Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008. / 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. 125-129). / In this thesis, we apply combinatorial means for proving and generalizing classical determinantal identities. In Chapter 1, we present some historical background and discuss the algebraic framework we employ throughout the thesis. In Chapter 2, we construct a fundamental bijection between certain monomials that proves crucial for most of the results that follow. Chapter 3 studies the first, and possibly the best-known, determinantal identity, the matrix inverse formula, both in the commutative case and in some non-commutative settings (Cartier-Foata variables, right-quantum variables, and their weighted generalizations). We give linear-algebraic and (new) bijective proofs; the latter also give an extension of the Jacobi ratio theorem. Chapter 4 is dedicated to the celebrated MacMahon master theorem. We present numerous generalizations and applications. In Chapter 5, we study another important result, Sylvester's determinantal identity. We not only generalize it to non-commutative cases, we also find a surprising extension that also generalizes the master theorem. Chapter 6 has a slightly different, representation theory flavor; it involves representations of the symmetric group, and also Hecke algebras and their characters. We extend a result on immanants due to Goulden and Jackson to a quantum setting, and reprove certain combinatorial interpretations of the characters of Hecke algebras due to Ram and Remmel. / by Matjaž Konvalinka. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1960. / Vita. / Includes bibliographical references (leaves 107-109). / by Hugo Rossi. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1991. / Includes bibliographical references (p. 101-103). / by Arthur M. Duval. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2014. / 27 / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 55-56). / In 2004, Seidel and Smith proved that the Liouville manifold associated to Ramanujams surface contains a Lagrangian torus which is not displaceable by Hamiltonian isotopy, and hence that higher products of this manifold provide non-standard symplectic structures on Euclidean space which are convex at infinity. I extend these techniques a wide class of smooth contractible affine surfaces of log-general type to produce a similar torus. I then show that the existence of this torus implies the non-vanishing of the symplectic cohomology of the Liouville manifolds associated to these surfaces, thus confirming a portion of McLeans conjecture that a smooth variety has vanishing symplectic cohomology if and only if it is affine ruled. / by David Sean Jackson-Hanen. / Ph. D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 107-108). / We study two gauge-theoretic Floer homologies associated to links, the singular instanton Floer homology introduced in [15] and the monopole Floer homology, which is explained in the book [16]. For both cases, we study in particular the spectral sequence that relates the Floer homologies to the Khovanov homologies of links. In our study of singular instanton Floer homology, we introduce a version of Khovanov homology for alternating links with marking data, W, inspired by singular instanton theory. We show that the analogue of the spectral sequence from Khovanov homology to singular instanton homology introduced in [15] for this marked Khovanov homology collapses on the E2 page for alternating links. We moreover show that for non-split links the Khovanov homology we introduce for alternating links does not depend on w; thus, the instanton homology also does not depend on W for non-split alternating links. We study a version of binary dihedral representations for links with markings, and show that for links of non-zero determinant, this also does not depend on w. In our study of monopole Floer homology, we construct families of metrics on the cobordisms that are used to construct differentials in the spectral sequence relating the Khovanov homology of a link to the monopole Floer homology of its double branched cover, such that each metric has positive scalar curvature. This allows us to conclude that the Seiberg-Witten equations for these families of metrics have no irreducible solutions, so the differentials in the spectral sequence can be computed from counting only the reducible solutions. / by Sherry Gong. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004. / Includes bibliographical references (p. 79-80). / This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections. / Classifying the irreducible unitary representations of a real reductive group is equivalent to the algebraic problem of classifying the Harish-Chandra modules admitting a positive definite invariant Hermitian form. Finding a formula for the signature of the Shapovalov form is a related problem which may be a necessary first step in such a classification. A Verma module may admit an invariant Hermitian form, which is unique up to multiplication by a real scalar when it exists. Suitably normalized, it is known as the Shapovalov form. The collection of highest weights decomposes under the affine Weyl group action into alcoves. The signature of the Shapovalov form for an irreducible Verma module depends only on the alcove in which the highest weight lies. We develop a formula for this signature, depending on the combinatorial structure of the affine Weyl group. / by Wai Ling Yee. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004. / Includes bibliographical references (p. 43-44). / In this thesis we explore the possibility of defining the p-local finite groups of Broto, Levi and Oliver in terms of their classifying spaces. More precisely, we consider the question posed by Haynes Miller, whether an equivalent theory can be recovered by studying maps f: BS --> X from the classifying space of a finite p-group S to a p-complete space X equipped with a stable retract t satisfying a form of Frobenius reciprocity. In the case where S is elementary abelian, we answer this question in the affirmative, by showing that under some finiteness conditions such a triple (f, t, X) does indeed induce a p-local finite group over S. We also discuss the converse in some detail for general S. / by Kaŕi Ragnarsson. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 65). / Recently, there has been much research on processes that are mostly random, but also have a small amount of deterministic choice; e.g., Achlioptas processes on graphs. This thesis builds on the balanced allocation algorithm first described by Azar, Broder, Karlin and Upfal. Their algorithm (and its relatives) uses randomness and some choice to distribute balls into bins in a balanced way. Here is a description of the opposite family of algorithms, with an analysis of exactly how unbalanced the distribution can become. / by Amanda E. Redlich. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2001. / Includes bibliographical references (p. 91). / By a Galton-Watson tree T we mean an infinite rooted tree that starts with one node and where each node has a random number of children independently of the rest of the tree. In the first chapter of this thesis, we prove a conjecture made in [7] for Galton-Watson trees where vertices have bounded number of children not equal to 1. The conjecture states that the electric conductance of such a tree has a continuous distribution. In the second chapter, we study rays in Galton-Watson trees. We establish what concentration of vertices with is given number of children is possible along a ray in a typical tree. We also gauge the size of the collection of all rays with given concentrations of vertices of given degrees. / by Alex Perlin. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1986. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Bibliography: p. 145-146. / by Sheila Sundaram. / Ph.D.