We construct the generalized Witt modular Lie superalgebra of Cartan type. We give a set of generators for and show that is an extension of a subalgebra of by an ideal . Finally, we describe the homogeneous derivations of Z-degree of and we determine the derivation superalgebra of .

The paper presents a proof of the Brylinski conjecture for compact Kähler orbifolds. The result is a corollary of the foliated version of the Mathieu theorem on symplectic harmonic representations of de Rham cohomology classes. The proofs are based on the idea of representing an orbifold as the leaf space of a Riemannian foliation and on the correspondence between foliated and holonomy invariant objects for foliated manifolds.

We investigate when the boundary of a multiply connected wandering domain of an entire function is uniformly perfect. We give a general criterion implying that it is not uniformly perfect. This criterion applies in particular to examples of multiply connected wandering domains given by Baker. We also provide examples of infinitely connected wandering domains whose boundary is uniformly perfect.

Let ℱ be a family of zero-free meromorphic functions in a domain D, let h be a holomorphic function in D, and let k be a positive integer. If the function f(k)−h has at most k distinct zeros (ignoring multiplicity) in D for each f∈ℱ, then ℱ is normal in D.

In this paper we extend the notion of FP-injective modules to that of complexes and characterize such complexes. We show that some characterizations similar to those for injective complexes exist for FP-injective complexes. We also introduce and study the notion of an FP-injective dimension associated to every complex of left R-modules over an arbitrary ring. We show that there is a close connection between the FP-injective dimension of complexes and flat dimension.

We prove that a germ of a one-dimensional holomorphic foliation with a generic singularity in dimension two or three that exhibits a Lie group transverse structure in the complement of some codimension one analytic subset is logarithmic, that is, given by a system of closed meromorphic one-forms with simple poles. In the global context, we prove that a foliation by curves in a three-dimensional complex manifold with generic singularities and a Lie group transverse structure off a codimension one analytic subset is logarithmic; that is, it is given by a system of closed meromorphic one-forms with simple poles.

Let Δ be an affine building of type and let 𝔸 be its fundamental apartment. We consider the set 𝕌0 of vertices of type 0 of 𝔸 and prove that the Hecke algebra of all W0-invariant difference operators with constant coefficients acting on 𝕌0 has three generators. This property leads us to define three Laplace operators on vertices of type 0 of Δ. We prove that there exists a joint eigenspace of these operators having dimension greater than ∣W0 ∣. This implies that there exist joint eigenfunctions of the Laplacians that cannot be expressed, via the Poisson transform, in terms of a finitely additive measure on the maximal boundary Ω of Δ.

We show that the Gelfand–Kirillov dimension for modules over quantum Laurent polynomials is additive with respect to tensor products over the base field. We determine the Brookes–Groves invariant associated with a tensor product of modules. We study strongly holonomic modules and show that there are nonholonomic simple modules.

In this article we consider whether, for integrable functions on the unit ball of ℂn, the mean-value property implies (α,β)-harmonicity. We find that the answer is affirmative when 0<n+α+β≤ρ0, but is negative when n+α+β>ρ0. Here ρ0 is a constant between 11.025 and 11.069.

In this paper we show that, in the stable case, when m≥2n−1, the cohomology ring H*(Repn(m)B) of the representation variety with Borel mold Repn(m)B and are isomorphic as algebras. Here the degree of si is 2m−3 when 1≤i<n. In the unstable cases, when m≤2n−2, we also calculate the cohomology group H*(Repn(m)B) when n=3,4 . In the most exotic case, when m=2 , Rep n (2)B is homotopy equivalent to Fn (ℂ2)×PGL n (ℂ) , where Fn (ℂ2)is the configuration space of n distinct points in ℂ2. We regard Rep n (2)B as a scheme over ℤ, and show that the Picard group Pic (Rep n (2)B)of Rep n (2)B is isomorphic to ℤ/nℤ. We give an explicit generator of the Picard group.

The support varieties for the induced modules or Weyl modules for a reductive algebraic group G were computed over the first Frobenius kernel G1 by Nakano, Parshall and Vella. A natural generalization of this computation is the calculation of the support varieties of Demazure modules over the first Frobenius kernel, B1, of the Borel subgroup B. In this paper we initiate the study of such computations. We complete the entire picture for reductive groups with underlying root systems A1 and A2. Moreover, we give complete answers for Demazure modules corresponding to a particular (standard) element in the Weyl group, and provide results relating support varieties between different Demazure modules which depend on the Bruhat order.

We study the geography and botany of symplectic spin four-manifolds with abelian fundamental group. By building on the constructions of J. Park and of B. D. Park and Szabó, we can give alternative proofs and extend several results on the geography of simply connected four-manifolds to the nonsimply connected realm.

We consider the contact between curves and hyperhorospheres in hyperbolic 4-space as an application of the theory of singularities of functions. We define the osculating hyperhorosphere and the horospherical hypersurface of the curve whose singular points correspond to the locus of polar vectors of osculating hyperhorospheres of the curve. One of the main results is a generic classification of singularities of horospherical hypersurfaces of curves.

Let R be a prime ring, let I be a nonzero ideal of R and let n be a fixed positive integer. We prove that if the characteristic of R is either 0 or a prime p that is greater than 2n, then an additive map d that satisfies d(xn+1)=∑ nj=0xn−jd(x)xj for all x∈I must be a derivation.

We give an explicit description of the free objects in the quasivariety of adequate semigroups, as sets of labelled directed trees under a natural combinatorial multiplication. The morphisms of the free adequate semigroup onto the free ample semigroup and into the free inverse semigroup are realised by a combinatorial ‘folding’ operation which transforms our trees into Munn trees. We use these results to show that free adequate semigroups and monoids are 𝒥-trivial and never finitely generated as semigroups, and that those which are finitely generated as (2,1,1)-algebras have decidable word problem.

We characterise regular bipartite locally primitive graphs of order 2pe, where p is prime. We show that either p=2 (this case is known by previous work), or the graph is a binormal Cayley graph or a normal cover of one of the basic locally primitive graphs; these are described in detail.

We study a canonical spanning surface obtained from a knot or link diagram, depending on a given Kauffman state. We give a sufficient condition for the surface to be essential. By using the essential surface, we can deduce the triviality and splittability of a knot or link from its diagrams. This has been done on the extended knot or link class that includes all semiadequate, homogeneous knots and links, and most algebraic knots and links. In order to prove the main theorem, we extend Gabai’s Murasugi sum theorem to the case of nonorientable spanning surfaces.

We give a new type of geometric construction that allows for the construction of families of quintic irrationalities, and is quite rich in algebraic properties. This construction may be considered as our first attempt at characterizing points constructible with compass and twice-notched ruler, a problem that seems to have been known in some form for more than two millennia.

We show that for any fixed base a, a positive proportion of primes become composite after any one of their digits in the base a expansion is altered; the case where a=2 has already been established by Cohen and Selfridge [‘Not every number is the sum or difference of two prime powers’, Math. Comput.29 (1975), 79–81] and Sun [‘On integers not of the form ±pa±qb’, Proc. Amer. Math. Soc.128 (2000), 997–1002], using some covering congruence ideas of Erdős. Our method is slightly different, using a partially covering set of congruences followed by an application of the Selberg sieve upper bound. As a consequence, it is not always possible to test whether a number is prime from its base a expansion without reading all of its digits. We also present some slight generalisations of these results.

We study the geometric properties of a base manifold whose unit tangent sphere bundle, equipped with the standard contact metric structure, is H-contact. We prove that a necessary and sufficient condition for the unit tangent sphere bundle of a four-dimensional Riemannian manifold to be H-contact is that the base manifold is 2-stein.