Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1987. / Bibliography: p. [163-168]. / by Günter M. Ziegler. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 171-175). / The Colemani integral is a p-adice line integral that can be used to encapsulate several quantities relevant, to a study of the arithmetic of varieties. In this thesis, I describe algorithms for computing Coleman integrals on hyperelliptic curves and discuss some immediate applications. I give algorithms to compute single and iterated integrals on odd models of hyperelliptic curves, as well as the necessary modifications to iplemieit these algorithms for even models. Furthermore, I show how these algorithinis can be used in various situations. The first application is the method of Chabatv to find rational points on curves of genus greater than 1. The second is Mlihyong Kim's recent nonabelian analogue of the Chabauty method for elliptic curves. The last two applications concern p-adic heights on Jacobians of hyperelliptic curves. necessary to formulate a p-adic analogue of the Birch and Swinnerton-Dyer conjecture. I conclude by stating the analogue of the Mazur-Tate-Teitelbaum conjecture iii our setting and presenting supporting data. / by Jennifer Sayaka Balakrishnan. / Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, February 1999. / Includes bibliographical references (p. 67). / by Andrei Radulescu-Banu. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2001. / Includes bibliographical references (p. [79]-[80]). / Inspired by the work of Lusztig, we apply the theory of character sheaves on a symmetric space (h la Ginzburg and Grojnowski) to the problem of determining the spherical functions (averages of the irreducible characters) of the associated symmetric space over a finite field. A crucial result about the filtration of character sheaves by cells enables us to compute the spherical functions explicitly in the cases GLn((Fq2)/GLn(]Fq) and GL2n(1Fq)/Sp2n((Fq), as well as other closely related examples. We include partial results about the symmetric space GLn/(GLm x GLn-m), to illustrate some of the obstacles to further generalization. / by Anthony Henderson. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2017. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 69-71). / This thesis studies the representation theory of Cherednik algebras associated to a complex algebraic varieties which carries the action of a finite group. First, we prove that the Knizhnik-Zamolodchikov functor from the category of P-coherent modules for the Cherednik algebra to finite dimensional modules over the Hecke algebra is essentially surjective. Then we begin to use this result to study the analog of category 0 for Cherednik algebras on Riemann surfaces and on products of elliptic curves. In particular we give conditions on the parameters under which this category 0 analog is nonzero. Our next goal is to generalize several basic results from the theory of D-modules to the representation theory of rational Cherednik algebras. We relate characterizations of holonomic modules in terms of singular support and Gelfand-Kirillov dimension. We study pullback, pushforward, and dual on the derived category of (holonomic) Cherednik modules for certain classes of maps between varieties. We prove, in the case of generic parameters for the rational Cherednik algebra, that pushforward with respect to an open affine inclusion preserves holonomicity. Finally, we relate the global sections algebra of the sheaf of Cherednik algebras associated to a smooth quadric hypersurface of Pn to the Dunkl angular momentum algebra of Feigin-Hakobyan. In particular, this lets us relate the angular momentum algebra for a rank 3 Coxeter group to the rank 2 symplectic reflection algebra for a corresponding finite subgroup of SL2. / by Daniel Thompson. / Ph. D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 117-128). / Coherent, large scale dynamics in many nonequilibrium physical, biological, or information transport networks are driven by small-scale local energy input. In the first part of this thesis, we introduce and explore two analytically tractable nonlinear models for such active flow networks, drawing motivation from recent microfluidic experiments on bacterial and other microbial suspensions. In contrast to equipartition with thermal driving, we find that active friction selects discrete states with only a limited number of modes excited at distinct fixed amplitudes. When the active transport network is incompressible, these modes are cycles with constant flow; when it is compressible, they are oscillatory. As is common in such network dynamical systems, the spectrum of the underlying graph Laplacian plays a key role in controlling the flow. Spectral graph theory has traditionally prioritized analyzing Laplacians of unweighted networks with specified adjacency properties. For the second part of the thesis, we introduce a complementary framework, providing a mathematically rigorous positively weighted graph construction that exactly realizes any desired spectrum. We illustrate the broad applicability of this approach by showing how designer spectra can be used to control the dynamics of three archetypal physical systems. Specifically, we demonstrate that a strategically placed gap induces weak chimera states in Kuramoto-type oscillator networks, tunes or suppresses pattern formation in a generic Swift-Hohenberg model, and leads to persistent localization in a discrete Gross-Pitaevskii quantum network. / by Aden Forrow. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1997. / Includes bibliographical references (p. 85-88). / by Alexander Postnikov. / Ph.D.

Massachusetts Institute of Technology. Dept. of Mathematics. Thesis. 1966. Ph.D. / Bibliography: leaf 66. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1995. / Includes bibliographical references (leaves 159-160). / by Tao Kai Lam. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011. / Cataloged from PDF version of thesis. In title on title page, double underscored "p̳" appears as script. / Includes bibliographical references (p. 121-124). / In this thesis, I first generalize Kisin's theory of finite slope subspaces to arbitrary p-adic fields, and then apply it to the generic fibers of Galois deformation spaces. I study the finite slope deformation rings in details by computing the dimensions of their Zariski cotangent spaces via Galois cohomologies. It turns out that the Galois cohomologies tell us not only the formal smoothness of finite slope deformation rings, but also the behavior of the Sen operator near a generic de Rham representation. Applying these results to the finite slope subspace of two dimensional Galois representations of the absolute Galois group of a p-adic field, we are able to show that a generic (indecomposible) de Rham representation lies in the finite slope subspace. It follows from the construction of the finite slope subspace that the complete local ring of a point in the finite slope subspace is closely related to the finite slope deformation ring at the same point. As a consequence, we manage to show the flatness of the weight map near generic de Rham points, and accumulation and smoothness of generic de Rham points. In particular, we have a precise dimension formula for the finite slope subspace. Taking into account twists by characters, we define the nearly finite slope subspace, which is believed to serve as the local eigenvariety, as is suggested by Colmez's theory of trianguline representation. Following Gouv~a- Mazur and Kisin, we construct an infinite fern in the local Galois deformation space. Moreover, we define the global eigenvariety for GL2 over any number field, and give a lower bound of its dimension. / by Fucheng Tan. / Ph.D.