Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005. / Includes bibliographical references (p. 93-96). / In the first part we study Double Affine Hecke algebra of type An-1 which is important tool in the theory of orthogonal polynomials. We prove that the spherical subalgebra eH(t, 1)e of the Double Affine Hecke algebra H(t, 1) of type An-1 is an integral Cohen-Macaulay algebra isomorphic to the center Z of H(t, 1), and H(t, 1)e is a Cohen-Macaulay eH(t, 1)e-module with the property H(t, 1) = EndeH(t,tl)(H(t, 1)e). This implies the classification of the finite dimensional representations of the algebras. In the second part we study the algebraic properties of the five-parameter family H(tl, t2, t3, t4; q) of double affine Hecke algebras of type CVC1, which control Askey- Wilson polynomials. We show that if q = 1, then the spectrum of the center of H is an affine cubic surface C, obtained from a projective one by removing a triangle consisting of smooth points. Moreover, any such surface is obtained as the spectrum of the center of H for some values of parameters. We prove that the only fiat de- formations of H come from variations of parameters. This explains from the point of view of noncommutative geometry why one cannot add more parameters into the theory of Askey-Wilson polynomials. We also prove several results on the universality of the five-parameter family H(tl, t2, t3, t4; q) of algebras. / by Alexei Oblomkov. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015. / 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 149-156). / A graph G is r-Ramsey minimal with respect to a graph H if every r-coloring of the edges of G yields a monochromatic copy of H, but the same is not true for any proper subgraph of G. The study of the properties of graphs that are Ramsey minimal with respect to some H and similar problems is known as graph Ramsey theory; we study several problems in this area. Burr, Erdös, and Lovász introduced s(H), the minimum over all G that are 2- Ramsey minimal for H of [delta](G), the minimum degree of G. We find the values of s(H) for several classes of graphs H, most notably for all 3-connected bipartite graphs which proves many cases of a conjecture due to Szabó, Zumstein, and Zürcher. One natural question when studying graph Ramsey theory is what happens when, rather than considering all 2-colorings of a graph G, we restrict to a subset of the possible 2-colorings. Erdös and Hajnal conjectured that, for any fixed color pattern C, there is some [epsilon] > 0 so that every 2-coloring of the edges of a Kn, the complete graph on n vertices, which doesn't contain a copy of C contains a monochromatic clique on n[epsilon] vertices. Hajnal generalized this conjecture to more than 2 colors and asked in particular about the case when the number of colors is 3 and C is a rainbow triangle (a K3 where each edge is a different color); we prove Hajnal's conjecture for rainbow triangles. One may also wonder what would happen if we wish to cover all of the vertices with monochromatic copies of graphs. Let F = {F₁, F₂, . . .} be a sequence of graphs such that Fn is a graph on n vertices with maximum degree at most [delta]. If each Fn is bipartite, then the vertices of any 2-edge-colored complete graph can be partitioned into at most 2C[delta] vertex disjoint monochromatic copies of graphs from F, where C is an absolute constant. This result is best possible, up to the constant C. / by Andrey Vadim Grinshpun. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2006. / Includes bibliographical references (p. 149-151). / We define and study the structure of SUSY Lie conformal and vertex algebras. This leads to effective rules for computations with superfields. Given a strongly conformal SUSY vertex algebra V and a supercurve X, we construct a vector bundle [ ... ] on X, the fiber of which, is isomorphic to V. Moreover, the state-field correspondence of V canonically gives rise to (local) sections of these vector bundles. We also define chiral algebras on any supercurve X, and show that the vector bundle [ ... ] corresponding to a SUSY vertex algebra, carries the structure of a chiral algebra. / by Reimundo Heluani. / Ph.D.

Massachusetts Institute of Technology. Dept. of Mathematics. Thesis. 1966. Ph.D. / Bibliography: leaves 83-85. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2003. / Includes bibliographical references (p. 59-60). / Motivated by results and conjectures of Stanley concerning minimal border strip tableaux of partitions, we present three results. First we generalize the rank of a partition [lambda] to the rank of a shifted partition S([lambda]).We show that the number of bars required in a minimal bar tableau of S([lambda]) is max(o, e + (ℓ([lambda]) mod 2)), where o and e are the number of odd and even rows of [lambda]. As a consequence we show that the irreducible negative characters of [tilde]S[sub]n vanish on certain conjugacy classes. Another corollary is a lower bound on the degree of the terms in the expansion of Schur's Q[sub][lambda] symmetric functions in terms of the power sum symmetric functions. The second result gives a basis for the space spanned by the lowest degree terms in the expansion of the Schur symmetric functions in terms of the power sum symmetric functions. These lowest degree terms studied by Stanley correspond to minimal border strip tableaux of [lambda]. The Hilbert series of these spaces is the generating function giving the number of partitions of n into parts differing by at least 2. Applying the Rogers-Ramanujan identity, the generating function also counts the number of partitions of n into parts 5k + 1 and 5k - 1. Finally for each [lambda] we give a relation between the power sum symmetric functions and the monomial symmetric functions; the terms are indexed by the types of minimal border strip tableaux of [lambda]. / by Peter Clifford. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1962. / Vita. / Includes bibliographical references (leaf 31). / by Andrew Seth Kahr. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2003. / Includes bibliographical references (p. 80). / In this thesis, I study pseudo-holomorphic curves in symplectisation of the unit cotangent bundle of a Riemann surface of genus greater than 1. The contact form and compatible almost complex structure are both constructed from a metric on the Riemann surface whose curvature is constant -1. I related the pseudo-holomorphic curve equation to harmonic map equations and a Cauchy-Riemann type equation perturbed with quadratic terms for functions on a punctured Riemann sphere. Then I prove a Theorem that gives one to one correspondence between solutions to the perturbed Cauchy-Riemann equation and finite energy pseudo-holomorphic curves. / by Wei Luo. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2003. / Includes bibliographical references (p. 44). / This thesis is organized into two papers. All results are proven over an algebraically closed field of characteristic zero. Paper 1 concerns morphisms between hypersurfaces in Pn, n =/> 4. We show that if the two hypersurfaces involved in the morphism are of general type, then the morphism of hypersurfaces extends to an everywhere-defined endomorphism of Pn. A corollary is that if X [right arrow] Y is a nonconstant morphism of hypersurfaces of large dimension and large degree, then deg Y divides deg X. The main tool used to analyze morphism between hypersurfaces is an inequality of Chern classes analogous to the Hurwitz-inequality. Paper 2 is a long example. We check that every morphism from a quintic hypersurface in I4 to a nonsingular cubic hypersurface in P4 is constant. In the process, we classify morphisms froin the projective plane to nonsingular cubic threefolds. / by David C. Sheppard. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 151-167). / Motivated by various applications such as seriation, network alignment and ranking from pairwise comparisons, we study the problem of estimating a structured matrix with rows and columns shuffled by latent permutations, given noisy and incomplete observations of its entries. This problem is at the intersection of shape constrained estimation which has a long history in statistics, and latent permutation learning which has driven a recent surge of interest in the machine learning community. Shape constraints on matrices, such as monotonicity and smoothness, are generally more robust than parametric assumptions, and often allow for adaptive and efficient estimation in high dimensions. On the other hand, latent permutations underlie many graph matching and assignment problems that are computationally intractable in the worst-case and not yet well-understood in the average-case. Therefore, it is of significant interest to both develop statistical approaches and design efficient algorithms for problems where shape constraints meet latent permutations. In this work, we consider three specific models: the statistical seriation model, the noisy sorting model and the strong stochastic transitivity model. First, statistical seriation consists in permuting the rows of a noisy matrix in such a way that all its columns are approximately monotone, or more generally, unimodal. We study both global and adaptive rates of estimation for this model, and introduce an efficient algorithm for the monotone case. Next, we move on to ranking from pairwise comparisons, and consider the noisy sorting model. We establish the minimax rates of estimation for noisy sorting, and propose a near-linear time multistage algorithm that achieves a near-optimal rate. Finally, we study the strong stochastic transitivity model that significantly generalizes the noisy sorting model for estimation from pairwise comparisons. Our efficient algorithm achieves the rate (n- 3 /4 ), narrowing a gap between the statistically optimal rate Õ(n-1 ) and the state-of-the-art computationally efficient rate [Theta] (n- 1/ 2 ). In addition, we consider the scenario where a fixed subset of pairwise comparisons is given. A dichotomy exists between the worst-case design, where consistent estimation is often impossible, and an average-case design, where we show that the optimal rate of estimation depends on the degree sequence of the comparison topology. / by Cheng Mao. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1991. / Includes bibliographical references (leaves 82-86). / by Mark Engel. / Ph.D.