vii, 54 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / We investigate the existence of a stable homotopy category (SHC) associated to the category of p -complete abelian groups [Special characters omitted]. First we examine [Special characters omitted] and prove [Special characters omitted] satisfies all but one of the axioms of an abelian category. The connections between an SHC and homology functors are then exploited to draw conclusions about possible SHC structures for [Special characters omitted]. In particular, let [Special characters omitted] denote the category whose objects are chain complexes of [Special characters omitted] and morphisms are chain homotopy classes of maps. We show that any homology functor from any subcategory of [Special characters omitted] containing the p-adic integers and satisfying the axioms of an SHC will not agree with standard homology on free, finitely generated (as modules over the p -adic integers) chain complexes. Explicit examples of common functors are included to highlight troubles that arrise when working with [Special characters omitted]. We make some first attempts at classifying small objects in [Special characters omitted]. / Committee in charge: Hal Sadofsky, Chairperson, Mathematics; Boris Botvinnik, Member, Mathematics; Daniel Dugger, Member, Mathematics; Sergey Yuzvinsky, Member, Mathematics; Elizabeth Reis, Outside Member, Womens and Gender Studies

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Bibliography: pages 90-94. / Ordered spaces were introduced by Leopoldo Nachbin [1948 a, b, c, 1950, 1965]. We will be primarily concerned with completely regular ordered spaces, because they are precisely those ordered spaces which admit quasi-uniform structures. A recent and convenient study of these spaces is in the book by P. Fletcher and W.F. Lindgren [1982]. In this thesis we consider functorial quasi-uniformities over (partially) ordered spaces. The functorial methods which we use were developed by Brummer [1971, 1977, 1979, 1982] and Brummer and Hager [1984, 1987] in the context of functorial uniformities over completely regular topological spaces, and of functorial quasi-uniformities over pairwise. completely regular bitopological spaces. We obtain results which are to a large extent analogous to results in those papers. We also introduce some functors which relate our functorial quasi-uniformities to the structures studied by Brummer and others (e.g. Salbany [1984]).

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In the first part of the dissertation, we construct two-dimensional TQFTs which categorify the evaluations of circles in Kuperberg’s 𝐵₂ spider. We give a purely combinatorial evaluation formula for these TQFTs and show that it is compatible with the trace map on the corresponding commutative Frobenius algebras. Furthermore, we develop a theory of Θ-foams and their combinatorial evaluations to lift the ungraded evaluation of the Θ-web, thus paving a way for categorifying 𝐵₂ webs to 𝐵₂ foams. In the second part of the dissertation, we study the calculus of unoriented 𝔰𝔩₃ webs and foams. We focus on webs with a small number of boundary points. We obtain reducible collections and consider bilinear forms on these collections given by pairings of webs. We give web categories stable under the action of certain endofunctors and derive relations between compositions of these endofunctors.

In this thesis, we discuss the formulation of variational problems on supermanifolds. Supermanifolds incorporate bosonic as well as fermionic degrees of freedom. Fermionic fields take values in the odd part of an appropriate Grassmann algebra and are thus showing an anticommutative behaviour. However, a systematic treatment of these Grassmann parameters requires a description of spaces as functors, e.g. from the category of Grassmann algberas into the category of sets (or topological spaces, manifolds). After an introduction to the general ideas of this approach, we use it to give a description of the resulting supermanifolds of fields/maps. We show that each map is uniquely characterized by a family of differential operators of appropriate order. Moreover, we demonstrate that each of this maps is uniquely characterized by its component fields, i.e. by the coefficients in a Taylor expansion w.r.t. the odd coordinates. In general, the component fields are only locally defined. We present a way how to circumvent this limitation. In fact, by enlarging the supermanifold in question, we show that it is possible to work with globally defined components. We eventually use this formalism to study variational problems. More precisely, we study a super version of the geodesic and a generalization of harmonic maps to supermanifolds. Equations of motion are derived from an energy functional and we show how to decompose them into components. Finally, in special cases, we can prove the existence of critical points by reducing the problem to equations from ordinary geometric analysis. After solving these component equations, it is possible to show that their solutions give rise to critical points in the functor spaces of fields. / In dieser Dissertation wird die Formulierung von Variationsproblemen auf Supermannigfaltigkeiten diskutiert. Supermannigfaltigkeiten enthalten sowohl bosonische als auch fermionische Freiheitsgrade. Fermionische Felder nehmen Werte im ungeraden Teil einer Grassmannalgebra an, sie antikommutieren deshalb untereinander. Eine systematische Behandlung dieser Grassmann-Parameter erfordert jedoch die Beschreibung von Räumen durch Funktoren, z.B. von der Kategorie der Grassmannalgebren in diejenige der Mengen (der topologischen Räume, Mannigfaltigkeiten, ...). Nach einer Einführung in das allgemeine Konzept dieses Zugangs verwenden wir es um eine Beschreibung der resultierenden Supermannigfaltigkeit der Felder bzw. Abbildungen anzugeben. Wir zeigen, dass jede Abbildung eindeutig durch eine Familie von Differentialoperatoren geeigneter Ordnung charakterisiert wird. Darüber hinaus beweisen wir, dass jede solche Abbildung eineindeutig durch ihre Komponentenfelder, d.h. durch die Koeffizienten einer Taylorentwickelung bzgl. von ungeraden Koordinaten bestimmt ist. Im Allgemeinen sind Komponentenfelder nur lokal definiert. Wir stellen einen Weg vor, der diese Einschränkung umgeht: Durch das Vergrößern der betreffenden Supermannigfaltigkeit ist es immer möglich, mit globalen Koordinaten zu arbeiten. Schließlich wenden wir diesen Formalismus an, um Variationsprobleme zu untersuchen, genauer betrachten wir eine super-Version der Geodäte und eine Verallgemeinerung von harmonischen Abbildungen auf Supermannigfaltigkeiten. Bewegungsgleichungen werden von Energiefunktionalen abgeleitet und wir zeigen, wie sie sich in Komponenten zerlegen lassen. Schließlich kann in Spezialfällen die Existenz von kritischen Punkten gezeigt werden, indem das Problem auf Gleichungen der gewöhnlichen geometrischen Analysis reduziert wird. Es kann dann gezeigt werden, dass die Lösungen dieser Gleichungen sich zu kritischen Punkten im betreffenden Funktor-Raum der Felder zusammensetzt.

We consider right modules over a ring, as models of a first order theory. We explorethe definable sets and the definable bijections between them. We employ the notionsof Euler characteristic and Grothendieck ring for a first order structure, introduced byJ. Krajicek and T. Scanlon in [24]. The Grothendieck ring is an algebraic structurethat captures certain properties of a model and its category of definable sets.If M is a module over a product of rings A and B, then M has a decomposition into a direct sum of an A-module and a B-module. Theorem 3.5.1 states that then the Grothendieck ring of M is the tensor product of the Grothendieck rings of the summands.Theorem 4.3.1 states that the Grothendieck ring of every infinite module over afield or skew field is isomorphic to Z[X].Proposition 5.2.4 states that for an elementary extension of models of anytheory, the elementary embedding induces an embedding of the corresponding Grothendieck rings. Theorem 5.3.1 is that for an elementary embedding of modules, we have the stronger result that the embedding induces an isomorphism of Grothendieck rings.We define a model-theoretic Grothendieck ring of the category Mod-R and explorethe relationship between this ring and the Grothendieck rings of general right R-modules. The category of pp-imaginaries, shown by K. Burke in [7] to be equivalentto the subcategory of finitely presented functors in (mod-R; Ab), provides a functorial approach to studying the generators of theGrothendieck rings of R-modules. It is shown in Theorem 6.3.5 that whenever R andS are Morita equivalent rings, the rings Grothendieck rings of the module categories Mod-R and Mod-S are isomorphic.Combining results from previous chapters, we derive Theorem 7.2.1 saying that theGrothendieck ring of any module over a semisimple ring is isomorphic to a polynomialring Z[X1,...,Xn] for some n.

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No description available.