Recent results of Jacobson and Barnes indicate that Lie, Jordan and alternative algebras may have a common Cartan theory. In this thesis, we show this is indeed the case. We also show that for certain classes of non-associative algebras, called E-classes, that possess an Engel function, a general Cartan theory is possible. In Chapter One, a generalization of nilpotence and solvability is introduced that permits our Cartan theory for E-classes. In Chapter Two, we construct Cartan subalgebras for alternative algebras based on a given Engel function. Jacobson's Cartan theory for Jordan algebras is given in Chapter Three along with our extensions of his results. We point out that the Engel function for alternative algebras and Jordan algebras coincides, and may be used to give the classical Cartan theory for Lie algebras Commutative power associative algebras are discussed in Chapter Four, and some results are obtained. / Science, Faculty of / Mathematics, Department of / Graduate

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Definition: Let k be an algebraically closed field. An algebraic monoid is a triple (E,m,l) such that E is an algebraic variety defined over k, m : ExE → E is an associative morphism and 1 € E is a two—sided unit for m. The object of this thesis is to expose several fundamental topics in the theory of algebraic monoids. My results may be divided into three types; general theory of irreducible affine monoids, structure and classification of semi—simple rank one reductive monoids, and theory of general monoid varieties (not necessarily affine). I General Theory of Affine Monoids II Reductive Monoids of Semi-simple Rank One III General Monoid Varieties [Please see document for entire abstract] / Science, Faculty of / Mathematics, Department of / Graduate

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This licentiate thesis consists of two papers related to configuration spaces of points. In paper I a general formula for the Euler characteristic of configuration spaces on any topologically stratified space X is obtained in terms of geometric and combinatorial data about the strata. More generally this paper provides a formula for the Euler characteristic of the cohomology with compact support of these configuration spaces with coefficients in a constructible complex of sheaves K on X. The formula for the classical Euler characteristic is then obtained by taking K to be the dualizing complex of X. This formula generalizes similar results about configuration spaces on a manifold or on a simplicial complex, as well as another formula for any Hausdorff space X when the complex of sheaves K is trivial. In paper II we study the cohomology with compact support of configuration spaces on a wedge sum of spheres X, with rational coefficients. We prove that these cohomology groups are the coefficients of an analytic functor computing the Hochschild--Pirashvili homology of X with certain coefficients. Moreover, we prove that, up to a filtration, these same cohomology groups are a polynomial functor in the reduced cohomology of X, with coefficients not depending on X. Contrasting the information provided by two different models we are able to partially compute these coefficients, and in particular we obtained a complete answer for configurations of at most 10 points. The coefficients thus obtained can be used to compute the weight 0 part of the cohomology with compact support of the moduli space M_{2,n}.

Denna uppsats behandlar en matematisk sats, oftast benämnd som Napoleons sats. I uppsatsen presenteras och bevisas denna sats med hjälp av ett flertal olika bevismetoder. Det presenteras även bevis för att satsen kan utvidgas till att gälla för fler geometriska objekt än en triangel, vilken är det objekt som satsen ursprungligen utgår från. Texten inleds med en kort redogörelse för historien bakom satsen.

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