Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2007. / Includes bibliographical references (p. 79-80). / In this thesis we use the method of moving planes to establish symmetry properties for positive solutions of semilinear elliptic equations. We give a detailed proof of the result due to Caffarelli, Gidas, and Spruck that a solution in the punctured ball, B\{0}, behaves asymptotically like its spherical average at the origin. We also show that a solution with an isolated singularity in the upper half space Rn+ must be cylindrically symmetric about some axis orthogonal to the boundary aRn+. / by Gregory Drugan. / S.M.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 89-91). / The theory of Whittaker functions is of crucial importance in the classical study of automorphic forms on adele groups. Motivated by the appearance of Whittaker functions for covers of reductive groups in the theory of multiple Dirichlet series, we provide a study of Whittaker functions on metaplectic covers of reductive groups over local fields. / by Peter James McNamara. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016. / Cataloged from PDF version of thesis. / Includes bibliographical references (page 87). / This thesis consists of two parts. In the first part we count antichains of interval orders and in particular semiorders. We associate a Dyck path to each interval order, and give a formula for the number of antichains of an interval order in terms of the corresponding Dyck path. We then use this formula to give a generating function for the total number of antichains of semiorders, enumerated by the sizes of the semiorders and the antichains. In the second part we expand the work of Liu and Stanley on Dilworth lattices. Let L be a distributive lattice, let -(L) be the maximum number of elements covered by a single element in L, and let K(L) be the subposet of L consisting of the elements that cover o-(L) elements. By a result of Dilworth, K(L) is also a distributive lattice. We compute o(L) and K(L) for various lattices L that arise as the coordinate-wise partial ordering on certain sets of semistandard Young tableaux. / by Efrat Engel Shaposhnik. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1996. / Includes bibliographical references (leaves 45-46). / by Jozsef Losonczy, Jr. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1998. / Includes bibliographical references (p. 71-72). / Peter Engel Trapa. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2009. / This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections. / Includes bibliographical references (p. 67). / The main result of this work is a q-analogue relationship between nilpotent transformations and spanning trees. For example, nilpotent endomorphisms on an n-dimensional vector space over Fq is a q-analogue of rooted spanning trees of the complete graph Kn. This relationship is based on two similar bijective proofs to calculate the number of spanning trees and nilpotent transformations, respectively. We also discuss more details about this bijection in the cases of complete graphs, complete bipartite graphs, and cycles. It gives some refinements of the q-analogue relationship. As a corollary, we find the total number of nilpotent transformations with some restrictions on Jordan block sizes. / by Jingbin Yin. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2007. / Includes bibliographical references (p. 71-74). / In this thesis a conjecture of Okounkov, a conjecture of Fomin-Fulton-Li-Poon, and a special case of Lascoux-Leclerc-Thibon's conjecture on Schur positivity of certain differences of products of Schur functions are proved. In the first part of the work a combinatorial method is developed that allows to prove weaker versions of those conjectures. In the second part a recent result of Rhoades and Skandera is used to provide a proof of actual Schur positivity results. Several further generalizations are stated and proved. In particular, an intriguing log-concavity property of Schur functions is observed. In addition, a stronger conjecture is stated in language of alcoved polytops. A weaker version of this conjecture is proved using a characterization of Klyachko cone and the theory of Temperley-Lieb immanants. / by Pavlo Pylyavskyy. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008. / Includes bibliographical references (p. 53-54). / A Sasakian manifold S is equipped with a unit-length, Killing vector field ( which generates a one-dimensional foliation with a transverse Kihler structure. A differential form a on S is called basic with respect to the foliation if it satisfies [iota][epsilon][alpha] = [iota][epsilon]d[alpha] = 0. If a compact Sasakian manifold S is regular, i.e. a circle bundle over a compact Kähler manifold, the results of Hodge theory in the Kahler case apply to basic forms on S. Even in the absence of a Kähler base, there is a basic version of Hodge theory due to El Kacimi-Alaoui. These results are useful in trying to imitate Kähler geometry on Sasakian manifolds; however, they have limitations. In the first part of this thesis, we will develop a "transverse Hodge theory" on a broader class of forms on S. When we restrict to basic forms, this will give us a simpler proof of some of El Kacimi-Alaoui's results, including the basic dd̄-lemma. In the second part, we will apply the basic dd̄-lemma and some results from our transverse Hodge theory to conclude (in the manner of Deligne, Griffiths, and Morgan) that the real homotopy type of a compact Sasakian manifold is a formal consequence of its basic cohomology ring and basic Kähler class. / by Aaron Michael Tievsky. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 77). / In this thesis we study the combinatorial objects that appear in the study of nonnegative part of the Grassmannian. The classical theory of total positivity studies matrices such that all minors are nonnegative. Lustzig extended this theory to arbitrary reductive groups and flag varieties. Postnikov studied the nonnegative part of the Grassmannian, showed that it has a nice cell decomposition using matroid strata, and introduced several combinatorial objects that encode such cells. In this thesis, we focus on the combinatorial aspects of such associated objects. In chapter 1, we review the definition of the cells in the totally nonnegative part of the Grassmannian, and the associated combinatorial objects. Each cell corresponds to a certain matroid called positroid. There are numerous combinatorial objects that can represent a positroid, such as a J-diagram, a Grassmann necklace or a decorated permutation. We will go over the definitions of such objects and check some of their properties. And for decorated permutations, there are certain planar graphs called plabic graphs, that plays the role of wiring diagrams for permutations, and this would serve as the main tool for our result in chapter 3. In chapter 2, we prove a conjecture by Postnikov, that allows us to give a purely combinatorial definition of positroids without relying on its realizability. We will show that positroids can be defined as certain collections that satisfy some cyclic inequalities. In other words, we express positroids using cyclically shifted Schubert matroids. Postnikov showed that each positroid cell is an intersection of the totally nonnegative Grassmannian and cyclically shifted Schubert cells. Combinatorially, this result implies that each positroid is included in an intersection of cyclically shifted Schubert matroids. We extend this result: each positroid is exactly an intersection of certain cyclically shifted Schubert niatroids. In chapter 3, we study maximal weakly separated collections. Weak separation is a condition on pair of sets that first appeared in Leclerc and Zelevinsky's work describing quasicommuting families of quantum minors. They conjectured that all maximal by inclusion weakly separated collections of minors have the same cardinality (the purity conjecture), and that they can be related to each other by a sequence of mutations. We link the study of weak separation with the totally nonnegative Grassmannian, by extending the notion of weak separation to positroids. By using plabic graphs, we generalize the results and conjectures of Leclerc and Zelevinsky, and prove them in this more general setup. This part of the thesis is based on joint work with Alexander Postnikov and David Speyer. In chapter 4, we prove a property on h-vector of positroids. The h-vector of a matroid is an interesting Tutte polynomial evaluation, which is originally defined as the h-vector of the corresponding independent complex of a matroid. Stanley conjectured that h-vector of any matroid is a pure O-sequence, which is a sequence coming froi a Hilbert function of a monomial Artinian level algebra. We show that the conjecture holds for positroids: that is, the h-vector of a positroid is a pure O-sequence. / by SuHo Oh. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1972. / On t.p., "[omega]" appears as the lower-case Greek letter. Vita. / Bibliography: leaves 56-58. / by W. Stephen Wilson. / Ph.D.