Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2009. / 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. 141-142). / This thesis primarily consists of results which can be used to simplify the computation of the equivariant cohomology of a GKM space. In particular we investigate the role that equivariant maps play in the computation of these cohomology rings. In the first part of the thesis, we describe some implications of the existence of an equivariant map p between an equivariantly formal T-manifold M and a GKM space fM. In particular we generalize the Chang-Skjelbred Theorem to this setting and derive some of its consequences. Then we consider the abstract setting of GKM graphs and define a category of objects which we refer to as GKM fiber bundles. For this class of bundles we prove a graph theoretical version of the Serre-Leray theorem. As an example, we study the projection maps from complete flag varieties to partial flag varieties from this combinatorial perspective. In the second part of the thesis we focus on GKM manifolds M which are also T-Hamiltonian manifolds. For these spaces, Guillemin and Zara ([GZ]), and Goldin and Tolman ([GT]), introduced a special basis for H* T (M), associated to a particular choice of a generic component ? of the moment map, the elements of this basis being called canonical classes. Since, for Hamiltonian T spaces, HT (M) can be viewed as a subring of the equivariant cohomology ring of the fixed point set, it is important to be able to compute the restriction of the elements of this basis to the fixed point set, and we investigate how one can use the existence of an equivariant map to simplify this computation. / (cont.) We also derive conditions under which the formulas we get are integral. Using the above results, we are able to prove, inter alia, positive integral formulas for the equivariant Schubert classes on a complete flag variety of type An,Bn,Cn and Dn. (These formulas are new, except in type An). More generally, we obtain positive integral formulas for the equivariant Schubert classes using fibrations of the complete flag variety over partial flag varieties, and when this fibration is a CP1-bundle one gets from these formulas the calculus of divided difference operators. [GT] Goldin, R. F. and Tolman, S., Towards generalizing Schubert calculus in the symplectic category, preprint. [GZ] Guillemin V. and Zara C., Combinatorial formulas for products of Thom classes. In Geometry, mechanics, and dynamics, pages 363-405, Springer NY, 2002. / by Silvia Sabatini. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1993. / Includes bibliographical references (leaf 92). / by Hsin-Fei Meng. / Ph.D.

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.