Thesis (Ph.D.) Massachusetts Institute of Technology. Dept. of Mathematics, 1946. / Vita. / Bibliography: leaf 27. / by Warren Simms Loud. / Ph.D.

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2009. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 29). / In their paper Metaplectic Forms, D. A. Kazhdan and S. J. Patterson developed a generalization of automorphic forms that are defined on metaplectic groups. These groups are non-trivial covering groups of usual algebraic groups, and the forms defined on them are representations that respect the covering. As in the case for automorphic forms, these representations fall into a principle series, indexed by characters on a torus of the metaplectic group, and there is an associated an L-function. In the final section of their paper, an equivalence is shown in the rank one case between this -function and an Dirichlet series defined using Gauss sums, in order to demonstrate the arithmetic content. In this paper we reexamine this connection in the particular case that was discussed in Metaplectic Forms. By looking through the scope of twisted multiplicativity, a property of L-series, the computation is simplified and more easily generalized. / by Sawyer Tabony. / S.M.

Thesis (M.S.) Massachusetts Institute of Technology. Dept. of Mathematics, 1947. / Bibliography: leaves 154-173. / by Beatrice Helen Worsley. / M.S.

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. 141-145). / Broken Lefschetz fibrations are a new way to depict smooth 4-manifolds and to investigate their topology; for instance, Perutz defines invariants of 4-manifolds by counting J-holomorphic sections of these fibrations. The first part of this thesis is about the calculus of these objects. In particular, based on earlier results we prove the existence of broken Lefschetz fibrations on any smooth oriented closed 4-manifold and describe certain topological manipulations of these objects, to construct new broken Lefschetz fibration, e.g. with better properties from other ones. The second part is about Perutz's invariants for broken Lefschetz fibrations, the corresponding invariants for 3-manifolds mapping to S1, and relating these invariants to Ozsváth-Szabó 3 and 4-manifold invariants. Specifically, we prove an isomorphism between two 3-manifold invariants, namely Perutz's quilted Floer homology and Ozsváth-Szabó Heegaard Floer homology for certain spinc structures. This yields interesting and in a sense simplified geometric interpretations of Ozsváth-Szabó invariants. In particular, we give new calculations of these invariants and other applications, e.g. a proof of Floer's excision theorem in the context of Heegaard Floer homology. / by Yankı Lekili. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 65-66). / In this thesis, we introduce the odd dimensional symplectic manifolds. In the first half we study the Hodge theory on the basic symplectic manifolds. We can define two cohomology theories on them, the standard basic de Rham cohomology gheory and a basic version of the Koszul-Brylinski-Mathieu 'harmonic' symplectic cohomology theory. Among our main results are a collection of examples for which these cohomology theories don't coincide, and, in fact, for which the usual basic cohomology theory is infinite dimensional and the symplectic cohomology theory is finite dimensional. On the other hand, we prove an odd version of the Mathieu theorem and the do-lemma: the two theories coincide if and only if a basic version of strong Lefschetz property holds. In the second half, we discuss the group actions on odd dimensional symplectic manifolds. In particular, we study the Hamiltonian group actions. Finally we use the Local-Global-Principle to prove a convexity theorem for the Hamiltonian torus actions on odd dimensional symplectic manifolds. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2009. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 117-123). / This thesis has two parts. In Part I, we present the results of a combined experimental and theoretical investigation of the vertical impact of spheres on a water surface. Particular attention is given to characterizing the shape of the resulting air cavity in the limit where cavity collapse is strongly influenced by surface tension. A parameter study reveals the dependence of the cavity structure on the governing dimensionless groups. A theoretical description is developed to describe the evolution of the cavity shape and yields an analytical solution for the pinch-off time and depth. We also examine low-density spheres that decelerate substantially following impact, and characterize the deceleration rate and resulting change in behavior of the associated water-entry cavities. Theoretical predictions compare favorably with our experimental observations. Finally, we present a theoretical model for the evolution of the splash curtain formed at high speeds, and couple it to the underlying cavity dynamics. In Part II, we present the results of a combined experimental and theoretical investigation of the motion of a sphere on an inclined flexible beam. A theoretical model based on Euler-Bernoulli beam theory is developed to describe the dynamics, and in the limit where the beam reacts instantaneously to the loading, we obtain exact solutions for the sphere trajectory and descent time. For the case of an initially horizontal beam, we calculate the period of the resulting oscillations. Theoretical predictions compare favorably with our experimental observations in this quasi-static regime. Inertial effects are also addressed. / (cont.) The time taken for descent along an elastic beam, the elastochrone, is shown to always exceed the classical brachistochrone, the shortest time between two points in a gravitational field. / by Jeffrey Michael Aristoff. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 117-120). / In this thesis, we describe a general theory of modules over an algebra over an operad. We also study functors between categories of modules. Specializing to the operad [epsilon]d of little d-dimensional disks, we show that each (d - 1) manifold gives rise to a theory of modules over [epsilon]d-algebras and each bordism gives rise to a functor from the category defined by its incoming boundary to the category defined by its outgoing boundary. Then, we describe a geometric construction of the homomorphisms objects in these categories of modules inspired by factorization homology (also called chiral homology). A particular case of this construction is higher Hochschild cohomology or Hochschild cohomology of Ed-algebras. We compute the higher Hochschild cohomology of the Lubin-Tate ring spectrum and prove a generalization of a theorem of Kontsevich and Soibelman about the action of higher Hochschild cohomology on factorization homology. / by Geoffroy Horel. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 205-210). / We study two instanton correction problems of Hitchin's moduli spaces along with their wall crossing formulas. The hyperkahler metric of a Hitchin's moduli space can be put into an instanton-corrected form according to physicists Gaiotto, Moore and Neitzke. The problem boils down to the construction of a set of special coordinates which can be constructed as Fock-Goncharov coordinates associated with foliations of quadratic differentials on a Riemann surface. A wall crossing formula of Kontsevich and Soibelman arises both as a crucial consistency condition and an effective computational tool. On the other hand Gross and Siebert have succeeded in determining instanton corrections of complex structures of Calabi-Yau varieties in the context of mirror symmetry from a singular affine structure with additional data. We will show that the two instanton correction problems are equivalent in an appropriate sense via the identification of the wall crossing formulas in the metric problem with consistency conditions in the complex structure problem. This is a nontrivial statement of mirror symmetry of Hitchin's moduli spaces which till now has been mostly studied in the framework of geometric Langlands duality. This result provides examples of Calabi-Yau varieties where the instanton correction (in the sense of mirror symmetry) of metrics and complex structures can be determined. This equivalence also relates certain enumerative problems in foliations to some gluing constructions of affine varieties. / by Wenxuan Lu. / Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2003. / Includes bibliographical references (p. 49). / This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections. / The purpose of this thesis is proving conjectures in [1] on the Khovanov invariant. Khovanov invariant [6] is an invariant of (relatively) oriented links which is a cohomology theory over the cube of the resolutions of a link diagram. Khovanov invariant specializes to the Jones polynomial by taking graded Euler characteristic. Bar-Natan [1] [2] computed this invariant for the prime knots of up to 11 crossings. From the data, Bar-Natan, Garoufalidis, and Khovanov formulated two conjectures on the value of the Khovanov invariant of an alternating knot [1][4]. We prove those conjectures by constructing a new map on Khovanov's chain complex which, with the original coboundary map, gives rise to a double complex structure on the chain complex. / by Eun Soo Lee. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1996. / Includes bibliographical references (leaves 23-24). / by Allen Ivar Knutson. / Ph.D.