Complex flat manifolds and their moduli spaces

Although most of the work in this thesis is algebraic, its starting point and examples come from differential topology and geometry. As essential background Chapter I includes sections which describe involuted algebras and Albert's classification of rational positively involuted algebras; representations of finite groups over fields; groups that embed in division algebras and Amitsur's classification. The differential topology of the thesis arises in the study of how one can give a flat compact Riemannian manifold a Kählerian/projective structure. In Chapter II we outline some differential geometry and the theory of Flat Riemannian manifolds, particularly holonomy, we include a description Charlap's classification. Also in this chapter we give a simple proof of a bound for the minimal dimension for a flat compact Riemannian manifold with predescribed holonomy (m(Φ) ≤ |Φ|); the proof requires Amitsur's classification. The notion of complex structures on real manifolds is introduced in Chapter III. Some work on Riemann matrices is required and given. In Chapter IV we parametrise the set of complex structures which give a (real) flat compact Riemannian manifold a Kählerian structure. A parametrisation is also given for complex structures which give a projective structure for certain manifolds with a fixed polarisation. This involves Siegel's generalised upper half plane. In Chapter V we give some examples and give the above parametrisations for certain holonomy groups and representations. Some of the working involves integral representations and cohomology of finite groups. Finally, the subject of Chapter VI is essentially independent of previous chapters in respect to the work we have done. The chapter concerns subgroups of a product of surface groups, by which we mean the fundamental group of an oriented surface of positive genus. We consider the simultaneous equivalence relations of commensurability and automorphism. In particular, we show that, in a product of two surface groups in which one factor has genus greater that one, there are infinitely many equivalence classes of normal subdirect products

Satisfiability in multi-valued circuits

Satisfiability of Boolean circuits is among the most known and important problems in theoretical computer science. This problem is NP-complete in general but becomes polynomial time when restricted either to monotone gates or linear gates. We go outside Boolean realm and consider circuits built of any fixed set of gates on an arbitrary large finite domain. From the complexity point of view this is strictly connected with the problems of solving equations (or systems of equations) over finite algebras. The research reported in this work was motivated by a desire to know for which finite algebras $\mathbf A$ there is a polynomial time algorithm that decides if an equation over $\mathbf A$ has a solution. We are also looking for polynomial time algorithms that decide if two circuits over a finite algebra compute the same function. Although we have not managed to solve these problems in the most general setting we have obtained such a characterization for a very broad class of algebras from congruence modular varieties. This class includes most known and well-studied algebras such as groups, rings, modules (and their generalizations like quasigroups, loops, near-rings, nonassociative rings, Lie algebras), lattices (and their extensions like Boolean algebras, Heyting algebras or other algebras connected with multi-valued logics including MV-algebras). This paper seems to be the first systematic study of the computational complexity of satisfiability of non-Boolean circuits and solving equations over finite algebras. The characterization results provided by the paper is given in terms of nice structural properties of algebras for which the problems are solvable in polynomial time.Comment: 50 page