Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1996. / Includes bibliographical references (p. 49-50). / by Olga Kravchenko. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2002. / Includes bibliographical references (p. 91-94). / We develop a new understanding of outliers and the behavior of linear programs under perturbation. Outliers are ubiquitous in scientific theory and practice. We analyze a simple algorithm for removal of outliers from a high-dimensional data set and show the algorithm to be asymptotically good. We extend this result to distributions that we can access only by sampling, and also to the optimization version of the problem. Our results cover both the discrete and continuous cases. This is joint work with Santosh Vempala. The complexity of solving linear programs has interested researchers for half a century now. We show that an arbitrary linear program subject to a small random relative perturbation has good condition number with high probability, and hence is easy to solve. This is joint work with Avrim Blum, Daniel Spielman, and Shang-Hua Teng. This result forms part of the smoothed analysis project initiated by Spielman and Teng to better explain mathematically the observed performance of algorithms. / by John D. Dunagan. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2002. / Includes bibliographical references (p. 163). / We find and describe the irreducible components of the space of rational curves on moduli spaces M of rank 2 stable vector bundles with odd determinant on curves C of genus g [greater than or equal to] 2. We prove that the maximally rationally connected quotient of such a component is either the Jacobian J(C) or a direct sum of two copies of the Jacobian. We show that moduli spaces of rational curves on M are in one-to-one correspondence with moduli of rank 2 vector bundles on the surface P[set]1 x C. / by Ana-Maria Castravet. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1993. / Includes bibliographical references (leaves 67-68). / by Mohamed Mekias. / Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2003. / Includes bibliographical references (p. 109-115). / This thesis consists of three projects on the enumerative and algebraic properties of matroids and hyperplane arrangements. In particular, a central object of study is the Tutte polynomial, which stores much of the enumerative information of these objects. The first project is the study of the Tutte polynomial of an arrangement and, more generally, of a semimatroid. It has two components: an enumerative one and a matroid-theoretic one. We start by considering purely enumerative questions about the Tutte polynomial of a hyperplane arrangement. We introduce a new method for computing it, which generalizes several known results. We apply our method to several specific arrangements, thus relating the computation of Tutte polynomials to problems in enumerative combinatorics. As a consequence, we obtain several new results about classical combinatorial objects such as labeled trees, Dyck paths, semiorders and alternating trees. We then address matroid-theoretic aspects of arrangements and their Tutte polynomials. We start by defining semimatroids, a class of objects which abstracts the dependence properties of an affine hyperplane arrangement. After discussing these objects in detail, we define and investigate their Tutte polynomial. In particular, we prove that it is the universal Tutte-Grothendieck invariant for semimatroids, and we give a combinatorial interpretation for its non-negative coefficients. The second project is the beginning of an attempt to study the Tutte polynomial from an algebraic point of view. / (cont.) Given a matroid representable over a field of characteristic zero, we construct a graded algebra whose Hilbert-Poincar6 series is a simple evaluation of the Tutte polynomial of the matroid. This construction is joint work with Alex Postnikov. The third project involves a class of matroids with very rich enumerative properties. We show how the set of Dyck paths of length 2n naturally gives rise to a matroid, which we call the Catalan matroid Cn. We describe this matroid in detail; among several other results, we show that Cn is self-dual, it is representable over the rationals but not over finite fields Fq with q < n - 2, and it has a nice Tutte polynomial. We then introduce a more general family of matroids, which we call shifted matroids. They are precisely the matroids whose independence complex is a shifted simplicial complex. / by Federico Ardila. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 265-273). / In this thesis, we study the growth of Sobolev norms of global solutions of solutions to nonlinear Schrödinger type equations which we can't bound from above by energy conservation. The growth of such norms gives a quantitative estimate on the low-to high frequency cascade which can occur due to the nonlinear evolution. In our work, we present two possible frequency decomposition methods which allow us to obtain polynomial bounds on the high Sobolev norms of the solutions to the equations we are considering. The first method is a high regularity version of the I-method previously used by Colliander, Keel, Staffilani, Takaoka, and Tao and it allows us to treat a wide range of equations, including the power type NLS equation and the Hartree equation with sufficiently regular convolution potential, as well as the Gross-Pitaevskii equation for dipolar quantum gases in the physically relevant 3D setting. The other method is based on a rough cut-off in frequency and it allows us to bound the growth of fractional Sobolev norms of the completely integrable defocusing cubic NLS on the real line. / by Vedran Sohinger. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1994. / Includes bibliographical references (leaves 79-80). / by Dan Port. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 157-164). / A decade ago, Yves Couder and coworkers discovered that millimetric droplets can walk on a vibrated fluid bath, and that these walking droplets or "walkers" display several features reminiscent of quantum particles. We first describe our experimental advances, that have allowed for a quantitative characterization of the system behavior, and guided the development of our accompanying theoretical models. We then detail our explorations of this rich dynamical system in several settings where the walker is confined, either by boundaries or an external force. Three particular cases are examined: a walker in a corral geometry, a walker in a rotating frame, and a walker passing through an aperture in a submerged barrier. In each setting, as the vibrational forcing is increased, progressively more complex trajectories arise. The manner in which multimodal statistics may emerge from the walker's chaotic dynamics is elucidated. / by Daniel Martin Harris. / Ph. D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2000. / Includes bibliographical references (p. 67-69). / by Baozhong Yang. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1992. / Includes bibliographical references (p. 51-52). / by Clara S. Chan. / Ph.D.