Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1987. / Includes bibliographical references (p. 85). / by James O. Schwartz. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1993. / Includes bibliographical references (p. 122-130). / by David Gillman. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 121-129). / In 2005, Yves Couder and coworkers discovered that a millimetric droplet of silicone oil may walk on the surface of a vertically-vibrating fluid bath, displaying features that were once thought to be peculiar to quantum mechanics. We here explore this hydrodynamic pilot-wave system through an integrated theoretical and experimental approach. We provide a theoretical characterization of the transition to chaos in orbital pilot-wave dynamics for droplets walking in the presence of a Coulomb, Coriolis, or central harmonic force. We proceed by investigating this hydrodynamic system above the Faraday threshold experimentally, with an aim of finding mechanisms to trap drops. We report a hydrodynamic analog of optical trapping with the Talbot effect, showing that drops may become trapped at the extrema of waves generated in the vicinity of a linear array of pillars. We also characterize the dynamics of droplets bouncing and walking above the Faraday threshold, indicating regimes of particle trapping and Brownian motion. We investigate the effect of bath topography in drop dynamics by considering a circular well that induces a circularly-symmetric Faraday wave pattern. In this regime, we show that droplets become trapped into stable circular orbits around the extrema of the well-induced wavefield. Finally, with a view to extending the phenomenological range of this hydrodynamic system, we consider a generalized pilot-wave framework, in which the relative magnitudes of dynamical parameters are altered relative to those relevant in the fluid system. In this generalized framework, we validate the theoretical result of Durey et al. relating the particle's mean wavefield to the emerging statistics, and characterizing the timescale of emergence of the statistically steady state for the chaotic pilot-wave dynamics. / by Lucas Dorigo Tambasco. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004. / Includes bibliographical references (leaves 61-64). / In this work, we study geometry of Ricci-flat Kähler manifolds, and also provide some counterexample constructions. We study asymptotic behavior of complete Ricci-flat metrics at infinity and consider a construction of approximate Ricci-flat metrics on quasiprojective manifolds with a divisor with normal crossings removed, by means of reducing torsion of a non-Kähler metric with the right volume form. Next, we study special Lagrangian fibrations using methods of geometric function theory. In particular, we generalize the method of extremal length and prove a generaliziation of the Teichmiiller theorem. We relate extremal problems to the existence of special Lagrangian fibrations in the large complex structure limit of Calabi-Yau manifolds. We proceed to some problems in the theory of minimal surfaces, disproving the Schoen-Yau conjecture and providing a first example of a proper harmonic map from the unit disk to a complex plane. In the end, we prove that the union closed set conjecture is equivalent to a strengthened version, giving a construction which might lead to a counterexample. / by Vladimir Božin. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2003. / Includes bibliographical references (p. 47). / In this thesis, the Hilbert scheme of lines on smooth hypersurfaces is studied. The main result is that the Hilbert scheme of lines on any smooth Fano hypersurface of degree d =/< 6 in ... has the expected dimension 2n - d - 3, if k is an algebraically closed field of characteristic zero. / by Roya Beheshti Zavareh. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 249-252). / In this thesis, we undertake an in-depth study of the Seiberg-Witten equations on manifolds with boundary. We divide our study into three parts. In Part One, we study the Seiberg-Witten equations on a compact 3-manifold with boundary. Here, we study the solution space of these equations without imposing any boundary conditions. We show that the boundary values of this solution space yield an infinite dimensional Lagrangian in the symplectic configuration space on the boundary. One of the main difficulties in this setup is that the three-dimensional Seiberg-Witten equations, being a dimensional reduction of an elliptic system, fail to be elliptic, and so there are resulting technical difficulties intertwining gauge-fixing, elliptic boundary value problems, and symplectic functional analysis. In Part Two, we study the Seiberg-Witten equations on a 3-manifold with cylindrical ends. Here, Morse-Bott techniques adapted to the infinite-dimensional setting allow us to understand topologically the space of solutions to the Seiberg-Witten equations on a semiinfinite cylinder in terms of the finite dimensional moduli space of vortices at the limiting end. By combining this work with the work of Part One, we make progress in understanding how cobordisms between Riemann surfaces may provide Lagrangian correspondences between their respective vortex moduli spaces. Moreover, we apply our results to provide analytic groundwork for Donaldson's TQFT approach to the Seiberg-Witten invariants of closed 3-manifolds. Finally, in Part Three, we study analytic aspects of the Seiberg-Witten equations on a cylindrical 4-manifold supplied with Lagrangian boundary conditions of the type coming from the first part of this thesis. The resulting system of equations constitute a nonlinear infinite-dimensional nonlocal boundary value problem and is highly nontrivial. We prove fundamental elliptic regularity and compactness type results for the corresponding equations, so that these results may therefore serve as foundational analysis for constructing a monopole Floer theory on 3-manifolds with boundary. / by Timothy Nguyen. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1986. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Bibliography: leaf 47. / by Marek Kierlanczyk. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 93-96). / In this thesis, we will study three topics related to Springer theory (specifically, the geometry of the exotic nilpotent cone, and two-block Springer fibers), and stability conditions for category 0. In the first chapter, we will be studying the geometry of the exotic nilpotent cone (which is a variant of the nilpotent cone in type C introduced by Kato). Bezrukavnikov has established a bijection between A+, the dominant weights for an arbitrary simple algebraic group H, and 0, the set of pairs consisting of a nilpotent orbit and a finite-dimensional irreducible representation of the isotropy group of the orbit (as originally conjectured by Lusztig and Vogan). Here we prove an analogous statement for the exotic nilpotent cone. In the second chapter (which is based on joint work with Rina Anno), we study the exotic t-structure for a two-block Springer fibre (i.e. for a nilpotent matrix of type (m + n, n) in type A). The exotic t-structure has been defined by Bezrukavnikov and Mirkovic for Springer theoretic varieties in order to study representations of Lie algebras in positive characteristic. Using techniques developed by Cautis and Kamnitzer, we show that the irreducible objects in the heart of the exotic t-structure are indexed by crossingless (m, m + 2n) matchings. We also show that the resulting Ext algebras resemble Khovanov's arc algebras (but placed on an annulus). In the third chapter, we study stability conditions on certain sub-quotients of category 0. Recently, Anno, Bezrukavnikov and Mirkovic have introduced the notion of a "real variation of stability conditions" (which are related to Bridgeland's stability conditions), and construct an example using categories of coherent sheaves on Springer fibers. Here we construct another example, by studying certain sub-quotients of category 0 with a fixed Gelfand-Kirillov dimension. We use the braid group action on the derived category of category 0, and certain leading coefficient polynomials coming from translation functors. / by Vinoth Nandakumar. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 165-172). / In this thesis, we address two problems involving partial differential equations. In the first problem, we reformulate the incompressible Navier-Stokes equations into an equivalent pressure Poisson system. The new system allows for the recovery of the pressure in terms of the fluid velocity, and consequently is ideal for efficient but also accurate numerical computations of the Navier-Stokes equations. The system may be discretized in theory to any order in space and time, while preserving the accuracy of solutions up to the domain boundary. We also devise a second order method to solve the recast system in curved geometries immersed within a regular grid. In the second problem, we examine the long time behavior of the Klein-Gordon equation with various nonlinearities. In the first case, we show that for a positive (repulsive) strong nonlinearity, the system thermalizes into a state which exhibits characteristics of linear waves. Through the introduction of a renormalized wave basis, we show that the waves exhibit a renormalized dispersion relation and a Planck-like energy spectrum. In the second case, we discuss the case of attractive nonlinearities. In comparison, here the waves develop oscillons as long lived, spatially localized oscillating fields. With an emphasis on their cosmological implications, we investigate oscillons in an expanding universe, and study their profiles and stability. The presence of a saturation nonlinearity results in flat-topped oscillons, which are relatively stable to long wavelength perturbations. / by David Shirokoff. / Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1999. / Includes bibliographical references (p. 181-190). / by Salil Pravin Vadhan. / Ph.D.