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.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005. / Includes bibliographical references (p. 93-96). / In the first part we study Double Affine Hecke algebra of type An-1 which is important tool in the theory of orthogonal polynomials. We prove that the spherical subalgebra eH(t, 1)e of the Double Affine Hecke algebra H(t, 1) of type An-1 is an integral Cohen-Macaulay algebra isomorphic to the center Z of H(t, 1), and H(t, 1)e is a Cohen-Macaulay eH(t, 1)e-module with the property H(t, 1) = EndeH(t,tl)(H(t, 1)e). This implies the classification of the finite dimensional representations of the algebras. In the second part we study the algebraic properties of the five-parameter family H(tl, t2, t3, t4; q) of double affine Hecke algebras of type CVC1, which control Askey- Wilson polynomials. We show that if q = 1, then the spectrum of the center of H is an affine cubic surface C, obtained from a projective one by removing a triangle consisting of smooth points. Moreover, any such surface is obtained as the spectrum of the center of H for some values of parameters. We prove that the only fiat de- formations of H come from variations of parameters. This explains from the point of view of noncommutative geometry why one cannot add more parameters into the theory of Askey-Wilson polynomials. We also prove several results on the universality of the five-parameter family H(tl, t2, t3, t4; q) of algebras. / by Alexei Oblomkov. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015. / This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections. / Cataloged from student-submitted PDF version of thesis. / Includes bibliographical references (pages 149-156). / A graph G is r-Ramsey minimal with respect to a graph H if every r-coloring of the edges of G yields a monochromatic copy of H, but the same is not true for any proper subgraph of G. The study of the properties of graphs that are Ramsey minimal with respect to some H and similar problems is known as graph Ramsey theory; we study several problems in this area. Burr, Erdös, and Lovász introduced s(H), the minimum over all G that are 2- Ramsey minimal for H of [delta](G), the minimum degree of G. We find the values of s(H) for several classes of graphs H, most notably for all 3-connected bipartite graphs which proves many cases of a conjecture due to Szabó, Zumstein, and Zürcher. One natural question when studying graph Ramsey theory is what happens when, rather than considering all 2-colorings of a graph G, we restrict to a subset of the possible 2-colorings. Erdös and Hajnal conjectured that, for any fixed color pattern C, there is some [epsilon] > 0 so that every 2-coloring of the edges of a Kn, the complete graph on n vertices, which doesn't contain a copy of C contains a monochromatic clique on n[epsilon] vertices. Hajnal generalized this conjecture to more than 2 colors and asked in particular about the case when the number of colors is 3 and C is a rainbow triangle (a K3 where each edge is a different color); we prove Hajnal's conjecture for rainbow triangles. One may also wonder what would happen if we wish to cover all of the vertices with monochromatic copies of graphs. Let F = {F₁, F₂, . . .} be a sequence of graphs such that Fn is a graph on n vertices with maximum degree at most [delta]. If each Fn is bipartite, then the vertices of any 2-edge-colored complete graph can be partitioned into at most 2C[delta] vertex disjoint monochromatic copies of graphs from F, where C is an absolute constant. This result is best possible, up to the constant C. / by Andrey Vadim Grinshpun. / Ph. D.