semigroups $A_t$ on a Banach space, a two-parameter family of operators $Q_{t,t+\Delta t}$ satisfying the relation: $\frac{\partial}{\partial \Delta t}Q_{t,t+\Delta t}|_{\Delta t = 0}=A_t$, whose products $Q_{t_i,t_{i+1}}... Q_{t_{k-1},t_k}$ are uniformly bounded for all subpartitions $s = t_0 < t_1 < >... < t_n = t$. The theorem states that $Q_{t_0,t_1}... Q_{t_{n-1},t_n}$ converges to an evolution family $U(s,t)$ solving a non-autonomous Cauchy problem. Furthermore, the theorem is formulated for a particular case when the generators $A_t$ are time dependent second order differential operators. Finally, an example of application of this theorem to a construction of time-inhomogeneous diffusions on a compact Riemannian manifold is given. Keywords: Chernoff's theorem, evolution family, strongly continuous semigroup, evolution families generated by manifold valued stochastic processes.


The classification of torsion endo-trivial modules

Abstract: This paper is a major step in the classification of endotrivial modules over p-groups. Let G be a finite p-group and k be a field of characteristic p. A kG-module M is an endo-trivial module if {\End_k(M)\cong k\oplus F} as kG-modules, where F is a free module. The classification of endo-trivial modules is the crucial step for understanding the more general class of endo-permutation modules. The endo-permutation modules play an important role in module theory, in particular as source modules, and in block theory where they appear in the description of source algebras. Endo-trivial modules are also important in the study of both derived equivalences and stable equivalences of group algebras and block algebras. The collection of isomorphism classes of endo-trivial modules modulo projectives is an abelian group under tensor product. The main result of this paper is that this group is torsion free except in the case that G is cyclic, quaternion or semi-dihedral. Hence for any p-group which is not cyclic, quaternion or semi-dihedral and any finitely generated kG-module M, if M \otimes_k M \otimes_k ... \otimes_k M \cong k \oplus P for some projective module P and some finite number of tensor products, then M \cong k \oplus Q for some projective module Q. The proof uses a reduction to the cases in which G is an extraspecial or almost extraspecial p-group, proved in a previous paper of the authors, and makes extensive use of the theory of support varieties for modules.


Inf-sup estimates for the Stokes problem in a periodic channel

Abstract: We derive estimates of the Babu\u{s}ka-Brezzi inf-sup constant $\beta$ for two-dimensional incompressible flow in a periodic channel with one flat boundary and the other given by a periodic, Lipschitz continuous function $h$. If $h$ is a constant function (so the domain is rectangular), we show that periodicity in one direction but not the other leads to an interesting connection between $\beta$ and the unitary operator mapping the Fourier sine coefficients of a function to its Fourier cosine coefficients. We exploit this connection to determine the dependence of $\beta$ on the aspect ratio of the rectangle. We then show how to transfer this result to the case that $h$ is $C^{1,1}$ or even $C^{0,1}$ by a change of variables. We avoid non-constructive theorems of functional analysis in order to explicitly exhibit the dependence of $\beta$ on features of the geometry such as the aspect ratio, the maximum slope, and the minimum gap thickness (if $h$ passes near the substrate). We give an example to show that our estimates are optimal in their dependence on the minimum gap thickness in the $C^{1,1}$ case, and nearly optimal in the Lipschitz case.


Equivariant path fields on topological manifolds

Abstract: A classical theorem of H. Hopf asserts that a closed connected smooth manifold admits a nowhere vanishing vector field if and only if its Euler characteristic is zero. R. Brown generalized Hopf's result to topological manifolds, replacing vector fields with path fields. In this note, we give an equivariant analog of Brown's theorem for locally smooth $G$-manifolds where $G$ is a finite group.


On graphs with subgraphs of large independence numbers

Abstract: Let G be a graph on n vertices in which every induced subgraph on s=\log^3 n vertices has an independent set of size at least t=\log n. What is the largest q=q(n) so that every such G must contain an independent set of size at least q ? This is one of several related questions raised by Erdos and Hajnal. We show that q(n)=\Theta(\log^2 n/\log \log n), investigate the more general problem obtained by changing the parameters s and t, and discuss the connection to a related Ramsey-type problem.


Embedding nearly-spanning bounded degree trees

Abstract: We derive a sufficient condition for a sparse graph G on n vertices to contain a copy of a tree T of maximum degree at most d on (1-\epsilon)n vertices, in terms of the expansion properties of G. As a result we show that for fixed d\geq 2 and 0<\epsilon<1, there exists a constant c=c(d,\epsilon) such that a random graph G(n,c/n) contains almost surely a copy of every tree T on (1-\epsilon)n vertices with maximum degree at most d. We also prove that if an (n,D,\lambda)-graph G (i.e., a D-regular graph on n vertices all of whose eigenvalues, except the first one, are at most \lambda in their absolute values) has large enough spectral gap D/\lambda as a function of d and \epsilon, then G has a copy of every tree T as above.


Making a K_4-free graph bipartite

Abstract: We show that every K_4-free graph G with n vertices can be made bipartite by deleting at most n^2/9 edges. Moreover, the only extremal graph which requires deletion of that many edges is a complete 3-partite graph with parts of size n/3. This proves an old conjecture of P. Erdos.


Ramsey numbers and the size of graphs

Abstract: For two graph H and G, the Ramsey number r(H, G) is the smallest positive integer n such that every red-blue edge coloring of the complete graph K_n on n vertices contains either a red copy of H or a blue copy of G. Motivated by questions of Erdos and Harary, in this note we study how the Ramsey number r(K_s, G) depends on the size of the graph G. For s \geq 3, we prove that for every G with m edges, r(K_s,G) \geq c (m/\log m)^{\frac{s+1}{s+3}} for some positive constant c depending only on s. This lower bound improves an earlier result of Erdos, Faudree, Rousseau, and Schelp, and is tight up to a polylogarithmic factor when s=3. We also study the maximum value of r(K_s,G) as a function of m.


Local resilience of graphs

Abstract: In this paper, we initiate a systematic study of graph resilience. The (local) resilience of a graph G with respect to a property P measures how much one has to change G (locally) in order to destroy P. Estimating the resilience leads to many new and challenging problems. Here we focus on random and pseudo-random graphs and prove several sharp results.


Induced Ramsey-type theorems

Abstract: We present a unified approach to proving Ramsey-type theorems for graphs with a forbidden induced subgraph which can be used to extend and improve the earlier results of Rodl, Erdos-Hajnal, Promel-Rodl, Nikiforov, Chung-Graham, and Luczak-Rodl. The proofs are based on a simple lemma (generalizing one by Graham, Rodl, and Rucinski) that can be used as a replacement for Szemeredi's regularity lemma, thereby giving much better bounds. The same approach can be also used to show that pseudo-random graphs have strong induced Ramsey properties. This leads to explicit constructions for upper bounds on various induced Ramsey numbers.


Effective Termination of Kohn's Algorithm for Subelliptic Multipliers

Abstract: This note discusses the problem of the effective termination of Kohn's algorithm for subelliptic multipliers for bounded smooth weakly pseudoconvex domains of finite type. We give a complete proof for the case of special domains of finite type and indicate briefly how this method is to be extended to the case of general bounded smooth weakly pseudoconvex domains of finite type.


On some power sum problems of Montgomery and Turan

Abstract: We use an estimate for character sums over finite fields of Katz to solve open problems of Montgomery and Turan. Let h=>2 be an integer. We prove that inf_{|z_k| => 1} max_{v=1,...,n^h} |sum_{k=1}^n z_k^v| <= (h-1+o(1)) sqrt n. This gives the right order of magnitude for the quantity and improves on a bound of Erdos-Renyi by a factor of the order sqrt log n.


Bounds on the number of real solutions to polynomial equations

Abstract: We use Gale duality for polynomial complete intersections and adapt the proof of the fewnomial bound for positive solutions to obtain the bound (e^4+3) 2^(k choose 2) n^k/4 for the number of non-zero real solutions to a system of n polynomials in n variables having n+k+1 monomials whose exponent vectors generate a subgroup of Z^n of odd index. This bound exceeds the bound for positive solutions only by the constant factor (e^4+3)/(e^2+3) and it is asymptotically sharp for k fixed and n large.


Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization

Abstract: The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NP-hard. In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum rank solution can be recovered by solving a convex optimization problem, namely the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability. The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this pre-existing concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization.


The Borel conjecture for manifolds with virtually solvable fundamental groups

Abstract: The article has been withdrawn by the author. Wolfgang Lueck and Peter Linnell pointed out that the proof of Lemma 3.8 does not apply to the unrestricted case of wreath product. It is not clear at this stage how to complete the proof of Theorem 3.1 using the present version of Lemma 3.8. The valid results originating from this article will be added in a later paper.


Adaptive dynamics in logistic branching populations

Abstract: We consider a trait-structured population subject to mutation, birth and competition of logistic type, where the number of coexisting types may fluctuate. Applying a limit of rare mutations to this population while keeping the population size finite leads to a jump process, the so-called `trait substitution sequence', where evolution proceeds by successive invasions and fixations of mutant types. The probability of fixation of a mutant is interpreted as a fitness landscape that depends on the current state of the population. It was in adaptive dynamics that this kind of model was first invented and studied, under the additional assumption of large population. Assuming also small mutation steps, adaptive dynamics' theory provides a deterministic ODE approximating the evolutionary dynamics of the dominant trait of the population, called `canonical equation of adaptive dynamics'. In this work, we want to include genetic drift in this models by keeping the population finite. Rescaling mutation steps (weak selection) yields in this case a diffusion on the trait space that we call `canonical diffusion of adaptive dynamics', in which genetic drift (diffusive term) is combined with directional selection (deterministic term) driven by the fitness gradient. Finally, in order to compute the coefficients of this diffusion, we seek explicit first-order formulae for the probability of fixation of a nearly neutral mutant appearing in a resident population. These formulae are expressed in terms of `invasibility coefficients' associated with fertility, defense, aggressiveness and isolation, which measure the robustness (stability w.r.t. selective strengths) of the resident type. Some numerical results on the canonical diffusion are also given.


Decay estimates of a tangential derivative to the light cone for the wave equation and their application

Abstract: We consider wave equations in three space dimensions, and obtain new weighted $L^\infty$-$L^\infty$ estimates for a tangential derivative to the light cone. As an application, we give a new proof of the global existence theorem, which was originally proved by Klainerman and Christodoulou, for systems of nonlinear wave equations under the null condition. Our new proof has the advantage of using neither the scaling nor the pseudo-rotation operators.


Explicit formulas for biharmonic submanifolds in Sasakian space forms

Abstract: We classify the biharmonic Legendre curves in a Sasakian space form, and obtain their explicit parametric equations in the $(2n+1)$-dimensional unit sphere endowed with the canonical and deformed Sasakian structures defined by Tanno. Then, composing with the flow of the Reeb vector field, we transform a biharmonic integral submanifold into a biharmonic anti-invariant submanifold. Using this method we obtain new examples of biharmonic submanifolds in spheres and, in particular, in $\mathbb{S}^{7}$.


Harmonic analysis of additive Levy processes

Abstract: Let $X_1,...,X_N$ denote $N$ independent $d$-dimensional L\'evy processes, and consider the $N$-parameter random field \[\X(\bm{t}):= X_1(t_1)+...+X_N(t_N).\] First we demonstrate that for all nonrandom Borel sets $F\subseteq\R^d$, the Minkowski sum $\X(\R^N_+)\oplus F$, of the range $\X(\R^N_+)$ of $\X$ with $F$, can have positive $d$-dimensional Lebesgue measure if and only if a certain capacity of $F$ is positive. This improves our earlier joint effort with Yuquan Zhong \ycite{KXZ:03} by removing a symmetry-type condition there. Moreover, we show that under mild regularity conditions, our necessary and sufficient condition can be recast in terms of one-potential densities. This rests on developing results in classical [non-probabilistic] harmonic analysis that might be of independent interest. As was shown in \fullocite{KXZ:03}, the potential theory of the type studied here has a large number of consequences in the theory of L\'evy processes. We present a few new consequences here.


\C-flows A^z of linear maps A expressed in terms of A^{-1},A^{-2},...,A^{-n} and analytic functions of z

Abstract: Suppose A\in GL_n(\C) has a relation A^p=c_{p-1}A^{p-1}+.... + c_1 A+ c_0I where the c_i in \C. This article describes how to construct analytic functions c_i(z) such that A^z=c_{p-1}(z)A^{p-1}+... + c_1(z) A+ c_0(z)I . One of the theorems gives a possible description of the c_i(z): c_i(z)=C^z\alpha where C\in Mat_p(\C) is (similar to) the companion matrix of X^p-c_{p-1}X^{p-1}-... -c_1X-c_0I, and \alpha:= (c_{p-1},...,c_1,c_0)^t.


Lattice polytopes of degree 2

Abstract: A theorem of Scott gives an upper bound for the normalized volume of lattice polygons with exactly $i>0$ interior lattice points. We will show that the same bound is true for the normalized volume of lattice polytopes of degree 2 even in higher dimensions. In particular, there is only a finite number of quadratic polynomials with fixed leading coefficient being the $h^*$-polynomial of a lattice polytope.


A characterization property on field equivalent to algebraicity on Banach spaces

Abstract: In his article "A discrete form of the theorem that each field endomorphism of $\mathbb{R}$ ($\mathbb{Q}_p$) is the identity", Tyszka introduce a logical property which is equivalent to algebraicity in $\mathbb{R}$ and in $\mathbb{Q}_p$. Amazingly, the property is no longer equivalent to algebraicity in $\mathbb{C}$. This article present a similirar property which is equivalent to algebraicity in any field of characteristic zero which is also a Banach space, and prove a weaker equivalency for fields of positive charcteristic (which are also Banach spaces).


Infinitely generated Derksen and Makar-Limanov invariant

Abstract: In this paper, we give an example of a finitely generated 3-dimensional C-algebra which has infinitely generated Derksen invariant as well as Makar-Limaonv invariant.


Constructing (almost) rigid rings and a UFD having infinitely generated Derksen and Makar-Limanov invariant

Abstract: An example is given of a UFD which has infinitely generated Derksen invariant. The ring is \textquotedblleft almost rigid\textquotedblright\ meaning that the Derksen invariant is equal to the Makar-Limanov invariant. Techniques to show that a ring is (almost) rigid are discussed, among which is a generalization of Mason's abc-theorem.


Zero-Bidimension and Various Classes of Bitopological Spaces

Abstract: The sum theorem and its corollaries are proved for a countable family of zero-dimensional (in the sense of small and large inductive bidimensions) p-closed sets, using a new notion of relative normality whose topological correspondent is also new. The notion of almost $n$-dimensionality is considered from the bitopological point of view. Bitopological spaces in which every subset is i-open in its $j$-closure (i.e.,(i,j)-submaximal spaces) are introduced and their properties are studied. Based on the investigations begun in [5] and [14], sufficient conditions are found for bitopological spaces to be(1,2)-Baire in the class of p-normal spaces. Furthermore, (i,j)-I-spaces are introduced and both the relations between(i,j)-submaximal, (i,j)-nodec and (i,j)-I-spaces, and their properties are studied when two topologies on a set are either independent of each other or interconnected by the inclusion, S-, C- and N-relations or by their combinations. The final part of the paper deals with the questions of preservation of $(i,j)$-submaximal and $(2,1)\dd I$-spaces to an image, of $D$-spaces to an image and an inverse image for both the topological and the bitopological cases. Two theorems are formulated containing, on the one hand, topological conditions and, on the other hand, bitopological ones, under which a topological space is a $D$-space.


The Automorphism Group of Certain Factorial Threefolds and a Cancellation Problem

Abstract: The automorphism groups of certain factorial complex affine threefolds admitting locally trivial actions of the additive group are determined. As a consequence new counterexamples to a generalized cancellation problem are obtained.


SiZer for time series: A new approach to the analysis of trends

Abstract: Smoothing methods and SiZer are a useful statistical tool for discovering statistically significant structure in data. Based on scale space ideas originally developed in the computer vision literature, SiZer (SIgnificant ZERo crossing of the derivatives) is a graphical device to assess which observed features are `really there' and which are just spurious sampling artifacts. In this paper, we develop SiZer like ideas in time series analysis to address the important issue of significance of trends. This is not a straightforward extension, since one data set does not contain the information needed to distinguish `trend' from `dependence'. A new visualization is proposed, which shows the statistician the range of trade-offs that are available. Simulation and real data results illustrate the effectiveness of the method.


Transportation-information inequalities for Markov processes

Abstract: In this paper, one investigates the following type of transportation-information $T_cI$ inequalities: $\alpha(T_c(\nu,\mu))\le I(\nu|\mu)$ for all probability measures $\nu$ on some metric space $(\XX, d)$, where $\mu$ is a given probability measure, $T_c(\nu,\mu)$ is the transportation cost from $\nu$ to $\mu$ with respect to some cost function $c(x,y)$ on $\XX^2$, $I(\nu|\mu)$ is the Fisher-Donsker-Varadhan information of $\nu$ with respect to $\mu$ and $\alpha: [0,\infty)\to [0,\infty]$ is some left continuous increasing function. Using large deviation techniques, it is shown that $T_cI$ is equivalent to some concentration inequality for the occupation measure of a $\mu$-reversible ergodic Markov process related to $I(\cdot|\mu)$, a counterpart of the characterizations of transportation-entropy inequalities, recently obtained by Gozlan and L\'eonard in the i.i.d. case . Tensorization properties of $T_cI$ are also derived.


On the Theory of Relative Bitopological and Topological Properties

Abstract: In the first part of the work (Sections 2-6) a special attention is given to relative separation axioms and relative connectedness, in particular, many relative versions of p-T_0, p-T_1, p-T_2, (i,j)- and p-regularities, (i,j)- and p-complete regularities, p-real normality and p-normality are discussed. Moreover, relative properties of (i,j)- and p-compactness types, including relative versions of (i,j)- and p-paracompactness, (i,j)- and p-Lindeofness, (i,j)- and p-pseudocompactness are also introduced and investigated. The second part (Sections 7-12) is devoted, on the one hand, to relative bitopological inductive and covering dimension functions and, on the other hand, to relative versions of Baire spaces for both the topological and the bitopological case. At the end, note that relative (bi)topological properties play a special role not only in the development of respective theories, but also in the strengthening of the previously known results.


Optimal curves of low genus over finite fields

Abstract: The Hasse-Weil-Serre bound is improved for curves of low genera over finite fields with discriminant in {-3,-4,-7,-8,-11,-19} by studying optimal curves.


The extended Burnside ring and module categories

Abstract: In this note an `extended Burnside ring' is defined, generated by classes of semisimple module categories over Rep(G) with quasifibre functors. Here G is a finite group and representations are taken over an algebraically closed field of characteristic 0. It is shown that this is equivalent to a ring generated by centrally extended G-sets and hence the name. Ring homomorphisms into the multiplicative group of the field are computed with an explicit formula and tables of these homomorphisms are given for the groups S_4 and S_5 which are of particular interest in the context of reductive algebraic groups. ----- L'anneau de Burnside \'etendu et cat\'egories de modules. Dans cette note un `Anneau de Burnside \'etendu' est d\'efini, gener\'e par des classes de cat\'egories de modules semisimples sur Rep(G) avec des foncteurs quasifibres. Ici G est un groupe fini, et des repr\'esentations sont prises sur un corps alg\'ebriquement clos de caract\'eristique nulle. Il est demontr\'e que ceci \'equivaut \`a un anneau gener\'e par des G-ensembles centralement \'etendus, d'o\`u le nom. Des homomorphismes d'anneau dans le groupe multiplicatif du corps sont comput\'ees avec une formule explicite et des tableaux de ces homomorphismes sont fournis pour les groupes S_4 et S_5 qui sont d'un int\'er\^et particulier dans le contexte de groupes alg\'ebriques r\'eductifs.


Automorphic Forms and Reeb-Like Foliations on Three-Manifolds

Abstract: In this paper, we consider different ways of generating dynamical systems on 3-manifolds. We first derive explicit differential equations for dynamical systems defined on generic hyperbolic 3-manifolds by using automorphic function theory to uniformize the upper half-space model. It is achieved via the modification of the standard Poincare theta series to generate systems invariant within each individual fundamental region such that the solution trajectories match up on the appropriate sides after the identifications which generate a hyperbolic 3-manifold. Then we consider the gluing pattern in the conformal ball model. At the end we shall study the construction of dynamical systems by using the Reeb foliation.


Dynamical Systems On Three Manifolds Part II: 3-Manifolds,Heegaard Splittings and Three-Dimensional Systems

Abstract: The global behaviour of nonlinear systems is extremely important in control and systems theory since the usual local theories will only give information about a system in some neighbourhood of an operating point. Away from that point, the system may have totally different behaviour and so the theory developed for the local system will be useless for the global one. In this paper we shall consider the analytical and topological structure of systems on 2- and 3- manifolds and show that it is possible to obtain systems with 'arbitrarily strange' behaviour, i.e., arbitrary numbers of chaotic regimes which are knotted and linked in arbitrary ways. We shall do this by considering Heegaard Splittings of these manifolds and the resulting systems defined on the boundaries.


Topology of the regular part for infinitely renormalizable quadratic polynomials

Abstract: In this paper we describe the well studied process of renormalization of quadratic polynomials from the point of view of their natural extensions. In particular, we describe the topology of the inverse limit of infinitely renormalizable quadratic polynomials and prove that when they satisfy a-priori bounds, the topology is rigid modulo its combinatorics.


On the methods to construct UFD counterexamples to a cancellation problem

Abstract: In a previous paper, the author together with prof. dr. Finston constructed a class of UFDs A_{n,m} where n,m\in \N^*. These rings are all stably equivalent (A_{n,m}[T]\cong A_{p,q}[T] for all n,m,p,q) but are only isomorphic themselves if (n,m)=(p,q). These examples are the first UFD examples over a characteristically closed field satisfying this behavior. In this paper, we describe the methods used in this article, and show that they are very general, enabling the reader to construct many more such examples, based on the same principles.


On the set of complex points of a 2-sphere

Abstract: Let $G$ be a strictly pseudoconvex domain in $\mathbb{C}^2$ with $C^\infty$-smooth boundary $\partial G$. Let $S$ be a 2-dimensional sphere embedded into $\partial G$. Denote by $\mathcal{E}$ the set of all complex points on $S$. We study how the structure of the set $\mathcal{E}$ depends on the smoothness of $S$


Symmetry in semidefinite programs

Abstract: This paper is a tutorial in a general and explicit procedure to simplify semidefinite programs which are invariant under the action of a symmetry group. The procedure is based on basic notions of representation theory of finite groups. As an example we derive the block diagonalization of the Terwilliger algebra of the binary Hamming scheme in this framework. Here its connection to the orthogonal Hahn and Krawtchouk polynomials becomes visible.


Immediate Calculation of some Poisson Type Integrals Using Supermathematics Circular Ex-Centric Functions

Abstract: This article presents two methods, in parallel, of solving more complex integrals, among which is the Poisson's integral, in order to emphasize the obvious advantages of a new method of integration, which uses the supermathematics circular ex-centric functions.


Quantum Unique Ergodicity for Eisenstein Series on the Hilbert Modular Group over a Totally Real Field

Abstract: W. Luo and P. Sarnak have proved the quantum unique ergodicity property for Eisenstein series on $\rm{PSL}(2,\mathbb{Z}) \backslash H$. We extend their result to Eisenstein series on $\rm{PSL}(2,O) \backslash H^n$, where $O$ is the ring of integers in a totally real field of degree $n$ over $Q$ with narrow class number one, using the Eisenstein series considered by I. Efrat. We also give an expository treatment of the theory of Hecke operators on non-holomorphic Hilbert modular forms.


Subgroup separability in residually free groups

Abstract: We prove that the finitely presentable subgroups of residually free groups are separable and that the subgroups of type $\mathrm{FP}_\infty$ are virtual retracts. We describe a uniform solution to the membership problem for finitely presentable subgroups of residually free groups.


Isoperimetric profile and random walks on locally compact solvable groups

Abstract: We study a large class of amenable locally compact groups containing all solvable algebraic groups over a local field and their discrete subgroups. We show that the isoperimetric profile of these groups is in some sense optimal among amenable groups. We use this fact to compute the probability of return of symmetric random walks, and to derive various other geometric properties which are likely to be only satisfied by these groups.


A mixed problem for the infinity laplacian via Tug-of-War games

Abstract: In this paper we prove that a function $ u\in\mathcal{C}(\bar{\Omega})$ is the continuous value of the Tug-of-War game described in \cite{PSSW} if and only if it is the unique viscosity solution to the infinity laplacian with mixed boundary conditions {-\Delta_{\infty}u(x)=0\quad & \text{in} \Omega, \frac{\partial u}{\partial n}(x)=0\quad & \text{on} \Gamma_N, u(x)=F(x)\quad & \text{on} \Gamma_D. By using the results in \cite{PSSW}, it follows that this viscous PDE problem has a unique solution, which is the unique {\it absolutely minimizing Lipschitz extension} to the whole $\bar{\Omega}$ (in the sense of \cite{Aronsson} and \cite{PSSW}) of the boundary data $ F:\Gamma_D\to\R $.


Theory of the Siegel Modular Variety

Abstract: In this paper, we discuss the theory of the Siegel modular variety in the aspects of arithmetic and geometry. This article covers the theory of Siegel modular forms, the Hecke theory, a lifting of elliptic cusp forms, geometric properties of the Siegel modular variety, (hypothetical) motives attached to Siegel modular forms and a cohomology of the Siegel modular variety.


Diagonal fibrations are pointwise fibrations

Abstract: On the category of bisimplicial sets there are different Quillen closed model structures associated to various definitions of fibrations. In one of them, which is due to Bousfield and Kan and that consists of seeing a bisimplicial set as a simplicial object in the category of simplicial sets, fibrations are those bisimplicial set maps such that each of the induced simplicial set maps is a Kan fibration, that is, the pointwise fibrations. In another of them, introduced by Moerdijk, a bisimplicial map is a fibration if it induces a Kan fibration of associated diagonal simplicial sets, that is, the diagonal fibrations. In this note, we prove that every diagonal fibration is a pointwise fibration.


Riesz integral representation theory

Abstract: We present a Riesz integral representation theory in which functions, operators and measures take values in uniform commutative monoids (a commutative monoid with a uniformity making the binary operation of the monoid uniformly continuous). It describes the operators to which the theory can be applied and the finitely-additive measures they generate. Operators satisfying the conditions will be called ``Riesz integrals''. Given an underlying ``Riesz system'', it is shown that every Riesz integral generates a certain kind of finitely additive measure called here a ``Riesz measure''. The correspondence between Riesz integrals and Riesz measures is a bijection. A straightforward calculation shows that if an operator has such a representation, then it must have the Hammerstein property. For topological vector spaces, the theory yields necessary and sufficient conditions for operators with the Hammerstein property to be Riesz integrals. We note that uniform commutative monoids arise naturally when considering set-valued functions, and that the axioms of a Riesz system rule out certain spaces of infinitely differentiable functions.


On the Ring of Integer-valued Quasi-polynomials

Abstract: The paper studies some properties of the ring of integer-valued quasi-polynomials. On this ring, theory of generalized Euclidean division and generalized GCD are presented. Applications to finite simple continued fraction expansion and Smith normal form of integral matrices with integer parameters are also given.


Schwarzian Derivatives and Uniform Local Univalence

Abstract: Quantitative estimates are obtained for the (finite) valence of functions analytic in the unit disk with Schwarzian derivative that is bounded or of slow growth. A harmonic mapping is shown to be uniformly locally univalent with respect to the hyperbolic metric if and only if it has finite Schwarzian norm, thus generalizing a result of B. Schwarz for analytic functions. A numerical bound is obtained for the Schwarzian norms of univalent harmonic mappings.


Accelerated Projected Gradient Method for Linear Inverse Problems with Sparsity Constraints

Abstract: Regularization of ill-posed linear inverse problems via $\ell_1$ penalization has been proposed for cases where the solution is known to be (almost) sparse. One way to obtain the minimizer of such an $\ell_1$ penalized functional is via an iterative soft-thresholding algorithm. We propose an alternative implementation to $\ell_1$-constraints, using a gradient method, with projection on $\ell_1$-balls. The corresponding algorithm uses again iterative soft-thresholding, now with a variable thresholding parameter. We also propose accelerated versions of this iterative method, using ingredients of the (linear) steepest descent method. We prove convergence in norm for one of these projected gradient methods, without and with acceleration.


A Metric on Shape Space with Explicit Geodesics

Abstract: This paper studies a specific metric on plane curves that has the property of being isometric to classical manifold (sphere, complex projective, Stiefel, Grassmann) modulo change of parametrization, each of these classical manifolds being associated to specific qualifications of the space of curves (closed-open, modulo rotation etc...) Using these isometries, we are able to explicitely describe the geodesics, first in the parametric case, then by modding out the paremetrization and considering horizontal vectors. We also compute the sectional curvature for these spaces, and show, in particular, that the space of closed curves modulo rotation and change of parameter has positive curvature. Experimental results that explicitly compute minimizing geodesics between two closed curves are finally provided


Moments from their very truncations

Abstract: It is known that positive definiteness is not enough for the multidimensional moment problem to be solved. We would like throw in to the garden of existing in this matter so far results one more, a result which takes into considerations the utmost possible truncations.


Smooth models of quiver moduli

Abstract: For any moduli space of stable representations of quivers, certain smooth varieties, compactifying projective space fibrations over the moduli space, are constructed. The boundary of this compactification is analyzed. Explicit formulas for the Betti numbers of the smooth models are derived. In the case of moduli of simple representations, explicit cell decompositions of the smooth models are constructed.


Reducibility of the polynomial representation of the degenerate double affine Hecke algebra

Abstract: In this note we determine the values of parameters c for which the polynomial representation of the degenerate double affine Hecke algebra (DAHA), i.e. the trigonometric Cherednik algebra, is reducible. Namely, we show that c is a reducibility point for the polynomial representation of the trigonometric Cherednik algebra for a root system R if and only if it is a reducibility point for the rational Cherednik algebra for the Weyl group of some root subsystem R' of R of the same rank; such subsystems for any R are given by the well known Borel-de Siebenthal algorithm. This result has been proved by Cherednik using a case-by-case method.


Cutoff Resolvent Estimates and the Semilinear Schr\"odinger Equation

Abstract: This paper shows how abstract resolvent estimates imply local smoothing for solutions to the Schr\"odinger equation. If the resolvent estimate has a loss when compared to the optimal, non-trapping estimate, there is a corresponding loss in regularity in the local smoothing estimate. As an application, we apply well-known techniques to obtain well-posedness results for the semi-linear Schr\"odinger equation.


The age grading and the Chen-Ruan cup product

Abstract: We prove that the obstruction bundle used to define the cup-product in Chen-Ruan cohomology is determined by the so-called `age grading' or `degree-shifting numbers'. Indeed, the obstruction bundle can be directly computed using the age grading. We obtain a Kunneth Theorem for Chen-Ruan cohomology as a direct consequence of an elementary property of the age grading, and explain how several other results - including associativity of the cup-product - can be proved in a similar way.


Power Loss for Inhomogeneous Poisson Processes

Abstract: In this work, based on a realization of an inhomogeneous Poisson process whose intensity function depends on a real unknown parameter, we consider a simple hypothesis against a sequence of close (contiguous) alternatives. Under certain regularity conditions we obtain the power loss of the score test with respect to the Neyman-Pearson test. The power loss measures the performance of a second order efficient test by the help of third order asymptotic properties of the problem under consideration.


Convergence properties of Donaldson's $T$-iterations on the Riemann sphere

Abstract: In a recent paper Donaldson defines three operators on a space of Hermitian metrics on a complex projective manifold: $T, T_{\nu}, T_K.$ Iterations of these operators converge to balanced metrics, and these themselves approximate constant scalar curvature metrics. In this paper we investigate the convergence properties of these iterations by examining the case of the Riemann sphere as well as higher dimensional $\mathbb{CP}^n$.


A note on p-adic q-integrals associated with q-Euler numbers

Abstract: In this we give a detailed proof of fermionic p-adic q-measures on Z_p and we will treat some interesting formulae related q-extension of Euler numbers and polynomials.


Hausdorff Dimension and Hausdorff Measure for Non-integer based Cantor-type Sets

Abstract: We consider digits-deleted sets or Cantor-type sets with $\beta$-expansions. We calculate the Hausdorff dimension $d$ of these sets and show that $d$ is continuous with respect to $\beta$. The $d$-dimentional Hausdorff measure of these sets is finite and positive.


2-Selmer Groups and the Birch-Swinnerton-Dyer Conjecture for the Congruent Number Curve

Abstract: We take an approach toward counting the number of n for which the curves E_n: y^2=x^3-n^2x have 2-Selmer groups of a given size. This question was also discussed in a pair of Invent. Math. papers by Roger Heath-Brown. We discuss the connection between computing the size of these Selmer groups and verifying cases of the Birch and Swinnerton-Dyer Conjecture. The key ingredient for the asymptotic formulae is the ``independence'' of the Legendre symbol evaluated at the prime divisors of an integer with exactly k prime factors.


A fast direct solver for network matrices

Abstract: A fast direct inversion scheme for the large sparse systems of linear equations resulting from the discretization of elliptic partial differential equations in two dimensions is given. The scheme is described for the particular case of a discretization on a uniform square grid, but can be generalized to more general geometries. For a grid containing $N$ points, the scheme requires $O(N \log^{2}N)$ arithmetic operations and $O(N \log N)$ storage to compute an approximate inverse. If only a single solve is required, then the scheme requires only $O(\sqrt{N} \log N)$ storage; the same storage is sufficient for computing the Dirichlet-to-Neumann operator as well as other boundary-to-boundary operators. The scheme is illustrated with several numerical examples. For instance, a matrix of size $10^6 \times 10^6$ is inverted to seven digits accuracy in four minutes on a 2.8GHz P4 desktop PC.


Geometric Invariant Theory via Cox Rings

Abstract: We consider actions of reductive groups on a varieties with finitely generated Cox ring, e.g., the classical case of a diagonal action on a product of projective spaces. Given such an action, we construct via combinatorial data in the Cox ring all maximal open subsets such that the quotient is quasiprojective or embeddable into a toric variety. As applications, we obtain an explicit description of the chamber structure of the linearized ample cone and several Gelfand-MacPherson type correspondences relating quotients of reductive groups to quotients of torus actions. Moreover, our approach provides information on the geometry of many of the resulting quotient spaces.


A counterexample to generalizations of the Milnor-Bloch-Kato conjecture

Abstract: We construct an example of a torus $T$ over a field $K$ for which the Galois symbol $K(K; T,T)/n K(K; T,T) \to H^2(K, T[n]\otimes T[n])$ is not injective for some $n$. Here $K(K; T,T)$ is the Milnor $K$-group attached to $T$ introduced by Somekawa. We show also that the motive $M(T\times T)$ gives a counterexample to another generalization of the Milnor-Bloch-Kato conjecture (proposed by Beilinson).


On Wahl's proof of $\mu(6)=65$

Abstract: D. Jaffe and D. Ruberman proved in 1997 that a sextic hypersurface in $\mathbb{P}^3$ has at most 65 nodes (the bound is sharp by Barth's construction). Almost at the same time, J. Wahl proposed a much shorter proof of the same result, by proving that a linear code $V\subset \F^{66}$ with weights in $\{24,32,40\}$ has dimension $\dim(V)\leq12$. He claimed that Jaffe-Ruberman's theorem follows as a corollary since the code associated to a sextic with n nodes has dimension at least $n-53$ and an incorrect result stated by Casnati and Catanese asserted that the possible cardinalities of an even set of nodes on a sextic were only 24, 32 and 40. Recently Catanese and Tonoli showed that the possible cardinalities of an even set of nodes on a sextic are exactly 24, 32, 40, 56. According to the above cardinalities, the theorem of Jaffe and Ruberman reduces to the following: Let $V\subset \F^{66}$ be a code with weights in $\{24,32,40,56\}$. Then $\dim(V)\leq12$. In this short note we give an elementary proof of this theorem using and integrating Wahl's ideas.


Multi-dimensional BSDE with Oblique Reflection and Optimal Switching

Abstract: In this paper, we study a multi-dimensional backward stochastic differential equation (BSDE) with oblique reflection, which is a BSDE reflected on the boundary of a special unbounded convex domain along an oblique direction, and which arises naturally in the study of optimal switching problem. The existence of the adapted solution is obtained by the penalization method, the monotone convergence, and the a priori estimations. The uniqueness is obtained by a verification method (the first component of any adapted solution is shown to be the vector value of a switching problem for BSDEs). As applications, we apply the above results to solve the optimal switching problem for stochastic differential equations of functional type, and we give also a probabilistic interpretation of the viscosity solution to a system of variational inequalities.


Combinatorial and geometric methods in topology

Abstract: Starting from the (apparently) elementary problem of deciding how many different topological spaces can be obtained by gluing together in pairs the faces of an octahedron, we will describe the central role played by hyperbolic geometry within three-dimensional topology. We will also point out the striking difference with the two-dimensional case, and we will review some of the results of the combinatorial and computational approach to three-manifolds developed by different mathematicians over the last several years.


Quartic equations and 2-division on elliptic curves

Abstract: Let K be a field of characteristic different from 2 and C an elliptic curve over K given by a Weierstrass equation. To divide an element of the group C by 2, one must solve a certain quartic equation. We characterise the quartics arising from this procedure and find how far the quartic determines the curve and the point. We find the quartics coming from 2-division of 2- and 3-torsion points, and generalise this correspondence to singular plane cubics. We use these results to study the question of which degree 4 maps of curves can be realised as duplication of a multisection on an elliptic surface.


On Hamiltonian stationary Lagrangian spheres in non-Einstein Kaehler surfaces

Abstract: Hamiltonian stationary Lagrangian spheres in Kaehler-Einstein surfaces are minimal. We prove that in the family of non-Einstein Kaehler surfaces given by the product $\Sigma_1\times\Sigma_2$ of two complete orientable Riemannian surfaces of different constant Gauss curvatures, there is only a (non minimal) Hamiltonian stationary Lagrangian sphere. This example is defined when the surfaces $\Sigma_1$ and $ \Sigma_2$ are spheres.


Improvements on removing non-optimal support points in D-optimum design algorithms

Abstract: We improve the inequality used in Pronzato [2003. Removing non-optimal support points in D-optimum design algorithms. Statist. Probab. Lett. 63, 223-228] to remove points from the design space during the search for a $D$-optimum design. Let $\xi$ be any design on a compact space $\mathcal{X} \subset \mathbb{R}^m$ with a nonsingular information matrix, and let $m+\epsilon$ be the maximum of the variance function $d(\xi,\mathbf{x})$ over all $\mathbf{x} \in \mathcal{X}$. We prove that any support point $\mathbf{x}_{*}$ of a $D$-optimum design on $\mathcal{X}$ must satisfy the inequality $d(\xi,\mathbf{x}_{*}) \geq m(1+\epsilon/2-\sqrt{\epsilon(4+\epsilon-4/m)}/2)$. We show that this new lower bound on $d(\xi,\mathbf{x}_{*})$ is, in a sense, the best possible, and how it can be used to accelerate algorithms for $D$-optimum design.


Tightness of voter model interfaces

Abstract: Consider a long-range, one-dimensional voter model started with all zeroes on the negative integers and all ones on the positive integers. If the process obtained by identifying states that are translations of each other is positively recurrent, then it is said that the voter model exhibits interface tightness. In 1995, Cox and Durrett proved that one-dimensional voter models exhibit interface tightness if their infection rates have a finite third moment. Recently, Belhaouari, Mountford, and Valle have improved this by showing that a finite second moment suffices. The present paper gives a new short proof of this fact. We also prove interface tightness for a long range swapping voter model, which has a mixture of long range voter model and exclusion process dynamics.


Relaxation Enhancement by Time-Periodic Flows

Abstract: We study enhancement of diffusive mixing by fast incompressible time-periodic flows. The class of relaxation-enhancing flows that are especially efficient in speeding up mixing has been introduced in [2]. The relaxation-enhancing property of a flow has been shown to be intimately related to the properties of the dynamical system it generates. In particular, time-independent flows $u$ such that the operator $u \cdot \nabla$ has sufficiently smooth eigenfunctions are not relaxation-enhancing. Here we extend results of [2] to time-periodic flows $u(x,t)$ and in particular show that there exist flows such that for each fixed time the flow is Hamiltonian, but the resulting time-dependent flow is relaxation-enhancing. Thus we confirm the physical intuition that time dependence of a flow may aid mixing. We also provide an extension of our results to the case of a nonlinear diffusion model. The proofs are based on a general criterion for the decay of a semigroup generated by an operator of the form $\Gamma+iAL(t)$ with a negative unbounded self-adjoint operator $\Gamma$, a time-periodic self-adjoint operator-valued function $L(t)$, and a parameter $A>>1$.


Some Two Color, Four Variable Rado Numbers

Abstract: There exists a minimum integer $N$ such that any 2-coloring of $\{1,2,...,N\}$ admits a monochromatic solution to $x+y+kz =\ell w$ for $k,\ell \in \mathbb{Z}^+$, where $N$ depends on $k$ and $\ell$. We determine $N$ when $\ell-k \in \{0,1,2,3,4,5\}$, for all $k,\ell$ for which ${1/2}((\ell-k)^2-2)(\ell-k+1)\leq k \leq \ell-4$, as well as for arbitrary $k$ when $\ell=2$.


Bounds on Van der Waerden Numbers and Some Related Functions

Abstract: For positive integers $s$ and $k_1, k_2, ..., k_s$, let $w(k_1,k_2,...,k_s)$ be the minimum integer $n$ such that any $s$-coloring $\{1,2,...,n\} \to \{1,2,...,s\}$ admits a $k_i$-term arithmetic progression of color $i$ for some $i$, $1 \leq i \leq s$. In the case when $k_1=k_2=...=k_s=k$ we simply write $w(k;s)$. That such a minimum integer exists follows from van der Waerden's theorem on arithmetic progressions. In the present paper we give a lower bound for $w(k,m)$ for each fixed $m$. We include a table with values of $w(k,3)$ which match this lower bound closely for $5 \leq k \leq 16$. We also give an upper bound for $w(k,4)$, an upper bound for $w(4;s)$, and a lower bound for $w(k;s)$ for an arbitrary fixed $k$. We discuss a number of other functions that are closely related to the van der Waerden function.


A presentation for Hilden's subgroup of the braid group

Abstract: Consider the unit ball, B = D x [0,1], containing n unknotted arcs a_1, ... a_n such that the boundary of each a_i lies in D x {0}. We give a finite presentation for the mapping class group of B fixing the arcs {a_1, ..., a_n} setwise and fixing D x {1} pointwise. This presentation is calculated using the action of this group on a simply-connected complex.