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. 141-142). / Let X be a smooth projective surface and choose a curve C on X. Let VC be the set of all irreducible divisors on X linearly equivalent to C whose normalization is a rational curve. The Severi problem for rational curves on X with divisor class [C] consists of studying the irreducibility of the spaces VC as C varies among all curves on X. In this thesis, we prove that all the spaces VC are irreducible in the case where X is a del Pezzo surface of degree at least two. If the degree of X is one, then we prove the same result only for a general X, with the exception of V-KX, where KX is the canonical divisor of X. It is well known that for general del Pezzo surface of degree one, V-KX consists of twelve points, and thus cannot be irreducible. / by Damiano Testa. / Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1999. / Includes bibliographical references (p. 67-68). / by Daniel Sai-Ping Chan. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1989. / Includes bibliographical references (leaves 72-74). / by Fabian Waleffe. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2003. / Includes bibliographical references (p. 57-58). / We examine the regularity properties of solutions to an elliptic free boundary problem, near a Neumann fixed boundary. Consider a nonnegative function u which minimizes the functional ... on a bounded, convex domain ... This function u is harmonic in its positive phase and satisfies ... along the free boundary ... , in a weak sense. We prove various basic properties of such a minimizer near the portion of the boundary ... on which ... weakly. These results include up-to-the boundary gradient estimates on harmonic functions with Neumann boundary conditions on convex domains. The main result is that the minimizer u is Lipschitz continuous. The proof in dimension 2 is by means of conformal mapping as well as a simplified monotonicity formula. In higher dimensions, the proof is via a maximum principle estimate for ... / by Sarah Groff Raynor. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1974. / Vita. / Bibliography: leaves 177-179. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2002. / Includes bibliographical references (p. 37-38). / In this thesis, I establish a homotopy equivalence between the matroid Grassmannian [parallel] MacP(k, n) [parallel] and the real Grassmannian G(k, n) of k-planes in [Real set]n. This is accomplished by finding a Schubert stratification of the former space and analyzing its relationship to the ordinary Schubert cell decomposition of the Grassmannian. Since the classifying spaces for rank k matroid bundles and rank k vector bundles are, respectively, obtained by taking colimits of the above spaces as n grows, this result provides a natural equivalence between the functors of matroid bundles and vector bundles. This, in turn, has implications for the interplay between combinatorics and topology, particularly concerning the Gelfand-MacPherson combinatorial formula for rational Pontrjagin classes. / by Daniel Kálmán Biss. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1970. / Vita. / Bibliography: leaves 133-134. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1973. / Vita. / Bibliography: leaves 120-123. / by William H. Barker. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1948. / Vita. / Includes bibliographical references (leaves 88-89). / by Raymond Moos Redheffer. / Ph.D.

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 (leaves 121-123). / Given an exact symplectic manifold, can we find topological constraints to the existence of exact Lagrangian submanifolds? I developed an approach using symplectic cohomology which provides such conditions for exact Lagrangians inside cotangent bundles and inside ALE hyperkähler spaces. For example, the only exact Lagrangians inside ALE hyperkähler spaces must be spheres. The vanishing of symplectic cohomology is an obstruction to the existence of exact Lagrangians. In the above applications even though the ordinary symplectic cohomology does not vanish, one can prove that a Novikov homology analogue for symplectic cohomology does vanish. / by Alexander F. Ritter. / Ph.D.