Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004. / Includes bibliographical references (p. 151-156). / We construct and describe compactified moduli stacks of Azumaya algebras on a smooth projective morphism X [right arrow] S. These stacks are the algebro-geometric version of the (suitably compactified) stacks of principal PGLn-bundles and they also have strong connections to arithmetic. A geometric approach to the problem leads one to study stacks of (semistable) twisted sheaves. We show that these stacks are very similar to the stacks of semistable sheaves. This gives a way of understanding the structure of the stack of principal PGLn-bundles and its coarse moduli space in terms of fairly well-understood spaces. In particular, when X [right arrow] S is a smooth projective curve or surface over an algebraically closed field, our method yields concrete theorems about the structure of these stacks (at least as certain natural invariants are allowed to increase without bound). On the arithmetic side, we use the geometry and rationality properties of these moduli spaces to study a classical question about the Brauer group of a function field K, known as the "period-index problem": for which classes o in Br(K) of order n does there exist a division algebra D of rank n2 with [D] = [alpha]? We give an answer to this question when K is the function field of a curve or surface over an algebraically closed, finite, or local field and when c is an unramified Brauer class of order prime to the characteristic of K. In the general case, we relate the unramified period-index problem to rationality questions on Galois twists of moduli spaces of semistable sheaves. / by Max Lieblich. / 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, 1983. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE / Bibliography: p. 100-101. / by Paul M. Hriljac. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2017. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 78-83). / The behavior of the spectrum of a large random matrix is a topic of great interest in probability theory and statistics. At a global level, the limiting spectra of certain random matrix models have been known for some time. For example, the limiting spectral measure of a Wigner matrix is a semicircle law and the limiting spectral measure of a sample covariance matrix under certain conditions is a Marc̆enko-Pastur law. The local behavior of eigenvalues for specific random matrix ensembles (GUE and GOE) have been known for some time as well and until recently, were conjectured to be universal. There have been many recents breakthroughs in the universality of this local behavior of eigenvalues for Wigner Matrices. Furthermore, these universality results laws have been proven for other probabilistic models of particle systems, such as Beta Ensembles. In this thesis we investigate the fluctuations of linear statistics of eigenvalues of Wigner Matrices and Beta Ensembles in regimes intermediate to the global regime and the microscopic regime (called the mesoscopic regime). We verify that these fluctuations are Gaussian and derive the covariance for a range of test functions and scales. On a separate line of investigation, we study the global spectral behavior of a random matrix arising in statistics, called Kendall's Tau and verify that it satisfies an analogue of the Marc̆enko-Pastur Law. / by Asad Lodhia. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1998. / Includes bibliographical references (p. 53-54). / by David Carlton. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1989. / Includes bibliographical references (leaves 38-39). / by Leonid Friedlander. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1970. / Vita. / Bibliography: leaves 351-355. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1971. / Vita. / Bibliography: leaf 106. / by Jeffrey H. Albert. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1994. / Includes bibliographical references (leaves 111-113). / by Mark R. Sepanski. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005. / 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. 105-110). / Proofs of Tsygan's formality conjectures for chains would unlock important algebraic tools which might lead to new generalizations of the Atiyah-Patodi-Singer index theorem and the Riemann-Roch-Hirzebruch theorem. Despite this pivotal role in the traditional investigations and the efforts of various people the most general version of Tsygan's formality conjecture has not yet been proven. In my thesis I propose Fedosov resolutions for the Hochschild cohomological and homological complexes of the algebra of functions on an arbitrary smooth manifold. Using these resolutions together with Kontsevich's formality quasi-isomorphism for Hochschild cochains of R((y1, . . . , yd)) and Shoikhet's formality quasi-isomorphism for Hochschild chains of R((y1, . . . , yd)) I prove Tsygan's formality conjecture for Hochschild chains of the algebra of functions on an arbitrary smooth manifold. The construction of the formality quasi-isomorphism for Hochschild chains is manifestly functorial for isomorphisms of the pairs (M,(vector differential)), where M is the manifold and (vector differential) is an affine connection on the tangent bundle. In my thesis I apply these results to equivariant quantization, computation of Hochschild homology of quantum algebras and description of traces in deformation quantization. / by Vasiliy A. Dolgushev. / Ph.D.