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1.
The main result of the first part of the paper is a generalization of the classical result of Menger-Urysohn : . Theorem. Suppose are subsets of a metrizable space and and are CW complexes. If is an absolute extensor for and is an absolute extensor for , then the join is an absolute extensor for .

As an application we prove the following analogue of the Menger-Urysohn Theorem for cohomological dimension: Theorem. Suppose are subsets of a metrizable space. Then

for any ring with unity and

for any abelian group .

The second part of the paper is devoted to the question of existence of universal spaces: Theorem. Suppose is a sequence of CW complexes homotopy dominated by finite CW complexes. Then
a.
Given a separable, metrizable space such that , , there exists a metrizable compactification of such that , .
b.
There is a universal space of the class of all compact metrizable spaces such that for all .
c.
There is a completely metrizable and separable space such that for all with the property that any completely metrizable and separable space with for all embeds in as a closed subset.

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2.
Let and be anisotropic quadratic forms over a field of characteristic not . Their function fields and are said to be equivalent (over ) if and are isotropic. We consider the case where and is divisible by an -fold Pfister form. We determine those forms for which becomes isotropic over if , and provide partial results for . These results imply that if and are equivalent and , then is similar to over . This together with already known results yields that if is of height and degree or , and if , then and are equivalent iff and are isomorphic over .

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3.
Let be the evaluation subgroup as defined by Gottlieb. Assume the Hurewicz map is non-trivial and is a field. We will prove: if is a Poincaré complex oriented in -coefficient, all the characteristic numbers of in -coefficient vanish. Similarly, if and is a -Poincaré complex, then all the mod Wu numbers vanish. We will also show that the existence of a non-trivial derivation on with some suitable conditions implies vanishing of mod Wu numbers.

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4.
Let be the set of holomorphic functions on the unit disc with and Dirichlet integral not exceeding one, and let be the set of complex-valued harmonic functions on the unit disc with and Dirichlet integral not exceeding one. For a (semi)continuous function , define the nonlinear functional on or by . We study the existence and regularity of extremal functions for these functionals, as well as the weak semicontinuity properties of the functionals. We also state a number of open problems.

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5.
A Banach space is known to be Arens regular if every continuous linear mapping from to is weakly compact. Let be an open subset of , and let denote the algebra of analytic functions on which are bounded on bounded subsets of lying at a positive distance from the boundary of We endow with the usual Fréchet topology. denotes the set of continuous homomorphisms . We study the relation between the Arens regularity of the space and the structure of .

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6.
We classify those finite simple groups whose Brauer graph (or decomposition matrix) has a -block with defect 0, completing an investigation of many authors. The only finite simple groups whose defect zero blocks remained unclassified were the alternating groups . Here we show that these all have a -block with defect 0 for every prime . This follows from proving the same result for every symmetric group , which in turn follows as a consequence of the -core partition conjecture, that every non-negative integer possesses at least one -core partition, for any . For , we reduce this problem to Lagrange's Theorem that every non-negative integer can be written as the sum of four squares. The only case with , that was not covered in previous work, was the case . This we prove with a very different argument, by interpreting the generating function for -core partitions in terms of modular forms, and then controlling the size of the coefficients using Deligne's Theorem (née the Weil Conjectures). We also consider congruences for the number of -blocks of , proving a conjecture of Garvan, that establishes certain multiplicative congruences when . By using a result of Serre concerning the divisibility of coefficients of modular forms, we show that for any given prime and positive integer , the number of blocks with defect 0 in is a multiple of for almost all . We also establish that any given prime divides the number of modularly irreducible representations of , for almost all .

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7.
Let be linearly independent positive functions in , let be the vector subspace generated by the and let denote the curve of determined by the function , where . We establish that is a vector lattice under the induced ordering from if and only if there exists a convex polygon of with vertices containing the curve and having its vertices in the closure of the range of . We also present an algorithm which determines whether or not is a vector lattice and in case is a vector lattice it constructs a positive basis of . The results are also shown to be valid for general normed vector lattices.

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8.
Let be a smooth projective variety. Every embedding is the linear projection of an embedding defined by a complete linear system. In this paper the geometry of such not necessarily complete embeddings is investigated in the special case of abelian varieites. To be more precise, the properties of complete embeddings are extended to arbitrary embeddings, and criteria for these properties to be satisfied are elaborated. These results are applied to abelian varieties. The main result is: Let be a general polarized abelian variety of type and , such that is even, and . The general subvector space of codimension satisfies the property .

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9.
Let be a fractional Brownian motion of index in If , then there exists a positive finite constant such that with probability 1,

where and - is the -packing measure of .

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10.
Let be a finite subgroup of is a field of characteristic and acting by linear substitution on a relatively free algebra of a variety of unitary associative algebras. The algebra of invariants is relatively free if and only if is a pseudo-reflection group and contains the polynomial

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