Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 41-45). / Recent work on equivariant aspects of mirror symmetry has discovered relations between the equivariant quantum cohomology of symplectic resolutions and Casimir-type connections (among many other objects). We provide a new example of this theory in the setting of the affine Grassmannian, a fundamental space in the geometric Langlands program. More precisely, we identify the equivariant quantum connection of certain symplectic resolutions of slices in the affine Grassmannian of a semisimple group G with a trigonometric Knizhnik-Zamolodchikov (KZ)-type connection of the Langlands dual group of G. These symplectic resolutions are expected to be symplectic duals of Nakajima quiver varieties, and thus our result is an analogue of (part of) the work of Maulik and Okounkov in the symplectic dual setting. / by Michael Viscardi. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 39-40). / One of the fundamental results in geometric representation theory is the geometric Satake equivalence, between the category of spherical perverse sheaves on the affine Grassmannian of a reductive group G and the category of representations of its Langlands dual group. The category of spherical perverse sheaves sits naturally in an equivariant derived category, and this larger category was described in terms of the dual group by Bezrukavnikov-Finkelberg. Recently, Finkelberg-Lysenko proved a "twisted" version of the geometric Satake equivalence, which involves perverse sheaves associated to twisted local systems on a line bundle over the affine Grassmannian. In this thesis we extend the Bezrukavnikov-Finkelberg description of the equivariant derived category to the twisted setting. Our method builds on theirs, but some additional subtleties arise. In particular, we cannot use Ginzburg's results on equivariant cohomology. We get around this by using localization techniques in equivariant cohomology in a more detailed way, allowing as to reduce certain computations to those of Ginzburg and Bezrukavnikov-Finkelberg. We also use show how our methods can be extended to explain an equivalence between Iwahori-equivariant peverse sheaves and twisted Iwahori-equivariant perverse sheaves on dual affine Grassmannians. This equivalence was observed earlier by Arkhipov-Bezrukavnikov-Ginzburg by combining several deep results, and they posed the problem of finding a more direct explanation. Finally, we explain how our results fit into the (quantum) geometric Langlands program. / by Bhairav Singh. / Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2002. / Includes bibliographical references (p. 89-97). / Support Vector Machines (SVMs) have attracted recent attention as a learning technique to attack classification problems. The goal of my thesis work is to improve computational algorithms as well as the mathematical understanding of SVMs, so that they can be easily applied to real problems. SVMs solve classification problems by learning from training examples. From the geometry, it is easy to formulate the finding of SVM classifiers as a linearly constrained Quadratic Programming (QP) problem. However, in practice its dual problem is actually computed. An important property of the dual QP problem is that its solution is sparse. The training examples that determine the SVM classifier are known as support vectors (SVs). Motivated by the geometric derivation of the primal QP problem, we investigate how the dual problem is related to the geometry of SVs. This investigation leads to a geometric interpretation of the scaling property of SVMs and an algorithm to further compress the SVs. A random model for the training examples connects the Hessian matrix of the dual QP problem to Wishart matrices. After deriving the distributions of the elements of the inverse Wishart matrix Wn-1(n, nI), we give a conjecture about the summation of the elements of Wn-1(n, nI). It becomes challenging to solve the dual QP problem when the training set is large. We develop a fast algorithm for solving this problem. Numerical experiments show that the MATLAB implementation of this projected Conjugate Gradient algorithm is competitive with benchmark C/C++ codes such as SVMlight and SvmFu. Furthermore, we apply SVMs to time series data. / (cont.) In this application, SVMs are used to predict the movement of the stock market. Our results show that using SVMs has the potential to outperform the solution based on the most widely used geometric Brownian motion model of stock prices. / by Tong Wen. / Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1999. / Includes bibliographical references (p. 60-61). / by Payman L. Kassaei. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1992. / Includes bibliographical references (leaf 47). / by Eugenio Garnica-Vigil. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1971. / Vita. / Bibliography: leaves 86-87. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2009. / Includes bibliographical references (p. 253-257). / In this thesis, I first systematically develop the theory of nonarchimedean differential modules, deducing fundamental theorems about the variation of generic radii of convergence for differential modules over polyannuli. The theorems assert that the log of subsidiary radii of convergence are convex, continuous, and piecewise affine functions of the log of the radii of the polyannuli. Then I apply these results to the ramification theory and deduce the fundamental result, Hasse-Arf theorem, for ramification filtrations defined by Abbes and Saito. Also, we include a comparison theorem to differential conductors and Borger's conductors in the equal characteristic case. Finally, I globalize this construction and give a new understanding of the ramification theory for smooth varieties, which provides some new insight to the global class field theory. We end the thesis with a series of conjectures as a starting point of a long going project on understanding global ramification. / by Liang Xiao. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1997. / Includes bibliographical references (p. 125-128). / by Harry Lewis Wolfgang III. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1973. / Vita. / Bibliography: leaves 132-136. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2006. / Includes bibliographical references (p. 183-190) and Index. / This thesis studies surfaces which minimize area, subject to a fixed boundary and to a free boundary with length constraint. Based on physical experiments, I make two conjectures. First, I conjecture that minimizers supported on generic wires have finitely many surface components. I approach this conjecture by proving that surface components of near-wire minimizers are Lipschitz graphs in wire Frenet coordinates, and appear near maxima of wire curvature. Second, I conjecture and prove that surface components of near-wire minimizers are C1 at corners where the thread touches the wire interior. Moreover, the limit of the surface normal field is the Frenet binormal of the wire at the corner point. This shows local wire geometry dominates global wire geometry in influencing the surface corner. Third, I show that these two conjectures are related: assuming additional regularity up to the corner, the finiteness conjecture follows. / by Benjamin K. Stephens. / Ph.D.