Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1994. / Includes bibliographical references (leaf 24). / by Dan Arnon. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1995. / Includes bibliographical references (p. 194-197). / by Shinya Watanabe. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1995. / Includes bibliographical references (p. 36). / by Siman Yat-Fai Wong. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1993. / Includes bibliographical references (p. 91-96). / by Lenore Jennifer Cowen. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2017. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 187-197). / Disparate biological datasets often exhibit similar well-defined structure; efficient algorithms can be designed to exploit this structure. In this doctoral thesis, we present a framework for similarity search based on entropy and fractal dimension; here, we prove that a clustered search algorithm scales in time with metric entropy number of covering hyperspheres-if the fractal dimension is low. Using these ideas, entropy-scaling versions of standard bioinformatics search tools can be designed, including for small-molecule, metagenomics, and protein structure search. This 'compressive acceleration' approach taking advantage of redundancy and sparsity in biological data can be leveraged also for next-generation sequencing (NGS) read mapping. By pairing together a clustered grouping over similar reads and a homology table for similarities in the human genome, our CORA framework can accelerate all-mapping by several orders of magnitude. Additionally, we also present work on filtering empirical base-calling quality scores from Next Generation Sequencing data. By using the sparsity of k-mers of sufficient length in the human genome and imposing a human prior through the use of frequent k-mers in a large corpus of human DNA reads, we are able to quickly discard over 90% of the information found in those quality scores while retaining or even improving downstream variant-calling accuracy. This filtering step allows for fast lossy compression of quality scores. / by Yun William Yu. / Ph. D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2014. / 69 / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 121-123). / We present two results relating to the symplectic geometry of the Milnor fibres of isolated affine hypersurface singularities. First, given two Lagrangian spheres in an exact symplectic manifold, we find conditions under which the Dehn twists about them generate a free non-abelian subgroup of the symplectic mapping class group. This extends a result of Ishida for Riemann surfaces. The proof generalises the categorical version of Seidel's long exact sequence to arbitrary powers of a fixed Dehn twist. We also show that the Milnor fibre of any isolated degenerate hypersurface singularity contains such pairs of spheres. In the second half of this thesis, we study exact Lagrangian tori in Milnor fibres. The Milnor fibre of any isolated hypersurface singularity contains many exact Lagrangian spheres: the vanishing cycles associated to a Morsification of the singularity. Moreover, for simple singularities, it is known that the only possible exact Lagrangians are spheres. We construct exact Lagrangian tori in the Milnor fibres of all non-simple singularities of real dimension four. This gives examples of Milnor fibres whose Fukaya categories are not generated by vanishing cycles. Also, this allows progress towards mirror symmetry for unimodal singularities, which are one level of complexity up from the simple ones. / by Ailsa Macgregor Keating. / Ph. D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 457-470). / A random planar map is a graph embedded in the sphere, viewed modulo orientation-preserving homeomorphisms. Random planar maps are the discrete analogues of random fractal surfaces called [gamma]-Liouville quantum gravity (LQG) surfaces with parameter [gamma] E (0, 2]. We study the large-scale structure of random planar maps (and statistical mechanics models on them) viewed as metric measure spaces equipped with the graph distance and the counting measure on vertices. In particular, we show that uniform random planar maps (which correspond to the case [gamma]= [square root of]8/3) decorated by a self-avoiding walk or a critical percolation interface converge in the scaling limit to [square root of]8/3- LQG surfaces decorated by SLE8/3 and SLE6, respectively, with respect to a generalization of the Gromov-Hausdorff topology. We also introduce an approach for analyzing certain random planar maps belonging to the [gamma]-LQG universality class for general [gamma] E (0, 2) and use this approach to prove several estimates for graph distances in such maps. / by Ewain Gwynne. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1962. / Vita. / Includes bibliographical references (leaf [62]). / by David I. Knee. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1997. / Includes bibliographical references (p. 95-97). / by David S. Metzler. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 131-136). / In this thesis we study wave and Klein-Gordon equations on Kerr exterior spacetimes. For the wave equation, we give a quantitative refinement and simple proofs of mode stability type statements on Kerr backgrounds in the full sub-extremal range ([absolute value of]a < M). As an application, we are able to quantitatively control the energy flux along the horizon for solutions to the wave equation in any bounded-frequency regime. This estimate plays a crucial role in the author's recent proof, joint with Mihalis Dafermos and Igor Rodnianski, of boundedness and decay for the solutions to the wave equation on the full range of sub-extremal Kerr spacetimes. For the Klein-Gordon equation, we show that given any Kerr exterior spacetime with non-zero angular momentum, we may find an open family of non-zero Klein-Gordon masses for which there exist smooth, finite energy, and exponentially growing solutions to the corresponding Klein-Gordon equation. If desired, for any non-zero integer m, an exponentially growing solution can be found with mass arbitrarily close to [absolute value of]am/2Mr+. / by Yakov Shlapentokh-Rothman. / Ph. D.