Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1970. / Vita. / Bibliography: leaves 135-136. / by Arthur L. Lieberman. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2003. / Includes bibliographical references (p. 89-91). / We analyze the evolution of Laplace eigenvalues on a domain induced by the motion of the boundary. We apply our analysis to two problems: 1. We study the equilibrium and stability of electron bubbles. Electron bubbles are cavities formed around electrons injected into liquid helium. They can be treated as simple mathematical systems that minimize the energy with three terms: the energy of the electron proportional to a Laplace eigenvalue, the surface energy proportional to the surface area of the cavity, and the hydrostatic pressure proportional to its volume. This system possesses a surprising result: an instability of the 2S electron bubbles. 2. We compute the simple eigenvalues on a regular polygon with N sides. The polygon is treated as a perturbation of the unit circle and its eigenvalues are approximated by a Taylor series. The accuracy of our approach is measured by comparison with finite element estimates. For the lowest eigenvalue, the first Taylor term provides an estimate within 10-5 of the true value. The second term reduces the error to 10-7. We discuss how to utilize the available symmetry to obtain better finite element estimates. Finally, we briefly discuss the expansion of simple eigenvalues on regular polygons in powers of 1/N. / by Pavel Greenfield. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1996. / Includes bibliographical references (p. 34-35). / by Marat Rovinsky. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2014. / This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections. / Cataloged from student-submitted PDF version of thesis. / Includes bibliographical references (pages 165-173). / In the last few decades, the study of human history has been fundamentally changed by our ability to detect the signatures left within our genomes by adaptations, migrations, population size changes, and other processes. Rapid advances in DNA sequencing technology have now made it possible to interrogate these signals at unprecedented levels of detail, but extracting more complex information about the past from patterns of genetic variation requires new and more sophisticated models. This thesis presents a suite of sensitive and efficient statistical tools for learning about human history and evolution from large-scale genetic data. We focus first on the problem of admixture inference and describe two new methods for determining the dates, sources, and proportions of ancestral mixtures between diverged populations. These methods have already been applied to a number of important historical questions, in particular that of tracing the course of the Austronesian expansion in Southeast Asia. We also report a new approach for estimating the human mutation rate, a fundamental parameter in evolutionary genetics, and provide evidence that it is higher than has been proposed in recent pedigree-based studies. / by Mark Lipson. / Ph. D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 96-99). / This thesis studies aspects of the set of acyclic orientations of a simple undirected graph. Acyclic orientations of a graph may be readily obtained from bijective labellings of its vertex-set with a totally ordered set, and they can be regarded as partially ordered sets. We will study this connection between acyclic orientations of a graph and the theory of linear extensions or topological sortings of a poset, from both the points of view of poset theory and enumerative combinatorics, and of the geometry of hyperplane arrangements and zonotopes. What can be said about the distribution of acyclic orientations obtained from a uniformly random selection of bijective labelling? What orientations are thence more probable? What can be said about the case of random graphs? These questions will begin to be answered during the first part of the thesis. Other types of labellings of the vertex-set, e.g. proper colorings, may be used to obtain acyclic orientations of a graph, as well. Motivated by our first results on bijective labellings, in the second part of the thesis, we will use eigenvectors of the Laplacian matrix of a graph, in particular, those corresponding to the largest eigenvalue, to label its vertex-set and to induce partial orientations of its edge-set. What information about the graph can be gathered from these partial orientations? Lastly, in the third part of the thesis, we will delve further into the structure of acyclic orientations of a graph by enhancing our understanding of the duality between the graphical zonotope and the graphical arrangement with the lens of Alexander duality. This will take us to non-crossing trees, which arguably vastly subsume the combinatorics of this geometric and algebraic duality. We will then combine all of these tools to obtain probabilistic results about the number of acyclic orientations of a random graph, and about the uniformly random choice of an acyclic orientation of a graph, among others. / by Benjamin Iriarte Giraldo. / Ph. D.

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005. / Includes bibliographical references (p. 91). / We provide two algorithms to solve branching from K to M for the real split reductive group of type A, one inductive and one related to semistandard Young tableaux. The results extend to branching from Ke to M Ke for the real split reductive groups of type Bn and Dn. / by Nicholas McCarthy. / S.M.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2009. / Includes bibliographical references (p. 87-88). / A grid shape is a set of boxes chosen from a square grid; any Young diagram is an example. We consider a notion of pattern-avoidance for 0-1 fillings of grid shapes, which generalizes permutation pattern-avoidance. A filling avoids a set of patterns if none of its sub-shapes, obtained by removing some rows and columns, equal any of the patterns. We focus on patterns that are pairs of 2 x 2 fillings. Totally nonnegative Grassmann cells are in bijection with Young shape fillings that avoid particular 2 x 2 pair, which are, in turn, equinumerous with fillings avoiding another 2 x 2 pair. The latter ones correspond to acyclic orientations of the shape's bipartite graph. Motivated by this result, due to Postnikov and Williams, we prove a number of such analogs of Wilf-equivalence for these objects - that is, we show that, in certain classes of shapes, some pattern-avoiding fillings are equinumerous with others. The equivalences in this paper follow from two very different bijections, and from a family of recurrences generalizing results of Postnikov and Williams. We used a computer to test each of the described equivalences on a diverse set of shapes. All our results are nearly tight, in the sense that we found no natural families of shapes, in which the equivalences hold, but the results' hypotheses do not. One of these bijections gives rise to some new combinatorics on tilings of skew Young shapes with rectangles, which we name Popeye diagrams. In a special case, they are exactly Hugh Thomas's snug partitions for d = 2. We show that Popeye diagrams are a lattice, and, moreover, each diagram is a sublattice of the Tamari lattice. We also give a simple enumerative result. / by Alexey Spiridonov. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2014. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 183-186). / In the first part of this thesis, we obtain some new results about infinitesimal Cherednik algebras. They have been introduced by Etingof-Gan-Ginzburg in [EGG] as appropriate analogues of the classical Cherednik algebras, corresponding to the reductive groups, rather than the finite ones. Our main result is the realization of those algebras as particular finite W-algebras of associated semisimple Lie algebras with nilpotent 1-block elements. To achieve this, we prove its Poisson counterpart first, which identifies the Poisson infinitesimal Cherednik algebras introduced in [DT] with the Poisson algebras of regular functions on the corresponding Slodowy slices. As a consequence, we obtain some new results about those algebras. We also generalize the classification results of [EGG] from the cases GL, and SP2n to SOl. In the second part of the thesis, we discuss the loop realization of the affine Yangian of gl₁. Similar objects were recently considered in the work of Maulik-Okounkov on the quantum cohomology theory, see [MO]. We present a purely algebraic realization of these algebras by generators and relations. We discuss some families of their representations. A similarity with the representation theory of the quantum toroidal algebra of gl₁ is explained by adapting a recent result of Gautam-Toledano Laredo, see [GTL], to the local setting. We also discuss some aspects of those two algebras such as the degeneration isomorphism, a shuffle presentation, and a geometric construction of the Whittaker vectors. / by Oleksandr Tsymbaliuk. / Ph. D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2014. / 18 / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 145-146). / This thesis studies rational curves in the complex projective plane that are homeomorphic to their normalizations. We derive some combinatorial constraints on such curves from a result of Borodzik-Livingston in Heegaard-Floer homology. Using these constraints and other tools from algebraic geometry, we offer a solution to certain cases of the Coolidge-Nagata problem on the rectifiability of planar rational cuspidal curves, that is, their equivalence to lines under the Cremona group of birational automorphisms of the plane. / by Tiankai Liu. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2002. / Includes bibliographical references (p. 35-37). / We show that for a large class of torsionfree classifying spaces, K-theory filtered ring is an invariant of the genus. We apply this result in two ways. First, we use it to show that the powerseries ring on n indeterminates over the integers admits uncountably many mutually non-isomorphic [lambda]-ring structures. Second, we use it to study the genus of infinite quaternionic projective space. In particular, we describe spaces in the genus of infinite quaternionic projective space which occur as targets of essential maps from infinite complex projective space, and we compute explicitly the homotopy classes of maps in these cases. / by Donald Y. Yau. / Ph.D.