No description available.

No description available.

In algebraic topology, we work to classify objects. My research aims to build a better understanding of one important notion of classification of differentiable manifolds called cobordism. Cobordism is an equivalence relation, and the equivalence classes in cobordism form a graded ring, with operations disjoint union and Cartesian product. My dissertation studies this graded ring in two ways: 1. by attempting to find preferred class representatives for each class in the ring. 2. by computing the image of the ring under an interesting ring homomorphism called the Witten Genus.

A long-standing problem in cobordism theory has been to find convenient manifolds to represent cobordism classes. For example, in the late 1950's, Hirzebruch asked which complex cobordism classes can be represented by smooth connected algebraic varieties. This question is still open. Progress can be made on this and related problems by studying certain convenient connected algebraic varieties, namely smooth projective toric varieties. The primary focus of this dissertation is to determine which complex cobordism classes can be represented by smooth projective toric varieties. A complete answer is given up to dimension six, and a partial answer is described in dimension eight. In addition, the role of smooth projective toric varieties in the polynomial ring structure of complex cobordism is examined. More specifically, smooth projective toric varieties are constructed as polynomial ring generators in most dimensions, and evidence is presented suggesting that a smooth projective toric variety can be chosen as a polynomial generator in every dimension. Finally, toric varieties with an additional fiber bundle structure are used to study some manifolds in oriented cobordism. In particular, manifolds with certain fiber bundle structures are shown to all be cobordant to zero in the oriented cobordism ring.

Symplectic cohomology is an algebraic invariant of filled symplectic cobordisms that encodes dynamical information. In this thesis we define a modified symplectic cohomology theory, called action-completed symplectic cohomology, that exhibits quantitative behavior. We illustrate the non-trivial nature of this invariant by computing it for annulus subbundles of line bundles over complex projective space. The proof relies on understanding the symplectic cohomology of the complex fibers and the quantum cohomology of the projective base. We connect this result to mirror symmetry and prove a non-vanishing result in the presence of Lagrangian submanifolds with non-vanishing Floer homology. The proof uses Lagrangian quantum cohomology in conjunction with a closed-open map.

Dans quelles conditions une application entre variétés différentiables est-elle homotope à un plongement lisse ? L'objet de la thèse est de compléter les obstructions rationnelles déjà connues, de façon à réduire le problème initial de topologie différentiable à un problème de calcul algébrique. Le théorème principal de la thèse permet de construire un plongement entre variétés lisses dans une classe d'homotopie rationnelle d'une application donnée, lorsque le problème algébrique a une solution. Plusieurs cas génériques de réalisabilité sont présentés, ainsi que des exemples mettant en évidence les nouvelles obstructions au plongement. Enfin, l'utilisation des techniques de chirurgie plongée dans le rang métastable aboutit à de nouveaux théorèmes de réalisation de plongements à cohomologie rationnelle près dans une variété fixée. / When does there exist a smooth embedding in a homotopy class of map between differentiable manifolds ? Rational homotopy theory provides computation machinery to such questions in differential topology. The purpose of this thesis is the completion of rational obstructions which prevent a map from being an embedding. More precisely, we show that a solution to the underlying algebraic problem gives rise to a smooth embedding with the same rational invariants. Several generic cases of realizability are detailed, as well as some examples which illustrate our new obstructions. We also use techniques of embedded surgery in metastable range, in order to state a theorem about realizability of an embedding up to rational cohomolgy into a fixed manifold.

Diss. (sammanfattning) Uppsala : Univ., 2003. / Härtill 3 uppsatser.

Thesis (Ph.D.)--Michigan State University. Mathematics, 2008. / Title from PDF t.p. (viewed on July 24, 2009) Includes bibliographical references (p. 40-41). Also issued in print.

Typescript. Includes vita and abstract. Includes bibliographical references (leaves 163-164) Also available online in Scholars' Bank; and in ProQuest, free to University of Oregon users.

A state sum is an expression approximating the partition function of a d-dimensional field theory on a closed d-manifold from a triangulation of that manifold. To consider state sums in completely local 2-dimensional topological field theories (TFT's), we introduce a mechanism for incorporating triangulations of surfaces into the cobordism ([infinity],2)-category. This serves to produce a state sum formula for any fully extended 2-dimensional TFT possibly with extra structure. We then follow the Cobordism Hypothesis in classifying fully extended 2-dimensional G-equivariant TFT's for a finite group G. These are oriented theories in which bordisms are equipped with principal G-bundles. Combining the mechanism mentioned above with our classification results, we derive Turaev's state sum formula for such theories. / text