Mathematics - Ph.D. / Sc.D.
http://hdl.handle.net/1721.1/7680
2015-11-07T15:03:03ZThe v₁-periodic part of the Adams spectral sequence at an odd prime/
http://hdl.handle.net/1721.1/99328
The v₁-periodic part of the Adams spectral sequence at an odd prime/
Andrews, Michael Joseph, Ph. D. Massachusetts Institute of Technology
We tell the story of the stable homotopy groups of spheres for odd primes at chromatic height 1 through the lens of the Adams spectral sequence. We find the "dancers to a discordant system." We calculate a Bockstein spectral sequence which converges to the 1-line of the chromatic spectral sequence for the odd primary Adams E₂-page. Furthermore, we calculate the associated algebraic Novikov spectral sequence converging to the 1-line of the BP chromatic spectral sequence. This result is also viewed as the calculation of a direct limit of localized modified Adams spectral sequences converging to the homotopy of the v1 -periodic sphere spectrum. As a consequence of this work, we obtain a thorough understanding of a collection of q₀-towers on the Adams E₂-page and we obtain information about the differentials between these towers. Moreover, above a line of slope 1/(p²-p-1) we can completely describe the E₂ and E₃ -pages of the mod p Adams spectral sequence, which accounts for almost all the spectral sequence in this range.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015.; In title on title-page, "v" is italicized, and "1" is subscript. Cataloged from PDF version of thesis.; Includes bibliographical references (pages 139-140).
2015-01-01T00:00:00ZUnstable operations in the Bousfield-Kan spectral sequence for simplicial commutative FF₂-algebras
http://hdl.handle.net/1721.1/99327
Unstable operations in the Bousfield-Kan spectral sequence for simplicial commutative FF₂-algebras
Donovan, Michael Jack
In this thesis we study the Bousfield-Kan spectral sequence (BKSS) in the Quillen model category sCom of simplicial commutative FF₂ -algebras. We develop a theory of unstable operations for this BKSS and relate these operations with the known unstable operations on the homotopy of the target. We also prove a completeness theorem and a vanishing line theorem which, together, show that the BKSS for a connected object of sCom converges strongly to the homotopy of that object. We approach the computation of the BKSS by deriving a composite functor spectral sequence (CFSS) which converges to the BKSS E2 -page. In fact, we generalize the construction of this CFSS to yield an infinite sequence of CFSSs, with each converging to the E2-page of the previous. We equip each of these CFSSs with a theory of unstable spectral sequence operations, after establishing the necessary chain-level structure on the resolutions defining the CFSSs. This technique may also yield operations on Blanc and Stover's generalized Grothendieck spectral sequences in other settings. We are able to compute the Bousfield-Kan E2-page in the most fundamental case, that of a connected sphere in sCom, using the structure defined on the CFSSs. We use this computation to describe the Ei-page of a May-Koszul spectral sequence which converges to the BKSS E2-page for any connected object of sCom. We conclude by making two conjectures which would, together, allow for a full computation of the BKSS for a connected sphere in sCom.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015.; Cataloged from PDF version of thesis.; Includes bibliographical references (pages 219-222).
2015-01-01T00:00:00ZDay convolution and the Hodge filtration on THH
http://hdl.handle.net/1721.1/99326
Day convolution and the Hodge filtration on THH
Glasman, Saul
This thesis is divided into two chapters. In the first, given symmetric monoidal oc-categories C and D, subject to mild hypotheses on D, we define an oc-categorical analog of the Day convolution symmetric monoidal structure on the functor category Fun(C, D). In the second, we develop a Hodge filtration on the topological Hochschild homolgy spectrum of a commutative ring spectrum and describe its elementary properties.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015.; Cataloged from PDF version of thesis.; Includes bibliographical references (pages 65-66).
2015-01-01T00:00:00ZCombinatorics of acyclic orientations of graphs : algebra, geometry and probability
http://hdl.handle.net/1721.1/99325
Combinatorics of acyclic orientations of graphs : algebra, geometry and probability
Iriarte Giraldo, Benjamin
This thesis studies aspects of the set of acyclic orientations of a simple undirected graph. Acyclic orientations of a graph may be readily obtained from bijective labellings of its vertex-set with a totally ordered set, and they can be regarded as partially ordered sets. We will study this connection between acyclic orientations of a graph and the theory of linear extensions or topological sortings of a poset, from both the points of view of poset theory and enumerative combinatorics, and of the geometry of hyperplane arrangements and zonotopes. What can be said about the distribution of acyclic orientations obtained from a uniformly random selection of bijective labelling? What orientations are thence more probable? What can be said about the case of random graphs? These questions will begin to be answered during the first part of the thesis. Other types of labellings of the vertex-set, e.g. proper colorings, may be used to obtain acyclic orientations of a graph, as well. Motivated by our first results on bijective labellings, in the second part of the thesis, we will use eigenvectors of the Laplacian matrix of a graph, in particular, those corresponding to the largest eigenvalue, to label its vertex-set and to induce partial orientations of its edge-set. What information about the graph can be gathered from these partial orientations? Lastly, in the third part of the thesis, we will delve further into the structure of acyclic orientations of a graph by enhancing our understanding of the duality between the graphical zonotope and the graphical arrangement with the lens of Alexander duality. This will take us to non-crossing trees, which arguably vastly subsume the combinatorics of this geometric and algebraic duality. We will then combine all of these tools to obtain probabilistic results about the number of acyclic orientations of a random graph, and about the uniformly random choice of an acyclic orientation of a graph, among others.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015.; Cataloged from PDF version of thesis.; Includes bibliographical references (pages 96-99).
2015-01-01T00:00:00Z