Mathematics - Ph.D. / Sc.D.
http://hdl.handle.net/1721.1/7843
Tue, 06 Oct 2015 01:43:09 GMT2015-10-06T01:43:09ZBounds on extremal functions of forbidden patterns
http://hdl.handle.net/1721.1/99069
Bounds on extremal functions of forbidden patterns
Geneson, Jesse (Jesse T.)
Extremal functions of forbidden sequences and 0 - 1 matrices have applications to many problems in discrete geometry and enumerative combinatorics. We present a new computational method for deriving upper bounds on extremal functions of forbidden sequences. Then we use this method to prove tight bounds on the extremal functions of sequences of the form (12 ... 1)t for 1 >/= 2 and t >/= 1, abc(acb)t for t >/= 0, and avav'a, such that a is a letter, v is a nonempty sequence excluding a with no repeated letters and v' is obtained from v by only moving the first letter of v to another place in v. We also prove the existence of infinitely many forbidden 0 - 1 matrices P with non-linear extremal functions for which every strict submatrix of P has a linear extremal function. Then we show that for every d-dimensional permutation matrix P with k ones, the maximum number of ones in a d-dimensional matrix of sidelength n that avoids P is 20(k) nd-1
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015.; Cataloged from PDF version of thesis.; Includes bibliographical references (pages 63-66).
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/1721.1/990692015-01-01T00:00:00ZThe pilot-wave dynamics of walking droplets in confinement
http://hdl.handle.net/1721.1/99068
The pilot-wave dynamics of walking droplets in confinement
Harris, Daniel Martin
A decade ago, Yves Couder and coworkers discovered that millimetric droplets can walk on a vibrated fluid bath, and that these walking droplets or "walkers" display several features reminiscent of quantum particles. We first describe our experimental advances, that have allowed for a quantitative characterization of the system behavior, and guided the development of our accompanying theoretical models. We then detail our explorations of this rich dynamical system in several settings where the walker is confined, either by boundaries or an external force. Three particular cases are examined: a walker in a corral geometry, a walker in a rotating frame, and a walker passing through an aperture in a submerged barrier. In each setting, as the vibrational forcing is increased, progressively more complex trajectories arise. The manner in which multimodal statistics may emerge from the walker's chaotic dynamics is elucidated.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015.; Cataloged from PDF version of thesis.; Includes bibliographical references (pages 157-164).
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/1721.1/990682015-01-01T00:00:00ZCoherent sheaves on varieties arising in Springer theory, and category 0
http://hdl.handle.net/1721.1/99067
Coherent sheaves on varieties arising in Springer theory, and category 0
Nandakumar, Vinoth
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.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015.; Cataloged from PDF version of thesis.; Includes bibliographical references (pages 93-96).
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/1721.1/990672015-01-01T00:00:00ZAnalyzing monotone space complexity via the switching network model
http://hdl.handle.net/1721.1/99066
Analyzing monotone space complexity via the switching network model
Potechin, Aaron H
Space complexity is the study of how much space/memory it takes to solve problems. Unfortunately, proving general lower bounds on space complexity is notoriously hard. Thus, we instead consider the restricted case of monotone algorithms, which only make deductions based on what is in the input and not what is missing. In this thesis, we develop techniques for analyzing monotone space complexity via a model called the monotone switching network model. Using these techniques, we prove tight bounds on the minimal size of monotone switching networks solving the directed connectivity, generation, and k-clique problems. These results separate monotone analgoues of L and NL and provide an alternative proof of the separation of the monotone NC hierarchy first proved by Raz and McKenzie. We then further develop these techniques for the directed connectivity problem in order to analyze the monotone space complexity of solving directed connectivity on particular input graphs.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015.; Cataloged from PDF version of thesis.; Includes bibliographical references (pages 177-179).
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/1721.1/990662015-01-01T00:00:00Z