Thesis (S. M.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008. / Includes bibliographical references (leaf 53). / In this thesis I study r-differential posets and dual graded graphs. Differential posets are partially ordered sets whose elements form the basis of a vector space that satisfies DU-UD=rI, where U and D are certain order-raising and order-lowering operators. New results are presented related to the growth and classification of differential posets. In particular, we prove that the rank sequence of an r-differential poset is bounded above by the Fibonacci sequence and that there is a unique poset with such a maximum rank sequence. We also prove that a 1-differential lattice is either Young's lattice or the Fibonacci lattice. In the second part of the thesis, we present a series of new examples of dual graded graphs that are not isomorphic to the ones presented in Fomin's original paper. / by Yulan Qing. / S.M.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 137-139). / In this thesis, we study the problem of asymptotic spectral flow for a family of coupled Dirac operators. We prove that the leading order term in the spectral flow on an n dimensional manifold is of order r n+1/2 followed by a remainder of O(r n/2). We perform computations of spectral flow on the sphere which show that O(r n-1/2) is the best possible estimate on the remainder. To obtain the sharp remainder we study a semiclassical Dirac operator and show that its odd functional trace exhibits cancellations in its first n+3/2 terms. A normal form result for this Dirac operator and a bound on its counting function are also obtained. / by Nikhil Savale. / Ph.D.

Massachusetts Institute of Technology. Dept. of Mathematics. Thesis. 1965. Ph.D. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1996. / Includes bibliographical references (p. 57-63). / by Alexander Craig Russell. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 59-63). / In this thesis, first, joint with Longzhi Lin, we establish estimates for the harmonic map heat flow from the unit circle into a closed manifold, and use it to construct sweepouts with the following good property: each curve in the tightened sweepout, whose energy is close to the maximal energy of curves in the sweepout, is itself close to a closed geodesic. Second, we prove the uniqueness for energy decreasing weak solutions of the harmonic map heat flow from the unit open disk into a closed manifold, given any H¹ initial data and boundary data, which is the restriction of the initial data on the boundary of the disk. Previously, under an additional assumption on boundary regularity, this uniqueness result was obtained by Rivière (when the target manifold is the round sphere and the energy of initial data is small) and Freire (for general target manifolds). The point of our uniqueness result is that no boundary regularity assumption is needed. Also, we prove the exponential convergence of the harmonic map heat flow, assuming that the energy is small at all times. Third, we prove that smooth self-shrinkers in the Euclidean space, that are entire graphs, are hyperplanes. This generalizes an earlier result by Ecker and Huisken: no polynomial growth assumption at infinity is needed. / by Lu Wang. / Ph.D.

Thesis (Ph.D.) Massachusetts Institute of Technology. Dept. of Mathematics, 1957. / Vita. / Bibliography: leaf [113]. / by Millard W. Johnson, Jr. / Ph.D.

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2007. / Includes bibliographical references (p. 53-54). / We present several graph polynomials, of which the most important one is the Tutte polynomial. These various polynomials have important applications in combinatorics and statistical physics. We generalize the Tutte polynomial and establish its correlations to the other graph polynomials. Finally, our result about the decomposition of planar graphs and its application to the ice-type model is presented. / by Jae Ill Kim. / S.M.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1992. / Includes bibliographical references (leaves 112-118). / by Evangelos Tabakis. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 345-354). / introduce a subset K2,A of the symplectic mapping class group, and an invariant ... that associates a characteristic class in Hochschild cohomology to every symplectomorphism ... K2,A. These are analogues to the familiar Johnson kernel X9 and second Johnson homomorphism - 2 from low-dimensional topology. The method is quite general, and unlike many abstract tools, explicitly computable in certain nice cases. As an application, we prove the existence of symplectomorphism ... of infinite order in symplectic mapping class group ... where Y is the blow-up of P3 at a genus 4 curve. The classical connection between such Fano varieties and cubic 3-folds allows us to factor ... as a product of six-dimensional generalized Dehn twists. / by Netanel S. Blaier. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010. / This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections. / In title on title page, double underscored "N" appears as upper case script. Cataloged from student submitted PDF version of thesis. / Includes bibliographical references (p. 118-121). / Given an oriented Riemannian four-manifold equipped with a principal bundle, we investigate the moduli spaceMVW of solutions to the Vafa-Witten equations. These equations arise from a twist of N = 4 supersymmetric Yang-Mills theory. Physicists believe that this theory has a well-defined partition function, which depends on a single complex parameter. On one hand, the S-duality conjecture predicts that this partition function is a modular form. On the other hand, the Fourier coefficients of the partition function are supposed to be the "Euler characteristics" of various moduli spacesMASD of compactified anti-self-dual instantons. For several algebraic surfaces, these Euler characteristics were verified to be modular forms. Except in certain special cases, it's unclear how to precisely define the partition function. If there is a mathematically sensible definition of the partition function, we expect it to arise as a gauge-theoretic invariant of the moduli spaces MVW. The aim of this thesis is to initiate the analysis necessary to define such invariants. We establish various properties, computations, and estimates for the Vafa-Witten equations. In particular, we give a partial Uhlenbeck compactification of the moduli space. / by Bernard A.Mares, Jr. / Ph.D.