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1.
Let be a simply connected complex Lie group with Lie algebra , a real form of , and the analytic subgroup of corresponding to . The symmetric space together with a -invariant partial order is referred to as an Olshanskii space. In a previous paper we constructed a family of integral spherical functions on the positive domain of . In this paper we determine all of those spherical functions on which are positive definite in a certain sense.

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2.
We work in the stable homotopy category of -complete connective spectra having mod homology of finite type. means cohomology with coefficients, and is a left module over the Steenrod algebra .

A spectrum is called spacelike if it is a wedge summand of a suspension spectrum, and a spectrum satisfies the Brown-Gitler property if the natural map is onto, for all spacelike .

It is known that there exist spectra satisfying the Brown-Gitler property, and with isomorphic to the injective envelope of in the category of unstable -modules.

Call a spectrum standard if it is a wedge of spectra of the form , where is a stable wedge summand of the classifying space of some elementary abelian -group. Such spectra have -injective cohomology, and all -injectives appear in this way.

Working directly with the two properties of stated above, we clarify and extend earlier work by many people on Brown-Gitler spectra. Our main theorem is that, if is a spectrum with -injective cohomology, the following conditions are equivalent:

(A) there exist a spectrum whose cohomology is a reduced -injective and a map that is epic in cohomology, (B) there exist a spacelike spectrum and a map that is epic in cohomology, (C) is monic in cohomology, (D) satisfies the Brown-Gitler property, (E) is spacelike, (F) is standard. ( is reduced if it has no nontrivial submodule which is a suspension.)

As an application, we prove that the Snaith summands of are Brown-Gitler spectra-a new result for the most interesting summands at odd primes. Another application combines the theorem with the second author's work on the Whitehead conjecture.

Of independent interest, enroute to proving that (B) implies (C), we prove that the homology suspension has the following property: if an -connected space admits a map to an -fold suspension that is monic in mod homology, then is onto in mod homology.

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3.
4.
5.
We classify all complex representations of the automorphism group of the free group of dimension Among those representations is a new representation of dimension which does not vanish on the group of inner automorphisms.

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6.
We prove that a ()-connected map from a compact PL -manifold to a generalized -manifold with the disjoint disks property, , is homotopic to a tame embedding. There is also a controlled version of this result, as well as a version for noncompact and proper maps that are properly ()-connected. The techniques developed lead to a general position result for arbitrary maps , , and a Whitney trick for separating submanifolds of that have intersection number 0, analogous to the well-known results when is a manifold.

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7.
At a given point , a convex function is differentiable in a certain subspace (the subspace along which has 0-breadth). This property opens the way to defining a suitably restricted second derivative of at . We do this via an intermediate function, convex on . We call this function the -Lagrangian; it coincides with the ordinary Lagrangian in composite cases: exact penalty, semidefinite programming. Also, we use this new theory to design a conceptual pattern for superlinearly convergent minimization algorithms. Finally, we establish a connection with the Moreau-Yosida regularization.

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8.
Let be an affine algebraic variety over (or any other real closed field ). We ask when it is true that every positive semidefinite (psd) polynomial function on is a sum of squares (sos). We show that for the answer is always negative if has a real point. Also, if is a smooth non-rational curve all of whose points at infinity are real, the answer is again negative. The same holds if is a smooth surface with only real divisors at infinity. The ``compact' case is harder. We completely settle the case of smooth curves of genus : If such a curve has a complex point at infinity, then every psd function is sos, provided the field is archimedean. If is not archimedean, there are counter-examples of genus .

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9.
The ``spin' L-function of an automorphic representation of is an Euler product of degree associated with the spin representation of the L-group . If or , and the automorphic representation is generic in the sense of having a Whittaker model, the analytic properties of these L-functions are studied by the Rankin-Selberg method.

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10.
We classify all pairs of reductive maximal connected subgroups of a classical algebraic group that have a dense double coset in . Using this, we show that for an arbitrary pair of reductive subgroups of a reductive group satisfying a certain mild technical condition, there is a dense -double coset in precisely when is a factorization.

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