Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2002. / Includes bibliographical references (p. 97-100). / The focus of this thesis is manifolds with group actions, in particular symplectic manifolds with Hamiltonian torus actions. We investigate the relationship between the equivariant cohomology of the manifold M and the fixed point data of the torus action. We are interested in understanding the topology of the space of T-orbits in M. In particular, we explore aspects of this topology which are determined by data from the image of a moment map [Phi] : M [right arrow] t* associated to the Hamiltonian action. To better understand the orbit space, we apply the algebraic techniques of equivariant cohomology to the study these systems further. Equivariant cohomology associates to a manifold with a G-action a ring H*G(M). Much of the topology of the orbit space is encoded in the equivariant cohomology ring H*G(M). In 1998, Goresky, Kottwitz and MacPherson provided a new method for computing this ring. Their method associates to this orbit space a graph [Gamma] whose vertices are the zero-dimensional orbits and edges the connected components of the set of one-dimensional orbits. The ring H*G(M) can then be computed combinatorially in terms of the data incorporated in [Gamma]. The strength of this construction is that it makes the computation of equivariant cohomology into a combinatorial computation, rather than a topological one. In the projects described herein, we apply the GKM theory to the case of homogeneous spaces by studying the combinatorics of their associated graphs. We exploit this theory to understand the geometry of homogeneous spaces with non-zero Euler characteristic. / (cont.) Next, we describe how to weaken the hypotheses of the GKM theorem. The spaces to which the GKM theorem applies must satisfy certain dimension conditions; however, there are many manifolds M with naturally arising T-actions that do not satisfy these conditions. We allow a more general situation, which includes some of these cases. Finally, we find a theory identical to the GKM theory in a setting suggested by work of Duistermaat. As in the GKM situation, this theory applies only when the spaces involved satisfy certain dimension conditions. / by Tara Suzanne Holm. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 71-73). / In the first part, we study pattern avoidance and permutation statistics. For a set of patterns n and a permutation statistic st, let Fst/n ([Pi]; q) be the polynomial that counts st on the permutations avoiding all patterns in [Pi]. Suppose [Pi] contains the pattern 312. For a class of permutation statistics (including inversion and descent statistics), we give a formula that expresses Fst/n ([Pi]; q) in terms of these st-polynomials for some subblocks of the patterns in [Pi]. Using this recursive formula, we construct examples of nontrivial st-Wilf equivalences. In particular, this disproves a conjecture by Dokos, Dwyer, Johnson, Sagan, and Selsor that all inv-Wilf equivalences are trivial. The second part is motivated by the problem of giving a bijective proof of the fact that the birational RSK correspondence satisfies the octahedron recurrence. We define interlacing networks to be certain planar directed networks with a rigid structure of sources and sinks. We describe an involution that swaps paths in these networks and leads to a three-term relations among path weights, which immediately implies the octahedron recurrences. Furthermore, this involution gives some interesting identities of Schur functions generalizing identities by Fulmek-Kleber. Then we study the balanced swap graphs, which encode a class of Schur function identities obtained this way. / by Wuttisak Trongsiriwat. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1995. / Includes bibliographical references (leaves 49-50). / by Alexander Astashkevich. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1996. / Includes bibliographical references (p. 63-68). / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008. / Includes bibliographical references (p. 51). / Let M be a riemannian manifold of dimension 3. We study the genus zero open rigid J-holomorphic curves in T*M with boundaries mapped in perturbations of the zero section. The perturbations of the zero section is defined fixing a. set of functions on M. We consider the graphs of the differential of the functions rescaled by an [epsilon] >/= 0. For a generic choice of the functions, we prove that, for E small enough, there exists a one to one correspondence between the J holomorphic curves and the planar Morse graphs of the functions. / by Vito Iacovino. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1992. / Includes bibliographical references (leaves 95-96). / by Esteban Gregorio Tabak. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004. / Includes bibliographical references (p. 59-61). / In this thesis, I studied the stability of local complex'singularity exponents (lcse) for holomorphic functions whose zero sets have only isolated singularities. For a given holomorphic function f defined on a neighborhood of the origin in C[to the power of]n, the lcse c[sub]0(f) is defined as the supremum of all positive real number [lambda] for which 1/[magnitude of]f[to the power of][2 lambda] is integrable on some neighborhood of the origin. It has been conjectured that c[sub]0(f) should not decrease if f is deformed small enough. Using J. Mather and S.S.T. Yau's result on the classification of isolated hypersurface singularities, together with a well known result on the stability of c[sub]0(f) when f is deformed in a finite dimension base space, I proved that if the zero set of f has only isolated singularity at the origin, then c[sub]0(g) >[or equal to][sub]0(f) for g close enough to f with respect to the C⁰ norm over a neighborhood of the origin, thus gave a partial solution to the conjecture. Using the stability results, I also computed the holomorphic invariant α(M) for some special Fano manifold M. / by Zuoliang Hou. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 54-56). / In this thesis, the author gives an explicit formula for the Fourier transform of the canonical measure on a nilpotent coadjoint orbit for GL(n, R). If G is a real, reductive algebraic group, and O C g* = Lie(G)* is a nilpotent coadjoint orbit, a necessary condition is given for 0 to appear in the wave front cycle of a tempered representation. In addition, the coefficients of the wave front cycle of a tempered representation of G are expressed in terms of volumes of precompact submanifolds of certain affine spaces. In the process of proving these results, we obtain several limit formulas for reductive Lie groups. / by Benjamin Harris. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2009. / 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 (p. 115-124). / In recent years there has been a great deal of new activity at the interface of biology and computation. This has largely been driven by the massive in flux of data from new experimental technologies, particularly high-throughput sequencing and array-based data. These new data sources require both computational power and new mathematics to properly piece them apart. This thesis discusses two problems in this field, network reconstruction and multiple network alignment, and draws the beginnings of a connection between information theory and population genetics. The first section addresses cellular signaling network inference. A central challenge in systems biology is the reconstruction of biological networks from high-throughput data sets, We introduce a new method based on parameterized modeling to infer signaling networks from perturbation data. We use this on Microarray data from RNAi knockout experiments to reconstruct the Rho signaling network in Drosophila. The second section addresses information theory and population genetics. While much has been proven about population genetics, a connection with information theory has never been drawn. We show that genetic drift is naturally measured in terms of the entropy of the allele distribution. We further sketch a structural connection between the two fields. The final section addresses multiple network alignment. With the increasing availability of large protein-protein interaction networks, the question of protein network alignment is becoming central to systems biology. / (cont.) We introduce a new algorithm, IsoRankN to compute a global alignment of multiple protein networks. We test this on the five known eukaryotic protein-protein interaction (PPI) networks and show that it outperforms existing techniques. / by Michael Hartmann Baym. / Ph.D.

Thesis (Ph.D.) Massachusetts Institute of Technology. Dept. of Mathematics, 1946. / Vita. / Bibliography: leaf 14. / by Stephen Harry Crandall. / Ph.D.