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Let $mathcal {M}(Sigma , mathcal {P})$ be the mapping class group of a punctured oriented surface $(Sigma ,mathcal {P})$ (where $mathcal {P}$ may be empty), and let $mathcal {T}_p(Sigma ,mathcal {P})$ be the kernel of the action of $mathcal {M} (Sigma , mathcal {P})$ on $H_1(Sigma setminus mathcal {P}, mathbb {F}_p)$. We prove that $mathcal {T}_p( Sigma ,mathcal {P})$ is residually $p$. In particular, this shows that $mathcal {M} (Sigma ,mathcal {P})$ is virtually residually $p$. For a group $G$ we denote by $mathcal {I}_p(G)$ the kernel of the natural action of $operatorname {Out}(G)$ on $H_1(G,mathbb {F}_p)$. In order to achieve our theorem, we prove that, under certain conditions ($G$ is conjugacy $p$-separable and has Property A), the group $mathcal {I}_p(G)$ is residually $p$. The fact that free groups and surface groups have Property A is due to Grossman. The fact that free groups are conjugacy $p$-separable is due to Lyndon and Schupp. The fact that surface groups are conjugacy $p$-separable is, from a technical point of view, the main result of the paper. [ABSTRACT FROM AUTHOR]

Let $G$ be a finite group, $S in mathit {Syl}_p(G)$, and $mathcal S$ be the set subgroups containing $S$. For $M in mathcal S$ and $V = Omega _1Z(O_p(M))$, the paper discusses the action of $M$ on $V$. Apart from other results, it is shown that for groups of parabolic characteristic $p$ either $S$ is contained in a unique maximal $p$-local subgroup, or there exists a maximal $p$-local subgroup in $M in mathcal S$ such that $V$ is a nearly quadratic 2F-module for $M$. [ABSTRACT FROM AUTHOR]

Let $S = K[x_1, ldots , x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $deg x_i = 1$. Let $Delta $ be a simplicial complex on $[n] = { 1, ldots , n }$ and $I_Delta subset S$ its Stanley--Reisner ideal. We write $Delta ^e$ for the exterior algebraic shifted complex of $Delta $ and $Delta ^c$ for a combinatorial shifted complex of $Delta $. Let $beta _{ii+j}(I_{Delta }) = dim _K mathrm {Tor}_i(K, I_Delta )_{i+j}$ denote the graded Betti numbers of $I_Delta $. In the present paper it will be proved that (i) $beta _{ii+j}(I_{Delta ^e}) leq beta _{ii+j}(I_{Delta ^c})$ for all $i$ and $j$, where the base field is infinite, and (ii) $beta _{ii+j}(I_{Delta }) leq beta _{ii+j}(I_{Delta ^c})$ for all $i$ and $j$, where the base field is arbitrary. Thus in particular one has $beta _{ii+j}(I_Delta ) leq beta _{ii+j}(I_{Delta ^{lex}})$ for all $i$ and $j$, where $Delta ^{operatorname {lex}}$ is the unique lexsegment simplicial complex with the same $f$-vector as $Delta $ and where the base field is arbitrary. [ABSTRACT FROM AUTHOR]

In this paper we settle a conjecture of Ball that uniform quasiconvexity and uniform positivity of the second variation are sufficient for a $C^{1}$ extremal to be a strong local minimizer. Our result holds for a class of variational functionals with a power law behavior at infinity. The proof is based on the decomposition of an arbitrary variation of the dependent variable into its purely strong and weak parts. We show that these two parts act independently on the functional. The action of the weak part can be described in terms of the second variation, whose uniform positivity prevents the weak part from decreasing the functional. The strong part ``localizes'', i.e. its action can be represented as a superposition of ``Weierstrass needles'', which cannot decrease the functional either, due to the uniform quasiconvexity conditions. [ABSTRACT FROM AUTHOR]

We prove that each invariant measure in a non-uniformly hyperbolic system can be approximated by atomic measures on hyperbolic periodic orbits. This contributes to our main result that the mean angle (Definition 1.10), independence number (Definition 1.6) and Oseledec splitting for an ergodic hyperbolic measure with simple spectrum can be approximated by those for atomic measures on hyperbolic periodic orbits, respectively. Combining this result with the approximation property of Lyapunov exponents by Wang and Sun, 2005 (Theorem 1.9), we strengthen Katok's closing lemma (1980) by presenting more extensive information not only about the state system but also its linearization. par In the present paper, we also study an ergodic theorem and a variational principle for mean angle, independence number and Liao's style number (Definition 1.3) which are bases for discussing the approximation properties in the main result. [ABSTRACT FROM AUTHOR]

In this paper, we investigate Murre's conjectures on the structure of rational Chow groups and exhibit explicit Chow--Künneth projectors for some examples. More precisely, the examples we study are the varieties which have a nef tangent bundle. For surfaces and threefolds which have a nef tangent bundle, explicit Chow--Künneth projectors are obtained which satisfy Murre's conjectures, and the motivic Hard Lefschetz theorem is verified. [ABSTRACT FROM AUTHOR]

This paper studies quadratic and Koszul duality for modules over positively graded categories. Typical examples are modules over a path algebra, which is graded by the path length, of a not necessarily finite quiver with relations. We present a very general definition of quadratic and Koszul duality functors backed up by explicit examples. This generalizes the work of Beilinson, Ginzburg, and Soergel, 1996, in two substantial ways: We work in the setup of graded categories, i.e. we allow infinitely many idempotents and also define a ``Koszul'' duality {it functor} for not necessarily Koszul categories. As an illustration of the techniques we reprove the Koszul duality (Ryom-Hansen, 2004) of translation and Zuckerman functors for the classical category $mathcal {O}$ in a quite elementary and explicit way. From this we deduce a conjecture of Bernstein, Frenkel, and Khovanov, 1999. As applications we propose a definition of a ``Koszul'' dual category for integral blocks of Harish-Chandra bimodules and for blocks outside the critical hyperplanes for the Kac-Moody category $mathcal {O}$. [ABSTRACT FROM AUTHOR]

Let $f$ be a germ of a holomorphic self-map of $mathbb {C}^2$ at the origin $O$ tangent to the identity, and with $O$ as a nondicritical isolated fixed point. A parabolic curve for $f$ is a holomorphic $f$-invariant curve, with $O$ on the boundary, attracted by $O$ under the action of $f$. It has been shown by M. Abate (2001) that if the characteristic direction $[v]in mathbb {P}(T_Omathbb {C}^2)$ has residual index not belonging to $mathbb {Q}^$, then there exist parabolic curves for $f$ tangent to $[v]$. In this paper we prove, using a different method, that the conclusion still holds just assuming that the residual index is not vanishing (at least when $f$ is regular along $[v]$). [ABSTRACT FROM AUTHOR]

In this paper we let $G$ be a simple algebraic group and $r$ be a natural number, and consider the codimension in $G$ of the variety of elements $g\in G$ satisfying $g^r=1$. We shall obtain a lower bound for this codimension which is independent of characteristic, and show that it is attained if $G$ is of adjoint type. [ABSTRACT FROM AUTHOR]

In this paper we show that the $\varrho$-th order variation operator, for both the Fejér and Poisson kernels, are bounded from $L^p$ to $L^p$, $12$. Counterexamples are given if $\varrho =2$. [ABSTRACT FROM AUTHOR]

This paper investigates varieties of tori and Cartan subalgebras of a finite-dimensional restricted Lie algebra $(\mathfrak{g},[p])$, defined over an algebraically closed field $k$ of positive characteristic $p$. We begin by showing that schemes of tori may be used as a tool to retrieve results by A. Premet on regular Cartan subalgebras. Moreover, they give rise to principal fibre bundles, whose structure groups coincide with the Weyl groups in case $\mathfrak{g}= \operatorname{Lie}(\mathcal{G})$ is the Lie algebra of a smooth group $\mathcal{G}$. For solvable Lie algebras, varieties of tori are full affine spaces, while simple Lie algebras of classical or Cartan type cannot have varieties of this type. In the final sections the quasi-projective variety of Cartan subalgebras of minimal dimension ${\rm rk}(\mathfrak{g})$ is shown to be irreducible of dimension $\dim_k\mathfrak{g}-{\rm rk}(\mathfrak{g})$, with Premet's regular Cartan subalgebras belonging to the regular locus. [ABSTRACT FROM AUTHOR]

Let $A=\oplus_{g\in G}A_g$ be a $G$-graded associative algebra over a field of characteristic zero. In this paper we develop a conjecture that relates the exponent of the growth of polynomial identities of the identity component $A_e$ to that of the whole of $A$, in the case where the support of the grading is finite. We prove the conjecture in several natural cases, one of them being the case where $A$ is finite dimensional and $A_e$ has polynomial growth. [ABSTRACT FROM AUTHOR]

A finite or infinite matrix $A$ is \textit{image partition regular\/} provided that whenever ${\mathbb N}$ is finitely colored, there must be some $\vec {x}$ with entries from ${\mathbb N}$ such that all entries of $A\vec {x}$ are in the same color class. In contrast to the finite case, infinite image partition regular matrices seem very hard to analyze: they do not enjoy the closure and consistency properties of the finite case, and it is difficult to construct new ones from old. In this paper we introduce the stronger notion of \textit{central image partition regularity\/}, meaning that $A$ must have images in every central subset of $\subben$. We describe some classes of centrally image partition regular matrices and investigate the extent to which they are better behaved than ordinary image partition regular matrices. It turns out that the centrally image partition regular matrices \textit{are\/} closed under some natural operations, and this allows us to give new examples of image partition regular matrices. In particular, we are able to solve a vexing open problem by showing that whenever ${\mathbb N}$ is finitely colored, there must exist injective sequences $\langle x_n\rangle_{n=0}^\infty$ and $\langle z_n\rangle_{n=0}^\infty$ in ${\mathbb N}$ with all sums of the forms $x_n+x_m$ and $z_n+2z_m$ with $n

Let $\mu$ be a Radon measure on $\mathbb{R}^d$, which may be nondoubling. The only condition that $\mu$ must satisfy is the size condition $\mu(B(x,r))\leq Cr^n$, for some fixed $0

In this paper, we take a new look at the representation theory of Lie triple systems. We consider both ordinary Lie triple systems and restricted Lie triple systems in the sense of [14]. In a final section, we begin a study of the cohomology of Lie triple systems. [ABSTRACT FROM AUTHOR]

The positive mass theorem is one of the fundamental results in geometric analysis and mathematical general relativity. It states that, assuming the dominant energy condition (i.e., nonnegative scalar curvature) then the total mass of an asymptotically flat spacetime is nonnegative. The Penrose inequality provides a lower bound on mass by the area of the black hole. Lu and Miao proved a quasi-local Penrose inequality for the quasi-local energy using the Schwarzschild manifold as a comparison manifold. In this article, we prove a quasi-local Penrose inequality for the quasi-local energy using any spherically symmetric static spacetime as the comparison manifold. Moreover, in the equality case, the manifold must be isometric to the comparison manifold, which has to be the Schwarzschild manifold. [ABSTRACT FROM AUTHOR]

We analyze the complexity of the bi-embeddability relations for countable torsion-free abelian groups and for countable torsion abelian groups. [ABSTRACT FROM AUTHOR]

We consider the non-local operator of variable order as follows: Lf(x) = ... (f(x + z) -- f(x) -- (∇ f(x), z) 1{|z|≤1})n(x, z)/... dz, where α(x) ∈ [α0, α2] for any x ∈ Rd and some constants 0 < α0 ≤ α2 < 2 and n(x, z) is a positive Borel measurable function bounded from above and below. Under mild continuity conditions on α(x) and n(x, z), we establish the H¨older regularity for the associated semigroups. The proof is based on a new construction of the coupling by reflection for non-local operator L, and the results successfully apply to both stable-like processes in the sense of Bass and time-change of symmetric stable processes. [ABSTRACT FROM AUTHOR]

Let Sg (g ≥ 2) be a closed surface of genus g. Let K be any real number field, and let A be any quaternion algebra over K such that A ⊗K ℝ = M2(ℝ.). We show that there exists a hyperbolic structure on Sg such that K and A arise as its invariant trace field and invariant quaternion algebra. [ABSTRACT FROM AUTHOR]

We construct a well-behaved compactification of the space of finite covers of a stacky curve using admissible cover degenerations. Using our construction, we compactify the space of tetragonal curves on Hirzebruch surfaces. As an application, we explicitly describe the boundary divisors of the closure in M6 of the locus of smooth plane quintic curves. [ABSTRACT FROM AUTHOR]

We consider sharp inequalities for Bergman spaces of the unit disc, establishing analogues of the inequality in Carleman's proof of the isoperimetric inequality and of Weissler's inequality for dilations. By contractivity and a standard tensorization procedure, the unit disc inequalities yield corresponding inequalities for the Bergman spaces of Dirichlet series. We use these results to study weighted multiplicative Hankel forms associated with the Bergman spaces of Dirichlet series, reproducing most of the known results on multiplicative Hankel forms associated with the Hardy spaces of Dirichlet series. In addition, we find a direct relationship between the two types of forms which does not exist in lower dimensions. Finally, we produce some counterexamples concerning Carleson measures on the infinite polydisc. [ABSTRACT FROM AUTHOR]

We strengthen the volume inequalities for Lp zonoids of even isotropic measures and for their duals, which are originally due to Ball, Barthe, and Lutwak, Yang, and Zhang. The special case p = ∞ yields a stability version of the reverse isoperimetric inequality for centrally symmetric convex bodies. Adding to known inequalities and stability results for the reverse isoperimetric inequality of arbitrary convex bodies, we state a conjecture on volume inequalities for Lp zonoids of general centered (non-symmetric) isotropic measures. We achieve our main results by strengthening Barthe's measure transportation proofs of the rank one case of the geometric Brascamp--Lieb and reverse Brascamp--Lieb inequalities by estimating the derivatives of the transportation maps for a special class of probability density functions. Based on our argument, we phrase a conjecture about a possible stability version of the Brascamp--Lieb and reverse Brascamp--Lieb inequalities. We also establish some geometric properties of the distribution of general isotropic measures that are essential to our argument. In particular, we prove a measure theoretic analog of the Dvoretzky--Rogers lemma. [ABSTRACT FROM AUTHOR]

For the Dunkl transforms associated with the weight functions Hk²(x) = ∏j=1d ..., k1, ⋯, kd ≥ 0 on ℝd, it is proved that if p ≥ 2 + 1/λk and λk := d-1/2 + ∑j=1dkj, the maximal Bochner-Riesz operator B*δ(hk²; f) order δ > 0 is bounded on the space Lp(ℝd; hk² dx) if and only if δ > δk(p) := max{(2λk +1)(1/2 -- 1/p ) -- 1/2, 0}. This extends a well known result of M. Christ for the classical Fourier transforms (Proc. Amer. Math. Soc. 95 (1985), 16-20). The proof relies on a new local restriction theorem for the Dunkl transforms, which is stronger than the corresponding global restriction theorem, but significantly more difficult to prove. [ABSTRACT FROM AUTHOR]

The automorphism group of a one dimensional shift space over a finite alphabet exhibits different types of behavior: for a large class with positive entropy, it contains a rich collection of subgroups, while for many shifts of zero entropy, there are strong constraints on the automorphism group. We view this from a different perspective, considering a single automorphism (and sometimes endomorphism) and studying the naturally associated twodimensional shift system. In particular, we describe the relation between nonexpansive subspaces in this two-dimensional system and dynamical properties of an automorphism of the shift. [ABSTRACT FROM AUTHOR]

Assume that D is a Krull-Schmidt, Hom-finite triangulated category with a Serre functor and a cluster-tilting object T. We introduce the notion of relative cluster tilting objects, and T[1]-cluster tilting objects in D, which are a generalization of cluster-tilting objects. When D is 2-Calabi-Yau, the relative cluster tilting objects are cluster-tilting. Let Λ = EndDop(T) be the opposite algebra of the endomorphism algebra of T. We show that there exists a bijection between T[1]-cluster tilting objects in D and support τ-tilting Λ- modules, which generalizes a result of Adachi--Iyama--Reiten [τ-tilting theory, Compos. Math. 150 (2014), no. 3, 415-452]. We develop a basic theory on T[1]-cluster tilting objects. In particular, we introduce a partial order on the set of T[1]-cluster tilting objects and mutation of T[1]-cluster tilting objects, which can be regarded as a generalization of "cluster-tilting mutation". As an application, we give a partial answer to a question posed in Adachi--Iyama--Reiten, loc. cit. [ABSTRACT FROM AUTHOR]

We prove a variant of Oesterl´e's conjecture describing p-adic periods of the modular curve X0(p), with an additional Γ(2)-structure (and also Γ(3) ∩ Γ0(p) if p ≡ 1 (mod 3)). We use de Shalit's techniques and p-adic uniformization of curves with semi-stable reduction. [ABSTRACT FROM AUTHOR]

Let K be a number field with algebraic closure ..., and let S be a finite set of places of K containing all the infinite ones. Let Γ0 be a finitely generated subgroup of 픾m..., and let Γ ⊂ 픾m... be the division group attached to Γ0. Here is an illustration of what we will prove in this article. Fix a proper closed subinterval I of [0, ∞) and a nonzero effective divisor D on 픾m which is not the translate of any torsion divisor on the algebraic torus 픾m by any point of Γ with height belonging to I. Then we prove a statement which easily implies that the set of "integral division points on 픾m with height near I", i.e., the set of points of Γ with (standard absolute logarithmic Weil) height in J which are S-integral on 픾m relative to D, is finite for some fixed subinterval J of [0, ∞) properly containing I. We propose a conjecture on the nondensity of integral division points on semi-abelian varieties with prescribed height values, which generalizes some previously known conjectures as well as this finiteness result for 픾m. Finally, we also propose an analogous version for a dynamical system on P¹. [ABSTRACT FROM AUTHOR]

We construct certain subgroups of hyperbolic triangle groups which we call "congruence" subgroups. These groups include the classical congruence subgroups of SL2(ℤ.), Hecke triangle groups, and 19 families of arithmetic triangle groups associated to Shimura curves. We determine the field of moduli of the curves associated to these groups and thereby realize the groups PSL2(Fq) and PGL2(픽q) regularly as Galois groups. [ABSTRACT FROM AUTHOR]

Let E be an elliptic curve over ℚ, such that End... (E) = ℤ and admitting a non-trivial cyclic ℚ,-isogeny. We prove that, for p > 37, the residual mod p Galois representation ...E,p : Gℚ → GL2( 픽p) is surjective. [ABSTRACT FROM AUTHOR]

All SL(n) covariant vector valuations on convex polytopes in Rn are completely classified without any continuity assumptions. The moment vector turns out to be the only such valuation if n = 3, while two new functionals show up in dimension two. [ABSTRACT FROM AUTHOR]

In this work we study the following three-space problem for operator spaces: if X is an operator space with base space isomorphic to a Hilbert space and X contains a completely isomorphic copy of the operator Hilbert space OH with the respective quotient also completely isomorphic to OH, must X be completely isomorphic to OH? This problem leads us to the study of short exact sequences of operator spaces, more specifically those induced by complex interpolation and their splitting. We show that the answer to the three-space problem is negative, giving two different solutions. [ABSTRACT FROM AUTHOR]

Subspace designs are a (large) collection of high-dimensional subspaces {Hi} of Fmq such that for any low-dimensional subspace W, only a small number of subspaces from the collection have non-trivial intersection with W; more precisely, the sum of dimensions of W nHi is at most some parameter L. The notion was put forth by Guruswami and Xing (STOC'13) with applications to list decoding variants of Reed-Solomon and algebraic-geometric codes and later also used for explicit rank-metric codes with optimal list decoding radius. Guruswami and Kopparty (FOCS'13, Combinatorica '16) gave an explicit construction of subspace designs with near-optimal parameters. This construction was based on polynomials and has close connections to folded Reed-Solomon codes and required large field size (specifically q ≥ m). Forbes and Guruswami (RANDOM'15) used this construction to give explicit constant degree "dimension expanders" over large fields and noted that subspace designs are a powerful tool in linear-algebraic pseudorandomness. Here, we construct subspace designs over any field, at the expense of a modest worsening of the bound L on total intersection dimension. Our approach is based on a (non-trivial) extension of the polynomial-based construction to algebraic function fields and instantiating the approach with cyclotomic function fields. Plugging in our new subspace designs in the construction of Forbes and Guruswami yields dimension expanders over Fn for any field F, with logarithmic degree and expansion guarantee for subspaces of dimension O(n/(log log n)). [ABSTRACT FROM AUTHOR]

It is known, since the seminal work [T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998)], that the solution map associated to a controlled differential equation is locally Lipschitz continuous in q-variation, resp., 1/q-Hölder-type metrics on the space of rough paths, for any regularity 1/q ∈ (0, 1]. We extend this to a new class of Besov-Nikolskii-type metrics, with arbitrary regularity 1/q ∈ (0, 1] and integrability p ∈ [q,8], where the case p ∈ {q,8} corresponds to the known cases. Interestingly, the result is obtained as a consequence of known q-variation rough path estimates. [ABSTRACT FROM AUTHOR]

For a curve C of genus 6 or 8 and a torsion bundle η of order ℓ we study the vanishing of the space of global sections of the twist EC ⊗ η of the rank 2 Mukai bundle EC of C. The bundle EC was used in a well-known construction of Mukai which exhibits general canonical curves of low genus as sections of Grassmannians in the Plücker embedding. Globalizing the vanishing condition, we obtain divisors on the moduli spaces R6,ℓ and R8,ℓ of pairs [C, η]. First we characterize these divisors by different conditions on linear series on the level curves, afterwards we calculate the divisor classes. As an application, we are able to prove that R8,3 is of general type. [ABSTRACT FROM AUTHOR]

We prove there is no sentence in the monadic second-order language MS0 that characterises when a matroid is representable over at least one field, and no sentence that characterises when a matroid is K-representable, for any infinite field K. By way of contrast, because Rota's Conjecture is true, there is a sentence that characterises F-representable matroids, for any finite field F. [ABSTRACT FROM AUTHOR]

The irreducible modules for the parafermion vertex operator algebra associated to any finite dimensional Lie algebra g and any positive integer k are classified, the quantum dimensions are computed and the fusion rules are determined. [ABSTRACT FROM AUTHOR]

We give a simplified proof of J. A. Wolf's classification of finite groups that can act freely and isometrically on a round sphere of some dimension. We slightly improve the classification by removing some nonobvious redundancy. The groups are the same as the Frobenius complements of finite group theory. [ABSTRACT FROM AUTHOR]

New Orlicz Brunn-Minkowski inequalities are established for rigid motion compatible Minkowski valuations of arbitrary degree. These extend classical log-concavity properties of intrinsic volumes and generalize seminal results of Lutwak and others. Two different approaches which refine previously employed techniques are explored. It is shown that both lead to the same class of Minkowski valuations for which these inequalities hold. An appendix by Semyon Alesker contains the proof of a new description of generalized translation invariant valuations. [ABSTRACT FROM AUTHOR]

We calculate the Krull-Gabriel dimension of the category of modules over any domestic string algebra, in particular showing that it is finite, thus confirming a conjecture of Schröer. We also compute the Cantor-Bendixson rank of each point of its Ziegler spectrum and determine the topology on this space. [ABSTRACT FROM AUTHOR]

Let (S¹, S²) be the two-sided random walks in Zd (d = 2, 3) conditionedso that S¹[0,∞)∩S²[1,∞) = ∅, which was constructed by the authorin 2012. We prove that the number of global cut times up to n grows liken38for d = 2. In particular, we show that each Sihas infinitely many globalcut times with probability one. Using this property, we prove that the simplerandom walk on S¹[0,∞) ∪ S²[0,∞) is subdiffusive for d = 2. We show thesame result for d = 3. [ABSTRACT FROM AUTHOR]

It is shown that there exists a compact set X in CN (N ≥ 2) such that X̂\ X is nonempty and the uniform algebra P(X) has a dense set of invertible elements, a large Gleason part, and an abundance of nonzero bounded point derivations. The existence of a Swiss cheese X such that R(X) has a Gleason part of full planar measure and a nonzero bounded point derivation at almost every point is established. An analogous result in CN is presented. The analogue for rational hulls of a result of Duval and Levenberg on polynomial hulls containing no analytic discs is established. The results presented address questions raised by Dales and Feinstein. [ABSTRACT FROM AUTHOR]

We show that for many pairs of infinite cardinals κ > μ+ > μ, (κ+, κ) ↠ (μ+, μ) is consistent relative to the consistency of a supercompact cardinal. We also show that it is consistent, relative to a huge cardinal, that (κ+, κ) ↠ (μ+, μ) for every successor cardinal κ and every μ < κ, answering a question of Foreman. [ABSTRACT FROM AUTHOR]

We study the ideals of the closure of the polynomial multipliers on the Drury-Arveson space. Structural results are obtained by investigating the relation between an ideal and its weak-* closure, much in the spirit of the corresponding classical facts for the disc algebra. Zero sets for multipliers are also considered and are deeply intertwined with the structure of ideals. Our approach is primarily based on duality arguments. [ABSTRACT FROM AUTHOR]

We introduce a family of piecewise isometries. This family is similar to the ones studied by Hooper and Schwartz. We prove that a renormalization scheme exists inside this family and compute the Hausdorff dimension of the discontinuity set. The methods use some cocycles and a continued fraction algorithm. [ABSTRACT FROM AUTHOR]

We show that various analogs of Hindman's theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets: Theorem 1. There exists a colouring c : ℝ → ..., such that for every X ⊆ ℝ with |X| = |ℝ|, and every colour γ ∈ ..., there are two distinct elements x0, x1 of X for which c(x0 + x1) = γ. This forms a simultaneous generalization of a theorem of Hindman, Leader and Strauss and a theorem of Galvin and Shelah. Theorem 2. For every abelian group G, there exists a colouring c : G → Q such that for every uncountable X ⊆ G and every colour γ, for some large enough integer n, there are pairwise distinct elements x0, ..., xn) of X such that c(x0 + ... + xn) = γ. In addition, it is consistent that the preceding statement remains valid even after enlarging the set of colours from ... to &#8477. Theorem 3. Let ...k assert that for every abelian group G of cardinality k, there exists a colouring c : G → G such that for every positive integer n, every X0, ..., Xn ∈ [G]k, and every γ ∈ G, there are x0 ∈ X0, ..., xn, ∈ Xn, such that c(x0 + ... + xn)= γ. Then ...k holds for unboundedly many uncountable cardinals k, and it is consistent that ...k holds for all regular uncountable cardinals k. [ABSTRACT FROM AUTHOR]

The dyadic problem ...n + λ2nun -λβnun-1² + λβ(n+1) u nun+1 = 0 with "smooth" initial data is considered. The uniqueness of the Leray-Hopf solution is proved. [ABSTRACT FROM AUTHOR]

We show that small normal subsets A of finite simple groups grow very rapidly; namely,|A²| ≥ |A|2-ε, where ε > 0 is arbitrarily small. Extensions, consequences, and a rapid growth result for simple algebraic groups are also given. [ABSTRACT FROM AUTHOR]

We construct a computable sequence of computable reals (Xi) such that any real that can compute a subsequence that is maximal with respect to the finite intersection property can also compute a Cohen 1-generic. This is extended to establish the same result with 2IP in place of FIP. This is the first example of a classical theorem of mathematics that has been found to be equivalent, both proof theoretically and in terms of its effective content, to computing a 1-generic. [ABSTRACT FROM AUTHOR]

Using methods of descriptive theory it is shown that the classification problem for wild knots is strictly harder than that for countable structures. [ABSTRACT FROM AUTHOR]

We prove that λg → e-t|g|r λg defines a multiplier on the von Neuman algebra of hyperbolic groups with a complete bound = r, for any 0 < t < ∞, 1 < r < ∞. In the proof we observe that a construction of Ozawa allows us to characterize the radial multipliers that are bounded on every hyperbolic graph, partially generalizing results of Haagerup-Steenstrup-Szwarc and Wysocza'nski. Our argument is also based on the work of Peller. [ABSTRACT FROM AUTHOR]