Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 219-222). / 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. / by Michael Jack Donovan. / Ph. D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 73-75). / Let G be a reductive group over a field k = k. Let P be a parabolic subgroup. We construct a functor Groupoid ... is a connected space, which induces an action of generalizing a classical result. It is also a part of a study of natural equivalences between ... for P, Q associated parabolic subgroups. / by D. Boger. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004. / Includes bibliographical references (leaves 69-72). / The Greene-Kleitman theorem says that the lengths of chains and antichains in any poset are intimately related via an integer partition, but very little is known about the partition [lambda](P) for most posets P. Our first goal is to develop a method for calculating values of [lambda]k(P) for certain posets. We find the size of the largest union of two or three chains in the lattice of partitions of n under dominance order, and in the Tamari lattice. Similar techniques are then applied to the k-equal partition lattice. We also present some partial results and conjectures on chains and antichains in these lattices. We give an elementary proof of the rank-unimodality of L(2, n, m), and find a symmetric chain decomposition of L(2, 2, m). We also present some partial results and conjectures about related posets, including a theorem on the size of the largest union of k chains in these posets and a bijective proof of the symmetry of the H-vector for 2 x n. We answer a question of Knuth about the existence of a Gray path for binary partitions, and generalize to b-ary partitions when b is even. We also discuss structural properties of the posets Rb(n), and compute some chain and antichain lengths in the subposet of join-irreducibles. / by Edward Fielding Early. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1996. / Includes bibliographical references (leaves 52-55). / by Christopher Thomas Woodward. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2013. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 131-133). / The overall goal of this thesis is to apply the theory of Goodwillie calculus to the category Algo of algebras over a spectral operad. Its first part generalizes many of the original results of Goodwillie in [14] so that they apply to a larger class of model categories and hence be applicable to Algo. The second part then applies that generalized theory to the Algo categories. The main results here are: an understanding of finitary homogeneous between such categories; identifying the Taylor tower of the identity in those categories; showing that finitary n-excisive functors can not distinguish between Algo and Algo,, the category of algebras over the truncated operad O<; and a weak form of the chain rule between the algebra categories, analogous to the one found in [1]. / by Luis Alexandre Meira Fernandes Alves Pereira. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 63-66). / Extremal functions of forbidden sequences and 0 - 1 matrices have applications to many problems in discrete geometry and enumerative combinatorics. We present a new computational method for deriving upper bounds on extremal functions of forbidden sequences. Then we use this method to prove tight bounds on the extremal functions of sequences of the form (12 ... 1)t for 1 >/= 2 and t >/= 1, abc(acb)t for t >/= 0, and avav'a, such that a is a letter, v is a nonempty sequence excluding a with no repeated letters and v' is obtained from v by only moving the first letter of v to another place in v. We also prove the existence of infinitely many forbidden 0 - 1 matrices P with non-linear extremal functions for which every strict submatrix of P has a linear extremal function. Then we show that for every d-dimensional permutation matrix P with k ones, the maximum number of ones in a d-dimensional matrix of sidelength n that avoids P is 20(k) nd-1 / by Jesse Geneson. / Ph. D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018. / In title on title page, "[beta]" appears as the lower case Greek letter. Cataloged from PDF version of thesis. / Includes bibliographical references (pages 137-142). / In this thesis, we investigate the local and global properties of the eigenvalues of [beta]-ensembles. A lot of attention has been drawn recently on the universal properties of [beta]-ensembles, and how their local statistics relate to those of Gaussian ensembles. We use transport methods to prove universality of the eigenvalue gaps in the bulk and at the edge, in the single cut and multicut regimes. In a different direction, we also prove Central Limit Theorems for the linear statistics of [beta]-ensembles at the macroscopic and mesoscopic scales. / by Florent Bekerman. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1989. / Includes bibliographical references (leaves 127-134). / by David A. Meyer. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2014. / 35 / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 51-54). / A Heegaard splitting of a 3-manifold gives rise to a natural set of sweepouts which by the Almgren-Pitts and Simon-Smith min-max theory generates a min-max sequence converging as varifolds to a smooth minimal surface (possibly disconnected, and with multiplicities). We prove a conjecture of Pitts-Rubinstein about how such a min-max sequence can degenerate; namely we show that after doing finitely many disk surgeries and isotopies on the sequence, and discarding some components, the remaining components are each isotopic to one component (or a double cover of one component) of the min-max limit. This convergence immediately gives rise to new genus bounds for min-max limits. Our results can be thought of as a min-max analog to the theorem of Meeks-Simon-Yau on convergence of a minimizing sequence of surfaces in an isotopy class. / by Daniel Ketover. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2002. / Includes bibliographical references (p. 239-240). / Enumerative geometry of algebraic varieties is a fascinating field of mathematics that dates back to the nineteenth century. We introduce new computational tools into this field that are motivated by recent progress in symplectic topology and its influence on enumerative geometry. The most straightforward applications of the methods developed are to enumeration of rational curves with a cusp of specified nature in projective spaces. A general approach for counting positive-genus curves with a fixed complex structure is also presented. The applications described include enumeration of rational curves with a (3,4)-cusp, genus-two and genus-three curves with a fixed complex structure in the two-dimensional complex projective space, and genus-two curves with a fixed complex structure in the three-dimensional complex projective space. Our constructions may be applicable to problems in symplectic topology as well. / by Aleksey Zinger. / Ph.D.