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1.
We investigate the relations between the cut number, and the first Betti number, of -manifolds We prove that the cut number of a ``generic' -manifold is at most This is a rather unexpected result since specific examples of -manifolds with large and are hard to construct. We also prove that for any complex semisimple Lie algebra there exists a -manifold with and Such manifolds can be explicitly constructed.

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2.
For an odd prime, we generalise the Glauberman-Thompson -nilpotency theorem (Gorenstein, 1980) to arbitrary fusion systems. We define a notion of -free fusion systems and show that if is a -free fusion system on some finite -group , then is controlled by for any Glauberman functor , generalising Glauberman's -theorem (Glauberman, 1968) to arbitrary fusion systems.

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3.
Let be a simple algebraic group over the complex numbers containing a Borel subgroup . Given a -stable ideal in the nilradical of the Lie algebra of , we define natural numbers which we call ideal exponents. We then propose two conjectures where these exponents arise, proving these conjectures in types and some other types.

When , we recover the usual exponents of by Kostant (1959), and one of our conjectures reduces to a well-known factorization of the Poincaré polynomial of the Weyl group. The other conjecture reduces to a well-known result of Arnold-Brieskorn on the factorization of the characteristic polynomial of the corresponding Coxeter hyperplane arrangement.

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4.
Let be the unital semigroup algebra of . We show that the cyclic cohomology groups vanish when is odd and are one dimensional when is even (). Using Connes' exact sequence, these results are used to show that the simplicial cohomology groups vanish for . The results obtained are extended to unital algebras for some other semigroups of .

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5.
We introduce a natural extension of the metric tensor and the Hodge star operator to the algebra of double forms to study some aspects of the structure of this algebra. These properties are then used to study new Riemannian curvature invariants, called the -curvatures. They are a generalization of the -curvature obtained by substituting the Gauss-Kronecker tensor to the Riemann curvature tensor. In particular, for , the -curvatures coincide with the H. Weyl curvature invariants, for the -curvatures are the curvatures of generalized Einstein tensors, and for the -curvatures coincide with the -curvatures.

Also, we prove that the second H. Weyl curvature invariant is nonnegative for an Einstein manifold of dimension , and it is nonpositive for a conformally flat manifold with zero scalar curvature. A similar result is proved for the higher H. Weyl curvature invariants.

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6.
Let denote the property: if is an -large set of natural numbers and is partitioned into parts, then there exists a -large subset of which is homogeneous for this partition. Here the notion of largeness is in the sense of the so-called Hardy hierarchy. We give a lower bound for in terms of for some specific .

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7.
We define , a substructure of (the lattice of classes), and show that a quotient structure of , , is isomorphic to . The result builds on the isomorphism machinery, and allows us to transfer invariant classes from to , though not, in general, orbits. Further properties of and ramifications of the isomorphism are explored, including degrees of equivalence classes and degree invariance.

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8.
Let , , be a bounded Lipschitz domain. For Laplace's equation in , we study the Dirichlet and Neumann problems with boundary data in the weighted space , where , is a fixed point on , and denotes the surface measure on . We prove that there exists such that the Dirichlet problem is uniquely solvable if , and the Neumann problem is uniquely solvable if . If is a domain, one may take . The regularity for the Dirichlet problem with data in the weighted Sobolev space is also considered. Finally we establish the weighted estimates with general weights for the Dirichlet and regularity problems.

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9.
We prove that for a general canonical curve of genus , the space of th (last) scrollar syzygies is isomorphic to the Brill-Noether locus . Schreyer has conjectured that these scrollar syzygies span the space of all th (last) syzygies of . Using Mukai varieties we prove this conjecture for genus , and .

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10.
In this paper, we study bounded solutions of on (where and sometimes ) and show that, for most 's, the weakly stable and finite Morse index solutions are quite simple. We then use this to obtain a very good understanding of the stable and bounded Morse index solutions of on with Dirichlet or Neumann boundary conditions for small .

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