Browsing by Subject "Algebraic topology"
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Item Item The Goodwillie tower of free augmented algebras over connective ring spectra(2014-12) Pancia, Matthew; Blumberg, Andrew J.Let R be a connective ring spectrum and let M be an R-bimodule. In this paper we prove several results that relate the K-theory of R⋉M and T[superscript M, subscript R] to a “topological Witt vectors” construction W(R; M), where R ⋉ M is the square-zero extension of R by M and T [superscript M, subscript R] is the tensor algebra on M. Our main results include a desciption of the Taylor tower of K(R ⋉ (−)) and the derived functor of K̃(TR(−)) on the category of R-bimodules in terms of the Taylor tower of W(R;−). W(R;−) has an easily described Taylor tower, given explicitly by Lindenstrauss and McCarthy in [17]. Our main results serve as generalizations of the results for discrete rings in [17, 18] and also extend the computations by Hesselholt and Madsen [15] showing that π₀(TR(R; p)) is isomorphic to the p-typical Witt vectors over R when R a commutative ring.Item State sums in two dimensional fully extended topological field theories(2011-05) Davidovich, Orit; Freed, Daniel S.; Ben-Zvi, David; Distler, Jacques; Reid, Alan; Rodriguez-Villegas, FernandoA 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.Item The Kauffman bracket Skein algebra of the punctured Torus(2012-08) Cho, Jea Pil; Gelca, Razvan; Lewis, Wayne; Toda, Magdalena D.This dissertation studies the Kauffman bracket skein algebra of the punctured torus. The first chapter contains the historical background on the Kauffman bracket skein algebra and its applications. The second chapter contains the multiplication rule for the Kauffman bracket skein algebra of the cylinder over the punctured torus. The explicit formula for the multiplicative rule for the case of the Kauffman bracket skein algebra of the cylinder over torus was found by Frohman and Gelca. In this work, we try to extend their result to the torus with a puncture. The punctured torus has a multiplicative structure of the Kauffman bracket skein algebra that is considerably more complicated than that of the torus, and we ilustrate this with examples for which the crossing number is small. In Chapter 3, we describe the action of the Kauffman bracket skein algebra on certain vector spaces that arise as relative Kauffman bracket skein modules of tangles in the torus. We analyze several particular cases, then we derive the general formula for the action of the Kauffman bracket skein algebra on the corresponding skein modules by using the geometric properties of the Jones-Wenzl idempotent, which is the main result of the dissertation. In Chapter 4, we show how the Reshetikhin-Turaev representation of the mapping class group of the punctured torus can be computed from the representation of the Kauffman bracket skein algebra, and based on this we derive explicit formulas for the matrices of the generators of the mapping class group of the punctured torus.