Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1999. / Includes bibliographical references (p. 55). / by Daniel Dugger. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2014. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 99-101). / Let G be a connected reductive group with a split maximal torus defined over a nonarchimedean local field. I evaluate a matrix coefficient of the unramified principal series of G known as the "Bessel function" at torus elements of dominant coweight. I show that the Bessel function shares many properties with the Macdonald spherical function of G, in particular the properties described in Casselman's 1980 evaluation of that function. The analogy I demonstrate between the Bessel and Macdonald spherical functions extends to an analogy between the spherical Whittaker function, evaluated by Casselman and Shalika in 1980, and a previously unstudied matrix coefficient. / by Mario A. DeFranco. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1996. / Includes bibliographical references (p. 101-106). / by Marcos Kiwi. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005. / Includes bibliographical references (p. 147-150) and index. / Classical random matrix models are formed from dense matrices with Gaussian entries. Their eigenvalues have features that have been observed in combinatorics, statistical mechanics, quantum mechanics, and even the zeros of the Riemann zeta function. However, their eigenvectors are Haar-distributed-completely random. Therefore, these classical random matrices are rarely considered as operators. The stochastic operator approach to random matrix theory, introduced here, shows that it is actually quite natural and quite useful to view random matrices as random operators. The first step is to perform a change of basis, replacing the traditional Gaussian random matrix models by carefully chosen distributions on structured, e.g., tridiagonal, matrices. These structured random matrix models were introduced by Dumitriu and Edelman, and of course have the same eigenvalue distributions as the classical models, since they are equivalent up to similarity transformation. This dissertation shows that these structured random matrix models, appropriately rescaled, are finite difference approximations to stochastic differential operators. Specifically, as the size of one of these matrices approaches infinity, it looks more and more like an operator constructed from either the Airy operator, ..., or one of the Bessel operators, ..., plus noise. One of the major advantages to the stochastic operator approach is a new method for working in "general [beta] " random matrix theory. In the stochastic operator approach, there is always a parameter [beta] which is inversely proportional to the variance of the noise. / (cont.) In contrast, the traditional Gaussian random matrix models identify the parameter [beta] with the real dimension of the division algebra of elements, limiting much study to the cases [beta] = 1 (real entries), [beta] = 2 (complex entries), and [beta] = 4 (quaternion entries). An application to general [beta] random matrix theory is presented, specifically regarding the universal largest eigenvalue distributions. In the cases [beta] = 1, 2, 4, Tracy and Widom derived exact formulas for these distributions. However, little is known about the general [beta] case. In this dissertation, the stochastic operator approach is used to derive a new asymptotic expansion for the mean, valid near [beta] = [infinity]. The expression is built from the eigendecomposition of the Airy operator, suggesting the intrinsic role of differential operators. This dissertation also introduces a new matrix model for the Jacobi ensemble, solving a problem posed by Dumitriu and Edelman, and enabling the extension of the stochastic operator approach to the Jacobi case. / by Brian D. Sutton. / Ph.D.

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005. / Includes bibliographical references (p. 65-66). / Delzant's theorem for symplectic toric manifolds says that there is a one-to-one correspondence between certain convex polytopes in ... and symplectic toric 2n-manifolds, realized by the image of the moment map. I present proofs of this theorem and the convexity theorem of Atiyah-Guillemin-Sternberg on which it relies. Then, I describe Honda's results on the local structure of near-symplectic 4-manifolds, and inspired by recent work of Gay-Symington, I describe a generalization of Delzant's theorem to near-symplectic toric 4-manifolds. One interesting feature of the generalization is the failure of convexity, which I discuss in detail. / by Samuel Kaufman. / S.M.

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1997. / Includes bibliographical references (p. 18). / by Julie Rehmeyer. / M.S.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1997. / Includes bibliographical references (p. 63-64). / by Eric Nathan Sommers. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, June 2011. / "June 2011." Cataloged from PDF version of thesis. / Includes bibliographical references (p. 57-59). / This thesis consists of two parts. Both parts are devoted to finding links between geometric/algebraic objects and combinatorial objects. In the first part of the thesis, we link Schubert varieties in the full flag variety with hyperplane arrangements. Schubert varieties are parameterized by elements of the Weyl group. For each element of the Weyl group, we construct certain hyperplane arrangement. We show that the generating function for regions of this arrangement coincides with the Poincaré polynomial if and only if the Schubert variety is rationally smooth. For classical types the arrangements are (signed) graphical arrangements coning from (signed) graphs. Using this description, we also find an explicit combinatorial formula for the Poincaré polynomial in type A. The second part is about Specht modules of general diagram. For each diagram, we define a new class of polytopes and conjecture that the normalized volume of the polytope coincides with the dimension of the corresponding Specht module in many cases. We give evidences to this conjecture including the proofs for skew partition shapes and forests, as well as the normalized volume of the polytope for the toric staircase diagrams. We also define new class of toric tableaux of certain shapes, and conjecture the generating function of the tableaux is the Frobenius character of the corresponding Specht module. For a toric ribbon diagram, this is consistent with the previous conjecture. We also show that our conjecture is intimately related to Postnikov's conjecture on toric Specht modules and McNamara's conjecture of cylindric Schur positivity. / by Hwanchul Yoo. / Ph.D.

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.