published_or_final_version / Mathematics / Master / Master of Philosophy

Thesis (M.Sc. (Mathematics)) --University of Limpopo, 2007 / The category Loc of locales and continuous maps is dual to the category Frm of frames and frame homomorphisms. Regular subobjects of a locale A are elements of the form Aj = fj : A ! A j j(a) = ag: The subobjects of this form are called sublocales of A. They arise from the lattice OX of open sets of a topological space X in a natural way. The right adjoint of a frame homomorphism maps closed (dually, open) sublocales to closed (dually, open) sublocales. Simple coverings and separated frames are studied and conditions under which they are closed (or open) are those that are related to coequalizers are shown. Under suitable conditions, simple coverings are regular epimorphisms. Extremal epimorphisms and strong epimorphisms in the setting of locales are studied and it is shown that strong epimorphisms compose. In the category Loc of locales and continuous maps, closed surjections are regular epimorphisms at least for those surjections with subfit domains. / National Research Foundation

Finite determinacy theorems are generalized to the class of C('k) maps where k (LESSTHEQ) (INFIN) is sufficiently large. For these maps, the concept of a combinatorial unfolding is defined. In the case k = (INFIN), the infinitesimal characterization of a combinatorial unfolding coincides with that of a universal unfolding. By representing a function by a polynomial in which the coefficients depend on parameters, each change of co-ordinates required in the demonstrations is obtained by variation of the coefficients in a polynomial.

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Thesis--Bonn. / Includes bibliographical references (p. 58-59).

Thesis--Bonn. / Vita. Includes bibliographical references (p. 91-94).

Thesis--Bonn. / Includes bibliographical references (p. 58-59).

Thesis--Bonn. / Vita. Includes bibliographical references (p. 91-94).

Let X and Y be topological spaces and C and D semigroups under composition of maps from X to X and Y to Y respectively. Let H be an isomorphism from C to D; it is shown that if both C and D contain the constant maps then there exists a bijection h from X to Y such that H(f) = h∘f∘h⁻¹, VfɛC. We investigate this situation and find sufficient conditions for this h to be a homeomorphism. In this regard we study the familiar semigroups of continuous, closed, and connected maps. An auxiliary problem is the case when C = D and H is an automorphism of D), We then ask when is every automorphism is inner. The question is answered for certain particular semigroups; e.g., the semigroup of differentiable maps on the reals has the property that all automorphisms are inner. / Science, Faculty of / Mathematics, Department of / Graduate

Given a triple (𝑋,𝘌,𝘴) of a smooth projective variety, a rank 𝘳 vector bundle and a regular section, we construct a moduli of stable maps to 𝑋 with fields together with a cosection localized virtual class. We show the class coincides up to a sign with the virtual fundamental class on the moduli space of stable maps to the vanishing locus 𝘡 of 𝘴. We show that this gives a generalization of the Quantum Lefschetz hyperplane principle, which relates the virtual classes of the moduli of stable maps to 𝑋 and that of the moduli of stable maps to 𝘡 if the bundle 𝘌 is convex. We further generalize this result by considering (𝒳,ɛ,s) where 𝒳is a smooth Deligne--Mumford stack with projective coarse moduli space. In this setting, we can construct a moduli space of twisted stable maps to 𝒳with fields. This moduli space will have (possibly disconnected) components of constant virtual dimension indexed by 𝓃-tuples of components of the inertia stack of 𝒳. We show that its cosection localized virtual class on each component agrees up to a sign with the virtual fundamental class of a corresponding component of the moduli of twisted stable maps to ƶ=s=0. This generalizes similar comparison results of Chang--Li, Kim--Oh and Chang--Li and presents a different approach from Chen--Janda--Webb.