We extend Culler and Shalen's work on constructing essential surfaces in 3-manifolds to orbifolds. A consequence of this work is that every valuation on the canonical component that detects an essential surface, detects an essential surface that is preserved by every orientation preserving symmetry on the manifold. This Theorem applies to orientable hyperbolic manifolds, with orientation preserving symmetry group, whose quotient by this group is an orbifold with a flexible cusp, which is the case for most hyperbolic 3-manifolds. We then look at a family of two bridge knots where our theorem shows it is impossible for every essential surface to be detected on the canonical component. We then prove that all surfaces that are preserved by the orientation preserving symmetries of these knots are detected by ideal points on the canonical component of the character variety by calculating the canonical component of the A-polynomial for the family of knots. We then prove that every essential surface in these knot that is not detected on the canonical component of the character variety is detected on another component. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Summer Semester 2018. / June 28, 2018. / Character Varieties, Essential Surfaces / Includes bibliographical references. / Kathleen Petersen, Professor Directing Dissertation; Dennis Duke, University Representative; Wolfgang Heil, Committee Member; Samuel Ballas, Committee Member.

<p> Kuramoto oscillators coupled through a graph provide one of the most influential models for the study of collective synchronizations. We propose the first sufficient synchronization conditions for lattice models. This sufficient condition is optimal as an <i>L</i>^&infin; synchronization condition. We also propose a novel continuum limit of the Kuramoto oscillators on lattices, by viewing the lattice as a discretization of the space. In the continuous model we have an analogous synchronization condition. To prove the continuous (discrete) synchronization conditions, we show the existence of solutions to some (discrete) elliptic PDE of divergence form. The two main ingredients in the proof are variational methods and gradient estimates for (discrete) elliptic PDE's.</p><p>

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 65-66). / This thesis is divided into two chapters. In the first, given symmetric monoidal oc-categories C and D, subject to mild hypotheses on D, we define an oc-categorical analog of the Day convolution symmetric monoidal structure on the functor category Fun(C, D). In the second, we develop a Hodge filtration on the topological Hochschild homolgy spectrum of a commutative ring spectrum and describe its elementary properties. / by Saul Glasman. / Ph. D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1984. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Vita. / Bibliography: leaves 83-85. / by Niles David Ritter. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1994. / Includes bibliographical references (p. 142-144). / by Daniel J. Mahoney. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015. / In title on title-page, "v" is italicized, and "1" is subscript. Cataloged from PDF version of thesis. / Includes bibliographical references (pages 139-140). / We tell the story of the stable homotopy groups of spheres for odd primes at chromatic height 1 through the lens of the Adams spectral sequence. We find the "dancers to a discordant system." We calculate a Bockstein spectral sequence which converges to the 1-line of the chromatic spectral sequence for the odd primary Adams E₂-page. Furthermore, we calculate the associated algebraic Novikov spectral sequence converging to the 1-line of the BP chromatic spectral sequence. This result is also viewed as the calculation of a direct limit of localized modified Adams spectral sequences converging to the homotopy of the v1 -periodic sphere spectrum. As a consequence of this work, we obtain a thorough understanding of a collection of q₀-towers on the Adams E₂-page and we obtain information about the differentials between these towers. Moreover, above a line of slope 1/(p²-p-1) we can completely describe the E₂ and E₃ -pages of the mod p Adams spectral sequence, which accounts for almost all the spectral sequence in this range. / by Michael Joseph Andrews. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004. / Includes bibliographical references (p. 59-60). / We present a smoothed analysis of Gaussian elimination, both with partial pivoting and without pivoting. Let A be any matrix and let A be a slight random perturbation of A. We prove that it is unlikely that A has large condition number. Using this result, we prove it is unlikely that A has large growth factor under Gaussian elimination without pivoting. By combining these results, we bound the smoothed precision needed to perform Gaussian elimination without pivoting. Our results improve the average-case analysis of Gaussian elimination without pivoting performed by Yeung and Chan (SIAM J. Matrix Anal. Appl., 1997). We then extend the result on the growth factor to the case of partial pivoting, and present the first analysis of partial pivoting that gives a sub-exponential bound on the growth factor. In particular, we show that if the random perturbation is Gaussian with variance [sigma]², then the growth factor is bounded by (n/[sigma])[to the power of] (o log n) with very high probability. / by Arvind Sankar. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1998. / Includes bibliographical references (p. 79). / by Paul Loya. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 175-194). / Systems biology, the study of biological systems in a holistic manner, has been catalyzed by a dramatic improvement in experimental techniques, coupled with a constantly increasing availability of biological data. The representation and analysis of this heterogeneous data is facilitated by the powerful abstraction of biological networks. This thesis examines several types of these networks and looks in detail at the kind of information their analysis can yield. The first part discusses protein interaction networks. We introduce a new algorithm for the pairwise alignment of these networks. We show that these alignments can provide important clues to the function of proteins as well as insights into the evolutionary history of the species under examination. The second part discusses regulatory networks. We present an approach for validating putative drug targets based on the information contained in these networks. We show how this approach can also be used to discover drug targets. The third part discusses metabolic networks. We provide new insights into the structure of constraint-based models of cell metabolism and describe a methodology for performing a complete analysis of a metabolic network. We also present an implementation of this methodology and discuss its application to a variety of problems related to the metabolism of bacteria. The final part describes an application of our methodology to Mycobacterium tuberculosis, the pathogen responsible for almost 2 million deaths around the world every year. We introduce a method for reconciling metabolic network reconstructions and apply it to merge the two published networks for tuberculosis. We analyze the merged network and show how it can be refined based on available experimental data to improve its predictive power. We conclude with a list of potential drug targets. / by Leonid Alexandrovich Chindelevitch. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1994. / Includes bibliographical references (p. 27-29). / by Eitan Bachmat. / Ph.D.