ALBERT

Description: We study the homological finiteness properties ${\rm FP}_m$ of wreath products $\Gamma = H \wr _X G$ . We show that, when $H$ has infinite abelianization, $\Gamma$ is of type ${\rm FP}_m$ if and only if both $G$ and $H$ have type ${\rm FP}_m$ and $G$ acts (diagonally) on $X^i$ with stabilizers of type ${\rm FP}_{m-i}$ and with finitely many orbits for all $1 \leq i \leq m$ . If furthermore $H$ is torsion-free, we give a criterion for $\Gamma$ to be Bredon- ${\rm FP}_m$ with respect to the class of finite subgroups of $\Gamma$ . Finally, when $H$ has infinite abelianization and $\chi : \Gamma \to \mathbb {R}$ is a non-zero homomorphism with $\chi (H) = 0$ , we classify when $[\chi ]$ belongs to the Bieri–Neumann–Strebel–Renz invariant $\Sigma ^m(\Gamma , \mathbb {Z})$ .

Description: We investigate the logarithmic bundles associated to arrangements of hypersurfaces with a fixed degree in a smooth projective variety. We then specialize to the case when the variety is a quadric hypersurface and a multiprojective space to prove a Torelli-type theorem in some cases.

Description: We study the higher differentiability of the very weak solution of the Cauchy–Dirichlet problem associated to the linear parabolic system with Hölder continuous coefficients and L 1 right-hand side of the form $$\begin{align*} \dfrac{\partial u}{\partial t} -{\rm div}(A(x,t)Du) & = \mu \quad \mbox{in } \Omega \,{\times}\,]0, T[,\\ u & = 0 \quad \mbox{on } \partial \Omega \,{\times}\,]0, T[,\\ u & = 0 \quad \mbox{on } \Omega \times \{ 0\}. \end{align*}$$

Description: In this paper, we introduce the generalized multi-parameter martingale $BMO$ spaces. The atomic decomposition of the multi-parameter martingale Hardy–Lorentz space $H_{p,q}^s$ is given. With the help of this, the dual space of $H_{p,q}^s$ is characterized as the generalized $BMO$ space. Finally, as an application, the John–Nirenberg inequality is generalized for multi-parameters.

Description: To a set $\mathcal {B} $ of 4-subsets of a set $\Omega $ of size $n,$ we introduce an invariant called the ‘hole stabilizer’ which generalizes a construction of Conway, Elkies and Martin of the Mathieu group $M_{12}$ based on Lloyd's ‘15-puzzle’. It is shown that hole stabilizers may be regarded as objects inside an objective partial group (in the sense of Chermak). We classify pairs $(\Omega ,\mathcal {B} )$ with a trivial hole stabilizer, and determine all hole stabilizers associated to 2- $(n,4,\lambda )$ designs with $\lambda \leqslant 2$ .

Description: The Erdös–Kac theorem is a celebrated result in number theory which says that the number of distinct prime factors of a uniformly chosen random integer satisfies a central limit theorem. In this paper, we establish the large deviations and moderate deviations for this problem in a very general setting for a wide class of additive functions.

Description: We study the metrizability of weak topologies when restricted to the unit sphere of some equivalent norm on a Banach space, and its relationships with other geometrical properties of norms. In case of dual Banach space $X^{\ast },$ we prove that there exists a dual norm such that its unit sphere is weak $^{\ast }$ metrizable if and only if $(\mathscr {B}_{X^{\ast }},w^{\ast })$ is a descriptive compact, which provides a complete characterization.

Description: We abstract Morimoto's construction of complex structures on product manifolds to pairs of certain generalized $F$ -structures on manifolds that are not necessarily global products. Using our Abstract Morimoto Theorem, we give a characterization of invariant generalized complex structures on products in which one factor is a Lie group. As a second application, we extend a theorem of Blair, Ludden and Yano on Hermitian bicontact manifolds to the non-commutative case.

Description: Let $\mathcal A$ be an elliptic tensor. A function $\mathbf {v}\in L^1(I;{\rm LD}_{{\rm div}\,}(B))$ is a solution to the non-stationary $\mathcal A $ -Stokes problem iff (0.1) \begin{equation} \int_Q \mathbf{v}\cdot\partial_t\boldsymbol{\varphi}\,{\rm d}x\,{\rm d}t-\int_Q \mathcal A(\varepsilon(\mathbf{v}),\varepsilon(\boldsymbol{\varphi}))\,{\rm d}x=0\quad\forall\boldsymbol{\varphi}\in C^{\infty}_{0,{\rm div}\,}(Q), \end{equation} where $Q:=I\times B$ , $B\subset \mathbb {R}^d$ bounded. If the l.h.s. of ( 0.1 ) is not zero but small, we talk about almost solutions. We present an approximation result in the fashion of the $\mathcal A$ -caloric approximation for the non-stationary $\mathcal A $ -Stokes problem. Precisely, we show that every almost solution $\mathbf {v}\in L^p(I;W^{1,p}_{{\rm div}\,}(B))$ , $1 〈 p 〈 \infty $ , to ( 0.1 ) can be approximated by a solution to ( 0.1 ) in the $L^s(I;W^{1,s}(B))$ -sense for all $s 〈 p$ . So, we extend the stationary $\mathcal A$ -Stokes approximation from [D. Breit, L. Diening and M. Fuchs, Solenoidal Lipschitz truncation and applications in fluid mechanics, J. Differential Equations 253 (2012), 1910–1942] to parabolic problems.

Description: The isogenous decomposition of the Jacobian variety of classical Fermat curve of prime degree $p \geq 5$ has been obtained by Aoki using techniques of number theory, by Barraza and Rojas in terms of decompositions of the algebra of groups, and by Hidalgo and Rodríguez using Kani–Rosen results. In the last, it was seen that all factors in the isogenous decomposition are Jacobian varieties of certain cyclic $p$ -gonal curves. The highest Abelian branched covers of an orbifold of genus 0 with exactly $n+1$ branch points, each one of order $p$ , are provided by the so-called generalized Fermat curves of type $(p,n)$ ; these being a suitable fiber product of $n-1$ Fermat curves of degree $p$ . In this paper, we provide a decomposition, up to isogeny, of the Jacobian variety of a generalized Fermat curve $S$ of type $(p,n)$ as a product of Jacobian varieties of certain cyclic $p$ -gonal curves, whose explicit equations are provided in terms of the equations for $S$ . As a consequence of this decomposition, we are able to provide explicit positive-dimensional families of closed Riemann surfaces whose Jacobian variety is isogenous to the product of elliptic curves.

Description: Let $\mathcal {A}\,{\bar {\otimes }}\,L(K)$ be the Fubini product of a monotone complete $C^{\ast }$ -algebra $\mathcal {A}$ and $L(K)$ , where $K$ is an arbitrary Hilbert space. Hamana defined a product ‘ $\circ $ ’ in $\mathcal {A}\,{\bar {\otimes }}\,L(K),$ which turns $(\mathcal {A}\,{\bar {\otimes }}\,L(K),\circ )$ into a monotone complete $C^{\ast }$ -algebra. To help him to do this, he introduced a new concept of order convergence, which is subtly different from Kadison–Pedersen convergence, and, made use of the Choi–Effros product related to a completely positive idempotent map on the injective envelope of $A$ , whose existence is a consequence of Zorn's lemma. When $K$ is separable, Saitô and Wright showed that the algebra $(\mathcal {A}\,{\bar {\otimes }}\,L(K),\circ )$ is the quotient algebra of the Borel tensor pro- duct $\mathcal {A}^{\infty } \,{\bar {\otimes }}\,L(K)$ of the Pedersen–Baire envelope $\mathcal {A}^{\infty }$ and $L(K)$ by a $\sigma $ -ideal $I$ . In this paper, when $K$ is separable, it is shown, by making use of Kadison–Pedersen convergence, that there is a natural monotone $\sigma $ -closed two-sided ideal $J$ of $\mathcal {A}^{\infty }\,{\bar {\otimes }}\,L(K)$ , such that the quotient algebra $\mathcal {B} =\mathcal {A}^{\infty }\,{\bar {\otimes }}\,L(K)/J$ is monotone $\sigma $ -complete, and the quotient map $q_{J}$ restricts to a unital iso- metric bijection $\Phi $ from $\mathcal {A}\,{\bar {\otimes }}\,L(K)$ to $\mathcal {B}$ . Via the map $\Phi $ , we can transplant the multiplication on $\mathcal {B}$ into the multiplication ‘ $\bullet $ ’ on $\mathcal {A}\bar {\otimes }L(K)$ so that $(\mathcal {A}\,{\bar {\otimes }}\,L(K),\bullet )$ is a monotone complete $C^{\ast }$ -algebra. It is also shown that the product ‘ $\bullet $ ’ coincides with ‘ $\circ $ ’ (and so $I=J$ ). So there is no need to appeal to Zorn's lemma here for our approach to defining the product ‘ $\circ $ ’. Our construction sheds some fresh light on classification problems of cross product algebras associated with generic dynamics.

Description: Given real Banach spaces $E$ and $F$ , we show that every isometric isomorphism from the space of approximable polynomials of degree at most $n$ on $E$ to the space of approximable polynomials of degree at most $n$ on $F$ has the form $T(P)(y) = \pm \bar {P} \circ r^t \circ J_F(y)$ , where $\bar {P}$ is the Aron–Berner extension of $P$ , $r$ is an isometry of $E'$ onto $F'$ and $J_F$ is the canonical inclusion of $F$ into $F''$ . With additional conditions on $E$ and $F$ , we are also able to characterize the isometric isomorphisms between the spaces of integral polynomials of degree at most $n$ and the isometric isomorphisms between the spaces of all polynomials of degree at most $n$ .

Description: Let $d_n := p_{n+1} - p_n$ , where $p_n$ denotes the $n$ th smallest prime, and let $R(T) := \log T \log _2 T\log _4 T/ (\log _3 T)^2$ (the ‘Erdös–Rankin’ function). We consider the sequence $(d_n/R(p_n))$ of normalized prime gaps, and show that its limit point set contains at least $25\%$ of non-negative real numbers. We also show that the same result holds if $R(T)$ is replaced by any ‘reasonable’ function that tends to infinity more slowly than $R(T)\log _3 T$ . We also consider ‘chains’ of normalized prime gaps. Our proof combines breakthrough work of Maynard and Tao on bounded gaps between primes with subsequent developments of Ford, Green, Konyagin, Maynard and Tao on long gaps between consecutive primes.

Description: Let $L(s,F)$ be the $L$ -function associated with a Hecke–Maass form $F$ for ${\rm SL}(3, {\mathbb {Z}})$ , with $A_F(n,1)$ denoting the $n$ th coefficient of the Dirichlet series for it. It is proved that, for $N\ge 2$ and any $\alpha \in {\mathbb {R}}$ , there exists an effective constant $c 〉 0$ such that \[ \sum_{n\le N} \Lambda(n)A_F(n,1)e(n\alpha)\ll N \exp(-c\sqrt{\log N}), \] where $\Lambda (n)$ is the von Mangoldt function, and the implied constant depends only on $F$ .

Description: We prove an asymptotic formula for the shifted convolution of the divisor functions $d_3(n)$ and $d(n)$ , which is uniform in the shift parameter and which has a power-saving error term. The method is also applied to give analogous estimates for the shifted convolution of $d_3(n)$ and Fourier coefficients of holomorphic cusp forms. These asymptotics improve previous results obtained by several different authors.

Description: We prove a characterization result in the spirit of the Kinderlehrer–Pedregal Theorem for Young measures generated by gradients of Sobolev maps satisfying the orientation-preserving constraint, that is, the pointwise Jacobian is positive almost everywhere. The argument to construct the appropriate generating sequences from such Young measures is based on a variant of convex integration in conjunction with an explicit lamination construction in matrix space. Our generating sequence is bounded in ${\rm L}^p$ for $p$ less than the space dimension, a regime in which the pointwise Jacobian behaves flexibly, as is illustrated by our results. On the other hand, for $p$ larger than or equal to the space dimension the situation necessarily becomes rigid and a construction as presented here cannot succeed. Applications to relaxation of integral functionals, the theory of semiconvex hulls and approximation of weakly orientation-preserving maps by strictly orientation-preserving ones in Sobolev spaces are given.

Description: In this paper, we define a congruence $\eta ^{\ast }$ on semigroups. For the finite semigroups $S$ , $\eta ^{\ast }$ is the smallest congruence relation such that $S/\eta ^{\ast }$ is a nilpotent semigroup (in the sense of Mal'cev). In order to study the congruence relation $\eta ^{\ast }$ on finite semigroups, we define a $\textbf {CS}$ -diagonal finite regular Rees matrix semigroup. We prove that if $S$ is a $\textbf {CS}$ -diagonal finite regular Rees matrix semigroup, then $S/\eta ^{\ast }$ is inverse. Also, if $S$ is a completely regular finite semigroup, then $S/\eta ^{\ast }$ is a Clifford semigroup. We show that, for every non-null principal factor $A/B$ of $S$ , there is a special principal factor $C/D$ such that every element of $A{\setminus } B$ is $\eta ^{\ast }$ -equivalent with some element of $C{\setminus } D$ . We call the principal factor $C/D$ the $\eta ^{\ast }$ -root of $A/B$ . All $\eta ^{\ast }$ -roots are $\textbf {CS}$ -diagonal. If certain elements of $S$ act in the special way on the $\textbf {R}$ -classes of a $\textbf {CS}$ -diagonal principal factor, then it is not an $\eta ^{\ast }$ -root. Some of these results are also expressed in terms of pseudovarieties of semigroups.

Description: The moduli space of stable pairs on a local surface $X=K_S$ is in general non-compact. The action of $ {\mathbb {C}}^{\ast }$ on the fibres of $X$ induces an action on the moduli space and the stable pair invariants of $X$ are defined by the virtual localization formula. We study the contribution to these invariants of stable pairs (scheme theoretically) supported in the zero section $S \subset X$ . Sometimes there are no other contributions, for example, when the curve class $\beta $ is irreducible. We relate these surface stable pair invariants to the Poincaré invariants of Dürr–Kabanov–Okonek. The latter are equal to the Seiberg–Witten invariants of $S$ by the work of Dürr–Kabanov–Okonek and Chang–Kiem. We give two applications of our result. (1) For irreducible curve classes the GW/PT correspondence for $X = K_S$ implies Taubes' GW/SW correspondence for $S$ . (2) When $p_g(S) = 0$ , the difference of surface stable pair invariants in class $\beta $ and $K_S - \beta $ is a universal topological expression.

Description: Let $K$ be a quadratic number field and $\zeta _K(s)$ be the associated Dedekind zeta-function. We show that there are infinitely many gaps between consecutive zeros of $\zeta _K(s)$ on the critical line which are $ 〉 $ 2.866 times the average spacing.

Description: Let $(R,\mathfrak {m},k)$ be a commutative noetherian local ring of Krull dimension $d$ . We prove that the cohomology annihilator $\mathsf {ca}\,(R)$ of $R$ contains a power of $\mathfrak {m}$ if and only if, for some $n\ge 0,$ the $n$ th syzygies in $\mathsf {mod}\,R$ are constructed from syzygies of $k$ by taking direct sums/summands and a fixed number of extensions. These conditions yield that $R$ is an isolated singularity such that the bounded derived category $\mathsf {D^b}\,(R)$ and the singularity category $\mathsf {D_{sg}}\,(R)$ have finite dimension, and the converse holds when $R$ is Gorenstein. We also show that the modules locally free on the punctured spectrum are constructed from syzygies of finite length modules by taking direct sums/summands and $d$ extensions. This result is exploited to investigate several ascent and descent problems between $R$ and its completion $\hat R$ .

Description: Let $C_{n}$ be the free center-by-metabelian Lie algebra of finite rank $n$ , with $n \geq 2$ , over a field $K$ of characteristic $0$ . We study the automorphism group ${\rm Aut}(C_{n})$ of $C_{n}$ by means of a topology. Let $T_{C_{n}}$ be the group of tame automorphisms of $C_{n}$ and ${\rm GL}_{n}(K)$ be the general linear group. It is shown that for any finite subset $X$ of IA-automorphisms of $C_{2}$ , the subgroup of ${\rm Aut}(C_{2})$ generated by ${\rm GL}_{2}(K)$ and $X$ is not dense in ${\rm Aut}(C_{2})$ . For $n = 3$ , the subgroup of ${\rm Aut}(C_{3})$ generated by $T_{C_{3}}$ and two more IA-automorphisms is dense in ${\rm Aut}(C_{3})$ . For $n \geq 4$ , the subgroup of ${\rm Aut}(C_{n})$ generated by $T_{C_{n}}$ and one more IA-automorphism is dense in ${\rm Aut}(C_{n})$ .

Description: We consider the homotopy category of complexes of projective modules over a Noetherian ring. Truncation at degree zero induces a fully faithful triangle functor from the totally acyclic complexes to the singularity category. We show that if the ring is either Artin or commutative Noetherian local, then the functor is dense if and only if the ring is Gorenstein. Motivated by this, we define the Gorenstein defect category of the ring, a category which in some sense measures how far the ring is from being Gorenstein.

Description: We extend the theory of the universal $\eta$ -invariant to the case of bordism groups of manifolds with boundaries. This allows the construction of secondary descendants of the universal $\eta$ -invariant. We obtain an interpretation of Laures’ $f$ -invariant as an example of this general construction. As an aside, we improve a recent result by Han–Zhang about the modularity of a certain formal power series of $\eta$ -invariants.

Description: In this note, we prove a generalization of the ‘strange formula’ of Freudenthal and de-Vries for compact homogeneous spaces with positive Euler characteristic. As a corollary, we obtain a new proof of the original formula. We also apply our results to computing a topological invariant used to study hyper-Kähler structures.

Description: We investigate the consequences of natural conjectures of Montgomery type on the non-vanishing of Dirichlet $L$ -functions at the central point. We first justify these conjectures using probabilistic arguments. We then show using a result of Bombieri, Friedlander and Iwaniec, and a result of the author that they imply that almost all Dirichlet $L$ -functions do not vanish at the central point. We also deduce a quantitative upper bound for the proportion of Dirichlet $L$ -functions for which $L(\frac {1}{2},\chi )=0$ .

Description: We prove a large sieve inequality with moduli of general polynomial form. The version with moduli being the product of $\ell$ many $k$ th powers has an application to counting $\ell$ -tuples of consecutive integers with high index of composition, a problem related to the abc-conjecture.

Description: Let $S: (V_{1}, F_{1}) \rightarrow (V_{2}, F_{2})$ be a foliated continuous map between smooth manifolds with codimension 1 smooth foliations, transversely oriented. If the transverse component of $S$ has bounded variation with derivative in the Lebesgue class of measure, then the image by $S^* : H^3(V_{2},F_{2}) \rightarrow H^3(V_{1},F_{1})$ of the Godbillon–Vey class of $V_{2}$ is the Godbillon–Vey class of $V_{1}$ . This solves partially a question raised by Etienne Ghys.

Description: We classify an action of the $n$ -strand braid group on the free group of rank $n$ , which is similar to the Artin representation in the sense that the $i$ th generator $\sigma _{i}$ of $B_{n}$ acts so that it fixes all free generators $x_{j}$ except $j = i,i+1$ . We determine all such representations and discuss knot invariants coming from such representations.

Description: We establish new estimates to compute the $\lambda$ -function of Aron and Lohman on the unit ball of a JB $^* $ -triple. It is established that for every Brown–Pedersen quasi-invertible element $a$ in a JB $^* $ -triple $E$ we have \[{\rm dist} (a, \mathfrak {E} (E_1)) = \max \{ 1- m_q (a), \|a\|-1\} ,\] where $\mathfrak {E} (E_1)$ denotes the set of extreme points of the closed unit ball $E_1$ of $E$ . It is proved that $\lambda (a) = ({1+m_q (a)})/{2},$ for every Brown–Pedersen quasi-invertible element $a$ in $E_1$ , where $m_q (a)$ is the square root of the quadratic conorm of $a$ . For an element $a$ in $E_1$ which is not Brown–Pedersen quasi-invertible, we can only estimate that $\lambda (a)\leq \frac 12 (1-\alpha _q (a))$ . A complete description of the $\lambda$ -function on the closed unit ball of every JBW $^* $ -triple is also provided, and as a consequence, we prove that every JBW $^* $ -triple satisfies the uniform $\lambda$ -property.

Description: In this paper, the generalized BMO martingale spaces are introduced as the dual spaces of martingale Hardy–Lorentz–Karamata spaces $H_{p,q,b}^s$ , which completes the very recent result [ Quart. J. Math. 65 (2014), 985–1009] in the case $0\lt p\leq 1, 1\lt q\lt \infty$ . Moreover, by duality we obtain a new John–Nirenberg theorem when the stochastic basis is regular.

Description: In this paper, we prove that more than 66.036% of zeros of the Riemann zeta-function are distinct.

Description: An integer $n$ is called practical if every $m\le n$ can be written as a sum of distinct divisors of $n$ . We show that the number of practical numbers below $x$ is asymptotic to $c x/\log x$ , as conjectured by Margenstern. We also give an asymptotic estimate for the number of integers below $x$ whose maximum ratio of consecutive divisors is at most $t$ , valid uniformly for $t\ge 2$ .

Description: We show that using the family of adapted Kähler polarizations of the phase space of a compact, simply connected, Riemannian symmetric space of rank 1, the field $H^{\rm corr}$ of quantum Hilbert spaces produced by geometric quantization, including the half-form correction, is flat if $M$ is the three-dimensional sphere and not even projectively flat otherwise.

Description: For an integer $r$ , a prime power $q$ and a polynomial $f$ over a finite field ${{\mathbb F}}_{q^r}$ of $q^r$ elements, we obtain an upper bound on the frequency of elements in an orbit generated by iterations of $f$ which fall in a proper subfield of ${{\mathbb F}}_{q^r}$ . We also obtain similar results for elements in affine subspaces of ${{\mathbb F}}_{q^r}$ , considered as a linear space over ${{\mathbb F}}_q$ .

Description: We define several new types of quantum chromatic numbers of a graph and characterize them in terms of operator system tensor products. We establish inequalities between these chromatic numbers and other parameters of graphs studied in the literature and exhibit a link between them and non-signalling correlation boxes.

Description: For $c 〉 1$ , $c \notin \mathbb Z$ and $\chi$ a primitive character $({\hbox{mod }}q)$ , we estimate the sum \[\sum _{x \lt n \le x +y} \chi ([n^c]).\] Our results complement earlier work of Banks and Shparlinski. We also give a new bound for the least $n$ for which $[n^c]$ is a quadratic non-residue $({\rm mod}\ q)$ , in the case of prime $q$ , for $1 \lt c \lt \tfrac {371}{309}$ .

Description: We are interested in existence and regularity results for weak solutions of parabolic equations of the type \[\partial _t u- \operatorname {div}\,a(x,t,Du)=0\] on a parabolic space time cylinder $\Omega _T$ . The vector field $a$ is assumed to satisfy a non-standard $p,q$ -growth condition. We treat the subquadratic case, where \[\frac {2n}{n+2}\lt p\lt 2 \quad {\text {and}} \quad p\leq q \lt p+ \frac {4}{n+2}\] holds. We show existence of weak s $u \in L^p(0,T;W^{1,p}(\Omega )) \cap L^q_{\mathrm {loc}}(0,T;W^{1,q}_{\mathrm {loc}}(\Omega ))$ for the Cauchy–Dirichlet problem associated to the parabolic equation from above. Further, a local bound for the spatial gradient $Du$ is established. The results cover for example equations of the type \[\partial _t u = \operatorname {div} (\alpha (x,t) (\mu ^2+ |Du|^2)^{(p-2)/2}Du) + \operatorname {div}(\beta (x,t)(\mu ^2 + |Du|^2)^{(q-2)/2}Du)\] with $\mu \in [0,1]$ and suitable functions $\alpha (x,t)$ and $\beta (x,t)$ . We emphasize that the results cover the singular case $\mu =0$ .

Description: Given a Calderón–Zygmund singular integral operator, it is an open question whether the Fefferman inequality holds true with u as an arbitrary weight. This inequality is known as the Muckenhoupt–Wheeden conjecture. In this paper we prove that, for a wider class of operators called Rubio de Francia operators, if || u || ≤1, then where the function is slightly bigger than the identity. Applications of this inequality in the setting of rearrangement invariant spaces are given.

Description: We show that whenever m ≤ n , the space of all continuous rational maps from CP m to CP n has the same homology as the space of all continuous maps between these spaces in dimensions smaller than d (2 n – 2 m + 1) – 1. This improves the result of the paper ‘Spaces of rational maps and the Stone–Weierstrass theorem’ ( Topology 45 (2006), 281–293).

Description: A theorem of Davenport, Mirsky, Newman and Rado shows that there does not exist an exact covering system with distinct moduli. Motivated by this, we raise the question whether or not this is true for covering systems in algebraic number fields. We provide an affirmative answer for certain quadratic fields.

Description: The caustic of a smooth surface in the Euclidean 3-space is the envelope of the normal rays to the surface. It is also the locus of the centres of curvature (the focal points) of the surface. This is why it is also referred to as the focal set of the surface. It has Lagrangian singularities and its generic models are given in [V. I. Arnol’d, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps , Vol. I, Birkhäuser, Boston, 1986]. The aim of this paper is to define the caustic C ( M ) of a smooth surface M embedded in the Minkowski 3-space and to study its geometry. We denote by the locus of degeneracy (LD) the locus of points on M where the metric is degenerate. If M is a closed surface then its LD is not empty. At a point on the LD the ‘normal’ line to M is lightlike and is tangent to M . Also, the focal set of M is not defined at points on the LD. We define the caustic of M as the bifurcation set of the family of distance-squared functions on M . Then C ( M ) coincides with the focal set of M \ LD and provides an extension of the focal set to the LD. We study the local behaviour of the metric on C ( M ).

Description: In a recent work of R. M. Bryant and the second author, a (partial) modular analogue of Klyachko's 1974 result on Lie powers of the natural module for GL( n , K ) was presented. There it was shown that nearly all of the indecomposable summands of the r th tensor power also occur up to isomorphism as summands of the r th Lie power, provided that r != p m and r !=2 p m , where p is the characteristic of K . In the current paper, we restrict attention to GL(2, K ) and consider the missing cases where r = p m and r =2 p m . In particular, we prove that the indecomposable summand of the r th tensor power of the natural module with highest weight ( r –1, 1) is a summand of the r th Lie power if and only if r = p or r is not a power of p .

Description: We consider Calderón's inverse problem on planar domains with conductivities in fractional Sobolev spaces. When is Lipschitz, the problem was shown to be stable in the L 2 -sense in Clop et al. [Stability of calderón's inverse conductivity problem in the plane for discontinuous conductivities, Inverse Probl. Imaging 4 (2010), 49–91]. We remove the Lipschitz condition on the boundary. To this end, we analyse the Sobolev regularity of the characteristic function of . For a quasiball, we compute || || W s , p (R d ) in terms of the -neighbourhoods of the boundary.

Description: In this paper, an explicit classification result for certain 5-manifolds with fundamental group Z/2 is obtained. These manifolds include total spaces of circle bundles over simply connected 4-manifolds.

Description: For irreducible polynomials of the form X d + c we establish the expected asymptotic formula for the number of k -free integers n d + c with n ≤ x , under the assumption that k ≥ (5 d + 3)/9. Previous techniques required k 〉 3 d /4. We also prove an entirely analogous result for p d + c with p prime.

Description: For a compact almost complex 4-manifold ( M , J ), we study the subgroups H ± J of H 2 ( M , R) consisting of cohomology classes representable by J -invariant, respectively, J -anti-invariant real 2-forms. If b + =1, we show that, for generic almost complex structures on M , the subgroup H – J is trivial. Computations of the subgroups and their dimensions h ± J are obtained for almost complex structures related to integrable ones. We also prove semi-continuity properties for h ± J .

Description: This paper considers the links between the geometry of the various flag manifolds of loop groups and bundles over families of rational curves. A topological stability result for the moduli on a rational ruled surface of G -bundles with additional flag structure along a line is proved for any reductive group; this gives the corresponding stability statement for any compact group K for the moduli of K -instantons over the four-sphere, and for the moduli of K -calorons over R 3 times the circle.

Description: In the first part of the paper, comprising Sections 1–5, we introduce a sequence of functions in the tangent bundle TM of any smooth two-dimensional manifold M with smooth Riemannian metric g that correspond to the higher order Schwarzians of the linearized geodesic flow. We use these functions and a classical theorem of Loewner on analytic continuation to characterize the existence of the adapted complex structure induced by g on the set T R M of vectors in TM of length up to R , that is, for M compact, the existence of a Grauert tube of radius R . The basic characterization obtained is expressed as a sequence of differential inequalities of increasing order polynomial in the derivatives of the Gauss curvature on M and in R –1 that should be regarded as the higher order versions of a curvature inequality by Lempert and Szoke. The second part of the paper, Sections 6–9, includes a discussion of the rank of the infinite Hankel matrix of the Schwarzians. In addition, a characterization of the existence of the adapted complex structure on T R M in terms of moment sequences with parameters R and v in TM is noted.

Description: In this paper, we investigate the asymptotic behavior of solutions to the initial boundary value problem for a theory of porous thermoviscoelastic mixtures. Our main result is to establish conditions that ensure the analyticity and the exponential stability of the corresponding semigroup. We show that under certain conditions over the coefficients, we obtain the lack of exponential stability.

Description: We continue the programme of Chinea, De León and Marrero who studied the topology of cosymplectic manifolds. We study 3-cosymplectic manifolds that are the closest odd-dimensional analogue of hyper-Kähler structures. We show that there is an action of the Lie algebra so (4, 1) on the basic cohomology spaces of a compact 3-cosymplectic manifold with respect to the Reeb foliation. This implies some topological obstructions to the existence of such structures which are expressed by bounds on the Betti numbers. It is known that every 3-cosymplectic manifold is a local Riemannian product of a hyper-Kähler factor and an abelian three-dimensional Lie group. Nevertheless, we present a non-trivial example of a compact 3-cosymplectic manifold that is not the global product of a hyper-Kähler manifold and a flat 3-torus.

Description: This paper computes the quantization of the moduli space of flat SO(3)-bundles over an oriented surface with boundary, with prescribed holonomies around the boundary circles. The result agrees with the generalized Verlinde formula conjectured by Fuchs and Schweigert.

Description: We prove that, for each positive integer n ≥2, there is an infinite arithmetic family of hyperelliptic curves of genus n violating the Hasse principle explained by the Brauer–Manin obstruction. Using these families of curves, we show that, for any positive integer k ≥1, there are infinitely many algebraic and arithmetic families of forms in three variables of degree 4 k +2 such that they are counterexamples to the Hasse principle explained by the Brauer–Manin obstruction.

Description: This paper consists of two related parts. In the first part we give a self-contained proof of homological stability for the spaces C n ( M ; X ) of configurations of n unordered points in a connected open manifold M with labels in a path-connected space X , with the best possible integral stability range of 2*≤ n . Along the way we give a new proof of the high connectivity of the complex of injective words. If the manifold has dimension at least 3, we show that in rational homology the stability range may be improved to *≤ n . In the second part we study to what extent the homology of the spaces C n ( M ) can be considered stable when M is a closed manifold. In this case there are no stabilization maps, but one may still ask if the dimensions of the homology groups over some fields stabilize with n . We prove that this is true when M is odd-dimensional, or when the field is F 2 or Q. It is known to be false in the remaining cases.

Description: In this paper, we consider the curves in the unit sphere. The unit sphere can be canonically embedded in the lightcone and de Sitter 3-space in Minkowski space-time. We investigate these curves in the framework of the theory of Legendrian dualities between pseudo-spheres in Minkowski space-time.

Description: Given ( X , 0)(C n , 0) a weighted homogeneous germ of hypersurface with isolated singularity and f :(C n , 0)-〉(C, 0) a germ of function finitely determined with respect to X , we show that μ BR ( X , f )=μ( f )+μ( X , f ), where μ( f ) and μ( X , f ) denote the Milnor numbers of f and of the fiber X f –1 (0), respectively, and μ BR ( X , f ) is the Bruce–Roberts number of f with respect to X . We show that the logarithmic characteristic subvariety LC( X ) is Cohen–Macaulay and we get relations between the Bruce–Roberts number and the Euler obstruction.

Description: We study the flat geometry of orthogonal projections of a generic space curve to planes. For a single projection, we do this by considering submersions on a (singular) plane curve. This is an alternative method to the classification of divergent diagrams carried out in Dias and Nuño Ballesteros [Plane curve diagrams and geometrical applications, Q. J. Math . 59 (2008), 287–310]. We redraw the bifurcation diagrams of the orthogonal projections of space curves adding the information about their flat geometry. We also study the duals of the projected curves and the way they bifurcate as the direction of projection varies locally in S 2 .

Description: We establish some upper bounds for the number of integer solutions to the Thue inequality $|F(x , y)| \leq m$ , where $F$ is a binary form of degree $n \geq 3$ and with non-zero discriminant $D$ , and $m$ is an integer. Our upper bounds are independent of $m$ , when $m$ is smaller than $|D|^{{1}/{4(n-1)}}$ . We also consider the Thue equation $|F(x , y)| = m$ and give some upper bounds for the number of its integral solutions. In the case of equation, our upper bounds will be independent of integer $m$ , when $m 〈 |D|^{{1}/{2(n-1)}}$ .

Description: We investigate the rearrangement of the Haar system induced by the postorder on the set of dyadic intervals in $[0,1]$ with length greater than or equal to $2^{-N}$ . By means of operator norms on $\hbox {BMO}_N$ we prove that the postorder has maximal distance to the usual lexicographic order.

Description: In this note, we prove that given a continuous Sobolev $W^{1,p}$ deformation $f$ , with $1 〈 p 〈 \infty $ , from a planar domain to $\mathbb {R}^2$ which is injective almost everywhere, we can find a sequence $f_k$ of diffeomorphisms with $f_k - f \in W^{1,p}_0$ such that $f_k \to f$ uniformly and in the Sobolev norm.

Description: We prove that C $^{\ast }$ -algebras which, as Banach spaces, are Grothendieck cannot be decomposed into a tensor product of two infinite-dimensional C $^{\ast }$ -algebras. By a result of Pfitzner, this class contains all von Neumann algebras and their norm-quotients. We thus complement a recent result of Ghasemi who established a similar conclusion for the class of SAW $^{\ast }$ -algebras.

Description: We prove that for any operator $T$ on bi–parameter BMO the identity factors through $T$ or $Id-T$ . Bourgain's localization method provides the conceptual framework of our proof. It consists of replacing the factorization problem on the non-separable bi-parameter BMO by its localized, finite dimensional counterpart. We solve the resulting finite dimensional factorization problems by exploiting the geometry and combinatorics of colored dyadic rectangles.

Description: We prove that for $1 〈 c 〈 4/3$ the subsequence of the Thue–Morse sequence $\mathbf t$ indexed by $\lfloor n^c\rfloor $ defines a normal sequence, that is, each finite sequence $(\varepsilon _0,\ldots ,\varepsilon _{T-1})\in \{0,1\}^T$ occurs as a contiguous subsequence of the sequence $n\mapsto \mathbf t(\lfloor n^c\rfloor )$ with asymptotic frequency $2^{-T}$ .

Description: Considering vectorial integrals in the multidimensional calculus of variations and quasilinear elliptic systems of partial differential equations, we prove gradient regularity of minimizers and weak solutions, respectively. In contrast to the classical theory, we impose our assumptions on the structure functions only locally (i.e. near a single point) or asymptotically (i.e. near infinity). In particular, we point out relations between the local and the asymptotic point of view, and we discuss notions of quasiconvexity at infinity and quasimonotonicity at infinity, which arise in this context.

Description: The average value of log s ( n )/ n taken over the first N even integers is shown to converge to a constant when N tends to infinity; moreover, the value of this constant is approximated and proved to be less than 0. Here, s ( n ) sums the divisors of n less than n . Thus, the geometric mean of s ( n )/ n , the growth factor of the function s , in the long run tends to be less than 1. This could be interpreted as probabilistic evidence that aliquot sequences tend to remain bounded.

Description: Natural metric structures on the tangent bundle and tangent sphere bundles S r M of a Riemannian manifold M with radius function r enclose many important unsolved problems. Admitting metric connections on M with torsion, we deduce the equations of induced metric connections on those bundles. Then the equations of reducibility of TM to the almost Hermitian category. Our purpose is the study of the natural contact structure on S r M and the G 2 -twistor space of any oriented Riemannian 4-manifold.

Description: In terms of fragmentability, we describe a new class of Banach spaces which may be c 0 -saturated but do not contain weak- G open bounded subsets. In particular, none of these spaces is isomorphic to a separable polyhedral space.

Description: A mathematical construction of the conformal field theory (CFT) associated to a compact torus, also called the ‘non-linear Sigma-model’ or ‘lattice-CFT’, is given. Underlying this approach to CFT is a unitary modular functor, the construction of which follows from a ‘quantization commutes with reduction’-type of theorem for unitary quantizations of the moduli spaces of holomorphic torus-bundles and actions of loop groups. This theorem in turn is a consequence of general constructions in the category of affine symplectic manifolds and their associated generalized Heisenberg groups.

Description: A longstanding question of Gromov asks whether every one-ended word-hyperbolic group contains a subgroup isomorphic to the fundamental group of a closed hyperbolic surface. An infinite family of word-hyperbolic groups can be obtained by taking doubles of free groups amalgamated along words that are not proper powers. We define the set of polygonal words in a free group of finite rank, and prove that polygonality of the amalgamating word guarantees that the associated square complex virtually contains a 1 -injective closed surface. We provide many concrete examples of classes of polygonal words. For instance, in the case when the rank is 2, we establish polygonality of words without an isolated generator, and also of almost all simple height-1 words, including the Baumslag–Solitar relator a p ( a q ) b for pq != 0.

Description: In our previous work, a unified description as polynomial Hamiltonian systems was established for a broad class of the Schlesinger systems including the sixth Painlevé equation and Garnier systems. The main purpose of this paper is to present particular solutions of this Hamiltonian system in terms of a certain generalization of Gauß's hypergeometric function. Key ingredients of the argument are the linear Pfaffian system derived from an integral representation of the hypergeometric function (with the aid of twisted de Rham theory) and Lax formalism of the Hamiltonian system.

Description: In this paper, we consider a certain class of closed Riemann surfaces, called generalized Fermat curves. These surfaces are the highest abelian branched cover of certain orbifolds. In this class of Riemann surfaces, we study the problem of deciding when the field of moduli is contained in R and when, in such a case, S can be defined over the reals.

Description: Given a univariate polynomial f ( x ) over a ring R , we examine when we can write f ( x ) as g ( h ( x )), where g and h are polynomials of degree at least 2. We answer two questions of Gusic regarding when the existence of such g and h over an extension of R implies the existence of such g and h over R .

Description: We establish the existence of stable manifolds for semiflows defined in Banach spaces by non-autonomous ordinary differential equations v ' = A ( t ) v + f ( t , v ) assuming that the non-autonomous linear equation v ' = A ( t ) v admits a type of non-uniform dichotomy that we call non-uniform polynomial dichotomy . We consider two families of perturbations f and obtain for one of them global C 1 stable manifolds and for the other local Lipschitz stable manifolds. In both cases we obtain the polynomial decay of the flow on the stable manifold and in the first case we also obtain the polynomial decay of its derivative. Additionally, we study how the manifolds obtained vary with the perturbations considered and present examples of linear non-autonomous differential equation admitting a non-uniform polynomial dichotomy that is not a uniform polynomial dichotomy.

Description: Every state on a C *-algebra A induces a *-symmetric semi-inner product ( x , y )↦ ( y * x ) + ( xy *) ( x , y A ). The main scope of the paper is to characterize those states for which the induced *-symmetric Gelfand–Naimark–Segal inner product space is complete. It is shown that this happens precisely when is a finite convex combination of pure states. (It is well known that the same conclusion follows if one considers the non-symmetric semi-inner product ( x , y ) ↦ ( y * x ).) In so doing, we exhibit an interesting connection between convexity properties of states, the transitivity of irreducible representations and Banach space properties of the quotients of C *-algebra s by kernels of states.

Description: We find two involutions on partitions that lead to partition identities for Ramanujan's third-order mock theta functions (– q ) and (– q ). We also give an involution for Fine's partition identity on the mock theta function f ( q ). The two classical identities of Ramanujan on third-order mock theta functions are consequences of these identities. Our combinatorial constructions also apply to Andrews’ generalizations of Ramanujan's identities.

Description: Let ( G , +) be a finite abelian group. Then, S ( G ) and ( G ) denote the smallest integer such that each sequence over G of length at least has a subsequence whose terms sum to 0 and whose length is equal to and at most, respectively, the exponent of the group. For groups of rank 2, we study the inverse problems associated to these constants, that is, we investigate the structure of sequences of length S ( G )–1 and ( G )–1 that do not have such a subsequence. On the one hand, we show that the structure of these sequences is in general richer than expected. On the other hand, assuming a well-supported conjecture on this problem for groups of the form C m C m , we give a complete characterization of all these sequences for general finite abelian groups of rank 2. In combination with partial results towards this conjecture, we get unconditional characterizations in special cases.

Description: We consider the completion ( ) of a countable commutative topological ring ( A , ) with a Hausdorff topology having a countable family of ideals as a system of neighborhoods of 0. The monoid ring A is inspired by Corner [On the existence of very decomposable abelian groups, Abelian Group Theory , Proceedings, Honolulu 1982/83, Lecture Notes in Mathematics 1006, Springer, Berlin (1983), 354–357.]. In addition, we consider as topological ring equipped with the discrete topology . Thus, ( , ) is trivially a complete ring, and is the discrete topology on . Both rings can be realized as topological endomorphism rings of torsion-free abelian groups with the finite topology , using a result from Corner and Göbel [Prescribing endomorphism algebras—a unified treatment, Proc. London Math. Soc. (3) 50 (1985), 447–479.]. Thus, the endomorphism rings of the groups M and H are algebraically isomorphic, but not isomorphic as topological rings. The interesting properties of , M and H are discussed in detail. They are used to answer an open problem on the entropy (as discussed in Section 1), showing in a strong sense that the rank-entropy [investigated in (L. Salce and P. Zanardo, A general notion of algebraic entropy and the rank-entropy, Forum Math. 21 (2009), 579–599.)] is a notion depending on the groups and not on their endomorphism rings: we get ent rk ( M ) = 0 and ent rk ( H ) = . The fact that a torsion-free abelian group M satisfies ent rk ( M ) = 0 is implied by the property of of being algebraic. As a byproduct of our results, we derive that the converse implication does not hold, since contains a lot of transcendental elements.

Description: We extend some theorems from the context of solid-angle sums over rational polytopes to the context of solid-angle sums over real polytopes. Moreover, we also consider any real dilation parameter, as opposed to the traditional integer dilation parameters. One of the main results is an extension of Macdonald's solid-angle quasi-polynomial for rational polytopes to a real analytic function of the dilation parameter, for any real convex polytope. Consequently, we find an extension of Macdonald's reciprocity law for real dilation parameters, over real polytopes.

Description: We consider the inclusion of the space of algebraic (regular) maps between real algebraic varieties in the space of all continuous maps. For a certain class of real algebraic varieties, which include real projective spaces, it is well known that the space of real algebraic maps is a dense subset of the space of all continuous maps. Our first result shows that, for this class of varieties, the inclusion is also a homotopy equivalence. After proving this, we restrict the class of varieties to real projective spaces. In this case, the space of algebraic maps has a ‘minimum degree’ filtration by finite-dimensional subspaces and it is natural to expect that the homotopy types of the terms of the filtration approximate closer and closer the homotopy type of the space of continuous mappings as the degree increases. We prove this and compute the lower bounds of this approximation of these spaces. This result can be seen as a generalization of the results of Mostovoy, Vassiliev and others on the topology of the space of real rational maps and the space of real polynomials without n -fold roots. It can also be viewed as a real analogue of Mostovoy's work on the topology of the space of holomorphic maps between complex projective spaces, which generalizes Segal's work on the space of complex rational maps.

Description: An ( r , ) overlap colouring of a graph G has r colours at each vertex, any two adjacent vertices sharing exactly colours. A theory analogous to multichromatic and fractional chromatic theory is developed. In particular, all the overlap chromatic numbers of cycle graphs are computed. It is shown that if a graph G contains an odd cycle C 2 p +1 and has the same p -fold chromatic number as C 2 p +1 , then all its overlap chromatic numbers are the same as those of C 2 p +1 . The core of a graph is the smallest induced subgraph to which it has a homomorphism. It is shown that some pairs of graphs with the same multichromatic numbers have different sets of overlap chromatic numbers, and that some graphs with non-isomorphic cores have the same sets of overlap chromatic numbers. (In particular, any non-bipartite series-parallel graph has the same overlap chromatic as its smallest odd cycle.) Thus classifying graphs by overlap chromatic properties is intermediate between classifying them by multichromatic properties and classifying them by cores.

Description: Given a prime p , we obtain upper bounds on single and bilinear character sums with Fermat quotients where gcd ( u , p )=1. We use these bounds to estimate character sums with Fermat quotients q p () at prime arguments .

Description: The purpose of this article is to describe connections between the loop space of the 2-sphere and Artin's braid groups. The current article exploits Lie algebras associated with Vassiliev invariants in the work of Kohno ( Linear representations of braid groups and classical Yang-Baxter equations , Cont. Math. 78 (1988), 339–369 and Vassiliev invariants and de Rham complex on the space of knots , Symplectic Geometry and Quantization, Contemp. Math. 179 (1994), Am. Math. Soc. Providence, RI, 123–138), and provides connections between these various topics. Two consequences are as follows: the homotopy groups of spheres are identified as ‘natural’ sub-quotients of free products of pure braid groups, and an axiomatization of certain simplicial groups arising from braid groups is shown to characterize the homotopy types of connected CW -complexes.

Description: A degeneration of a smooth projective curve to a strongly stable curve gives rise to a specialization map from divisors on curves to divisors on graphs. In this paper we show that this specialization behaves well under the presence of real structures. In particular we study real linear systems on graphs with a real structure and we prove results on them comparable to results in the classical theory of real curves. We also consider generalizations to metric graphs and tropical curves.

Description: We prove partial Hölder continuity, for the gradient of minimizers u W 1, p (, R N ), R n a bounded domain, of variational integrals of the form , where f is strictly quasi-convex and satisfies standard continuity and growth conditions, but where h is only a Carathéodory function of subcritical growth. The main focus is set on the presentation of a unified approach for the interior and the boundary estimates (provided that the boundary data are sufficiently regular) for all p (1, ). Furthermore, a corresponding lower-order Hölder regularity result for u is given in dimensions n ≤ p + 2 under the stronger assumption that f is strictly convex.

Description: Consider a 3-dimensional manifold N obtained by gluing a finite number of ideal hyperbolic tetrahedra via isometries along their faces. By varying the isometry type of each tetrahedron but keeping fixed the gluing pattern we define a space T of complete hyperbolic metrics on N with cone singularities along the edges of the tetrahedra. We prove that T is homeomorphic to a Euclidean space and we compute its dimension. By means of examples, we examine if the elements of T are uniquely determined by the angles around the edges of N .

Description: We consider Davenport-like series with coefficients in l 2 and discuss L 2 -convergence as well as almost-everywhere convergence. We give an example where both fail to hold. We next improve former sufficient conditions under which these convergences are true.

Description: It was proved by Beligiannis and Krause that over certain Artin algebras, there are Gorenstein flat modules which are not direct limits of finitely generated Gorenstein projective modules. That is, these algebras have no Gorenstein analogue of the Govorov–Lazard theorem. We show that, in fact, there is a large class of rings without such an analogue. Namely, let R be a commutative local noetherian ring. Then the analogue fails for R if it has a dualizing complex, is henselian, not Gorenstein, and has a finitely generated Gorenstein projective module which is not free. The proof is based on a theory of Gorenstein projective preenvelopes. We show, among other things, that the finitely generated Gorenstein projective modules form an enveloping class in mod R if and only if R is Gorenstein or has the property that each finitely generated Gorenstein projective module is free. This is analogous to a recent result on covers by Christensen, Piepmeyer, Striuli and Takahashi, and their methods are an important input to our work.

Description: We describe the structure of ‘ K -approximate subgroups’ of solvable subgroups of GL n (C), showing that they have a large nilpotent piece. By combining this with the main result of our recent paper on approximate subgroups of torsion-free nilpotent groups (E. Breuillard and B. J. Green, Approximate groups, I: the torsion-free nilpotent case , J. Inst. Math. Jussieu, to appear), we show that such approximate subgroups are efficiently controlled by nilpotent progressions.

Description: In this paper, we obtain analogues of Jørgensen's inequality for non-elementary groups of isometries of quaternionic hyperbolic n -space generated by two elements, one of which is loxodromic. Our result gives some improvement over earlier results of Kim and Markham. These results also apply to complex hyperbolic space and give improvements on results of Jiang, Kamiya and Parker. As applications, we use the quaternionic version of Jørgensen's inequalities to construct embedded collars about short, simple, closed geodesics in quaternionic hyperbolic manifolds. We show that these canonical collars are disjoint from each other. Our results give some improvement over earlier results of Markham and Parker and answer an open question that they posed.

Description: The number of unramified normal coverings of a closed Riemann surface C with groups of covering transformations isomorphic to Z 2 2 or Z 2 is well known. If C is hyperelliptic, then Horiouchi has given the explicit algebraic equations of the subset of those covers which turn out to be hyperelliptic themselves and, recently, González-Diez and the first author have obtained the remaining ones for both cases. In this paper,we obtain the algebraic equations for all unbranched degree four coverings of a hyperelliptic Riemann surface with monodromy group isomorphic to the dihedral group D 4 and such that the hyperelliptic involution lifts to the covering. As an immediate application we obtain examples of curves whose field of moduli is Q but they can not be defined over Q. The examples in this paper can be defined over a real quadratic extension of Q.

Description: Let A = 1 (B) be the semigroup algebra of B, the bicyclic semigroup. We give a resolution of (B) which simplifies the computation of the cohomology of 1 (B) dual bimodules. We apply this to the dual module (B) and show that the simplicial cohomology groups H n (A, A') vanish for n ≥ 2. Using the Connes–Tzygan exact sequence, these results are used to show that the cyclic cohomology groups HC n (A, A') vanish when n is odd and are one-dimensional when n is even ( n ≥ 2).

Description: A graph G is r-starred if, for some v V ( G ), a largest pairwise intersecting family of independent r -subsets of V ( G ) may be obtained by taking all such subsets containing v (the r - star at v ). Let G be the disjoint union of powers of cycles; Hilton and Spencer have studied the problem of determining the values of r for which G is r -starred. They conjectured that the property holds for all r , and made a weaker conjecture that this is so for the union of just two cycles. In this paper we prove the second conjecture, showing also that if G is the union of several graphs, each a power of a cycle, then G is α-starred (where α is the independence number of G ), provided that there is a homomorphism from some component of G to each of the other components.

Description: Let t ( n ) denote the n th normalized Fourier coefficient of a primitive holomorphic or Maass cusp form for the full modular group SL(2, Z). In this paper, we are concerned with the upper bound and omega results for the summatory function Asymptotic formulae for high power moments of t ( n ) are (conditionally) established.

Description: We show that the core function plays an important role in solving boundary value problems for semilinear elliptic equations.

Description: Sufficient conditions are given for discrete weighted admissibility of operators. However, counterexamples are given to show that these conditions are sharp. As a consequence of the sufficient conditions for weighted admissibility, sufficient new conditions are provided for a measure to be a Carleson measure for the weighted Dirichlet spaces (D) or for the classical Dirichlet space. Additionally, conditions are given for an analytic function to be a multiplier on (D) or a multiplier from the Hardy space H 2 (D) to a weighted Bergman space (D).

Description: The variational problem for the functional F =1/2 M ||*|| 2 is considered, where :( M , g )-〉( N , ) maps a Riemannian manifold to a symplectic manifold. This functional arises in theoretical physics as the strong coupling limit of the Faddeev–Hopf energy, and may be regarded as a symplectic analogue of the Dirichlet energy familiar from harmonic map theory. The Hopf fibration : S 3 -〉 S 2 is known to be a locally stable critical point of F . It is proved here that in fact minimizes F in its homotopy class and this result is extended to the case where S 3 is given the metric of the Berger's sphere. It is proved that if * is coclosed, F is in its homotopy class. If M is a compact Riemann surface, it is proved that every critical point of F has * coclosed. A family of holomorphic homogeneous projections into Hermitian symmetric spaces is constructed and it is proved that these too minimize F in their homotopy class.

Description: An analogue of the Montgomery–Hooley asymptotic formula is established for the variance of the number of primes in arithmetic progressions, in which the moduli are restricted to the values of a polynomial.

Description: Let p be a prime and e p (·) = e 2i·/ p . First, we make explicit the monomial sum bounds of Heath-Brown and Konyagin: where . Second, letting d = ( k , l , p – 1), we obtain the explicit binomial sum bound for any non-constant binomial ax k + bx l on Z p , by sharpening the estimate for the number of solutions of the system and . Finally, we apply the latter estimate to establish the Goresky–Klapper conjecture on the decimation of -sequences for p 〉 4.92 x 10 34 .

Description: We give a new moduli construction of the minimal resolution of the singularity of type 1/ r ;(1, a ) by introducing the Special McKay quiver. To demonstrate that our construction trumps that of the G -Hilbert scheme, we show that the induced tautological line bundles freely generate the bounded derived category of coherent sheaves on X by establishing a suitable derived equivalence. This gives a moduli construction of the Special McKay correspondence for abelian subgroups of GL(2).

Description: In 2007, assuming the Riemann hypothesis, Soundararajan (Moments of the Riemann zeta-function, Ann. Math. , 170 (2009) 981–993) proved that 0 T |(1/2 + i t )| 2k d t 〈〈 k , T (log T ) k 2 + for every positive real number k and every 〉0. In this paper, we generalize his methods to find upper bounds for shifted moments. We also obtained their lower bounds and conjectured asymptotic formulae based on the random matrix model, which is analogous to Keating and Snaith's work. These upper and lower bounds suggest that the correlation of |(1/2 + i t + iα 1 )| and |(1/2 + i t + iα 2 )| transition at |α 1 –α 2 | 1/log T . In particular, these distributions appear to be independent when |α 1 – α 2 | is much larger than 1/log T .