Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004. / Includes bibliographical references (p. 165-167). / This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections. / In this thesis we study tilings of regions on the square grid by polyominoes. A polyomino is any connected shape formed from a union of grid cells, and a tiling of a region is a collection of polyominoes lying in the region such that each square is covered exactly once. In particular, we focus on two main themes: local connectivity and tile invariants. Given a set of tiles T and a finite set L of local replacement moves, we say that a region [Delta] has local connectivity with respect to T and L if it is possible to convert any tiling of [Delta] into any other by means of these moves. If R is a set of regions (such as the set of all simply connected regions), then we say there is a local move property for T and R if there exists a finite set of moves L such that every r in R has local connectivity with respect to T and L. We use height function techniques to prove local move properties for several new tile sets. In addition, we provide explicit counterexamples to show the absence of a local move property for a number of tile sets where local move properties were conjectured to hold. We also provide several new results concerning tile invariants. If we let ai(t) denote the number of occurrences of the tile ti in a tiling t of a region [Delta], then a tile invariant is a linear combination of the ai's whose value depends only on t and not on r. / (cont.) We modify the boundary-word technique of Conway and Lagarias to prove tile invariants for several new sets of tiles and provide specific examples to show that the invariants we obtain are the best possible. In addition, we prove some new enumerative results, relating certain tiling problems to Baxter permutations, the Tutte polynomial, and alternating-sign matrices. / by Michael Robert Korn. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 69-71). / Suppose that X is a smooth variety and L is an effective divisor. One of the main goals of bi rational geometry is to understand the asymptotic behavior of the linear series... as m increases. The two most important features of the asymptotic behavior - the litaka dimension and the litaka fibration - are subtle and difficult to work with. In this thesis we will construct approximations to these objects that depend only on the numerical class of L. The main interest in such results arises from the Abundance Conjecture which predicts that the Iitaka fibration for Kx is determined by its numerical properties. In the second chapter we study a numerical approximation to the Iitaka dimension of L. For a nef divisor L, this quantity is a classical invariant known as the numerical dimension. There have been several proposed extensions of the numerical dimension to pseudo-effective divisors in [Nak04] and [BDPP04]. We show that these proposed definitions coincide and agree with many other natural notions. Just as in the nef case, the numerical dimension v(L) of a pseudo-effective divisor L should measure the maximum dimension of a subvariety ... such that the "positive restriction" of L is big along W. In the third chapter, we analyze how the properties of the Iitaka fibration OL for L are related to the numerical properties of L. Although the numerical dimension detects the existence of "virtual sections", it does not have a direct relationship with the Iitaka fibration. However, we do construct a rational map that only depends on the numerical class of L and approximates the Jitaka fibration. This rational map is the maximal possible fibration for which a general fiber F satisfies v(LIF) = 0. Thus, this chapter recovers and extends the work of [Eck05] from an algebraic viewpoint. Finally, we use the pseudo-effective reduction map to study the Abundance Conjecture. / by Brian Lehmann. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 47-48). / A zonotopal algebra is the quotient of a polynomial ring by an ideal generated by powers of linear forms which are derived from a zonotope, or dually it's hyperplane arrangement. In the case that the hyperplane arrangement is of Type A, we can rephrase the definition in terms of graphs. Using the symmetry of these ideals, we can find monomial ideals which preserve much of the structure of the zonotopal algebras while being computationally very efficient, in particular far faster than Gröbner basis techniques. We extend this monomization theory from the known case of the central zonotopal algebra to the other two main cases of the external and internal zonotopal algebras. / by Craig J. Desjardins. / Ph.D.

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1995. / Includes bibliographical references (leaf 23). / by Anthony Harkin. / M.S.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1999. / Includes bibliographical references (p. 123-126). / by Patricia Hersh. / Ph.D.

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1983. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Bibliography: leaves 68-69. / by David Michael Shera. / M.S.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2006. / This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections. / Includes bibliographical references (p. 119-125). / The majority of human genes are not encoded in contiguous segments in the genome but are rather punctuated by long interruptions known as introns. These introns are copied from generation to generation, and even from cell to cell as a person grows from an embryo into an adult. Each time a gene is activated, the cell must first accurately excise all the introns in a process known as splicing. This excision is determined by the sequence of the gene, but in a complicated way that is not fully understood. By analyzing gene sequences we can learn about how cells decide which sequences to splice. We have developed two new mathematical models, one for the end of introns, and another for long distance interactions between different parts of genes, that expose previously unknown elements potentially involved in the splicing reaction. However their boundaries are determined, introns are very ancient: although they are absent from bacteria they are found in almost all protists, fungi, plants and animals. It is therefore of great interest to explain their evolutionary origins. We have developed a probabilistic model for the evolution of introns and used it to perform a genome-wide analysis of the patterns of intron conservation in four euascomycete fungi, establishing that intron gain and loss are constantly reshaping the distribution of introns in genes. / by Brad Aaron Friedman. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 87-90). / In this thesis, we study the trigonometric and elliptic Cherednik algebras. In the first part, we give a Lie-theoretic construction of the trigonometric Cherednik algebras of type BC,. We construct a functor from the category of Harish- Chandra modules of the symmetric pair of type AIII to the category of representations of the degenerate affine and double affine Hecke algebra of type BC. We also study the images of some D-modules and the principal series modules. In the second part, we define the elliptic Dunkl operators on an abelian variety with a finite group action. Using these elliptic Dunkl operators, we construct a new family of quantum integrable systems. / by Xiaoguang Ma. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 149-150). / The derived category of a hypersurface has an action by "cohomology operations" k[[beta]], deg[beta] = 2, underlying the 2-periodic structure on its category of singularities (as matrix factorizations). We prove a Thom-Sebastiani type Theorem, identifying the k[[beta]]-linear tensor products of these dg categories with coherent complexes on the zero locus of the sum potential on the product (with a support condition), and identify the dg category of colimit-preserving k[[beta]]-linear functors between Ind-completions with Ind-coherent complexes on the zero locus of the difference potential (with a support condition). These results imply the analogous statements for the 2-periodic dg categories of matrix factorizations. We also present a viewpoint on matrix factorizations in terms of (formal) groups actions on categories that is conducive to formulating functorial statements and in particular to the computation of higher algebraic structures on Hochschild invariants. Some applications include: we refine and establish the expected computation of 2-periodic Hochschild invariants of matrix factorizations; we show that the category of matrix factorizations is smooth, and is proper when the critical locus is proper; we show how Calabi-Yau structures on matrix factorizations arise from volume forms on the total space; we establish a version of Knörrer Periodicity for eliminating metabolic quadratic bundles over a base. / by Anatoly Preygel. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2001. / Includes bibliographical references (p. 57-58). / We look at the singularity Cn/[Gamma], for [Gamma] finite subgroup of SU(n), from two perspectives. From a geometrical point of view, Cn/[Gamma] is an orbifold with boundary S2n-1/[Gamma]. We define and compute the corresponding orbifold [eta]-invariant. From an algebraic point of view, we look at the algebraic variety Cn/[Gamma] and we analyze the associated Molien series. The main result is formula which relates the two notions: [eta]-invariant and Molien series. Along the way computations of the spectrum of the Dirac operator on the sphere are performed. / by Anda Degeratu. / Ph.D.