Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013. / 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 133-139). / Rapid advances in next-generation sequencing technologies are revolutionizing genomics, with data sets at the scale of thousands of human genomes fast becoming the norm. These technological leaps promise to enable corresponding advances in biology and medicine, but the deluge of raw data poses substantial mathematical, computational and statistical challenges that must first be overcome. This thesis consists of two research thrusts along these lines. First, we propose an algorithmic framework, "compressive genomics," that accelerates bioinformatic computations through analysis-aware compression. We demonstrate this methodology with proof-of-concept implementations of compression-accelerated search (CaBLAST and CaBLAT). Second, we develop new computational tools for investigating population admixture, a phenomenon of importance in understanding demographic histories of human populations and facilitating association mapping of disease genes. Our recently released ALDER and MixMapper software packages provide fast, sensitive, and robust methods for detecting and analyzing signatures of admixture created by genetic drift and recombination on genome-wide, large-sample scales. / by Po-Ru Loh. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 99-100). / In this thesis a geometric way to understand the relations of certain noncommutative quadratic algebras defined by Anatol N. Kirillov is developed. These algebras are closely related to the Fomin-Kirillov algebra, which was introduced in the hopes of unraveling the main outstanding problem of modern Schubert calculus, that of finding a combinatorial interpretation for the structure constants of Schubert polynomials. Using a geometric understanding of the relations of Kirillov's algebras in terms of subdivisions of root polytopes, several conjectures of Kirillov about the reduced forms of monomials in the algebras are proved and generalized. Other than a way of understanding Kirillov's algebras, this polytope approach also yields new results about root polytopes, such as explicit triangulations and formulas for their volumes and Ehrhart polynomials. Using the polytope technique an explicit combinatorial description of the reduced forms of monomials is also given. Inspired by Kirillov's algebras, the relations of which can be interpreted as subdivisions of root polytopes, commutative subdivision algebras are defined, whose relations encode a variety of possible subdivisions, and which provide a systematic way of obtaining subdivisions and triangulations. / by Karola Mészáros. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1991. / Includes bibliographical references (leaf 81). / by Bo-Yin Yang. / Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1999. / Includes bibliographical references (p. 186-192). / by Lior Samuel Pachter. / Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1999. / Includes bibliographical references (p. 35). / by Gustavo Granja. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1968. / Vita. / Bibliography: leaves 116-117. / Peterson, Bent Edvard. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1985. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Includes index. / Bibliography: p. 151-153. / by John Reese Stembridge. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004. / Includes bibliographical references (p. 57-59). / In this thesis, we discuss the relation between the Euler number of a tight contact small Seifert space and the contact framing of Legendrian vertical circles in it, and apply this relation to classify up to isotopy tight contact structures on small Seifert spaces with e₀ [not equal to] 0, -1, -2. / by Hao Wu. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 55). / The Heegaard Floer hat invariant is defined on closed 3-manifolds, with a related invariant for 4-dimensional cobordisms, forming a 3+1 topological quantum field theory. Bordered Heegaard Floer homology generalizes this invariant to parametrized Riemann surfaces and to cobordisms between them, yielding a 2+1 TQFT. We discuss an approach to synthesizing these two theories to form a 2+1+1 TQFT, by defining Heegaard Floer invariants for Lefschetz fibrations with corners. / by Tova Helen Fell Brown. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 83). / Let g be a complex, reductive Lie algebra. We prove a theorem parametrizing the set of nilpotent orbits in g in terms of even nilpotent orbits of subalgebras of g and show how to determine these subalgebras and how to explicitly compute this correspondence. We prove a theorem parametrizing nilpotent orbits for strong involutions of G in terms of even nilpotent orbits of complex subalgebras of g and show how to explicitly compute this correspondence. / by Peter Speh. / Ph.D.