Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 211-221). / I first give an overview of the thesis and Matrix Product States (MPS) representation of quantum spin systems on a line with an improvement on the notation. The rest of this thesis is divided into two parts. The first part is devoted to eigenvalues of quantum many-body systems (QMBS). I introduce Isotropic Entanglement (IE) and show that the distribution of QMBS with generic interactions can be accurately obtained using IE. Next, I discuss the eigenvalue distribution of one particle hopping random Schrbdinger operator in one dimension from free probability theory in context of the Anderson model. The second part is devoted to ground states and gap of QMBS. I first give the necessary background on frustration free Hamiltonians, real and imaginary time evolution of quantum spin systems on a line within MPS representation and the numerical implementation. I then prove the degeneracy and unfrustration condition for quantum spin chains with generic local interactions. Following this, I summarize my efforts in proving lower bounds for the entanglement of the ground states, which includes partial results, with the hope that it will inspire future work resulting in solving the conjecture given. Next I discuss two interesting measure zero examples where the Hamiltonians are carefully constructed to give unique ground states with high entanglement. This includes exact calculations of Schmidt numbers, entanglement entropies and a novel technique for calculating the gap. The last chapter elaborates on one of the measure zero examples (i.e., d = 3) which is the first example of a Frustration Free translation-invariant spin-i chain that has a unique highly entangled ground state and exhibits signatures of a critical behavior. / by Ramis Movassagh. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004. / Includes bibliographical references (p. 243-248). / (cont.) yield a new proof of a result of Mochizuki yield a new proof of a result of Mochizuki Frobenius-unstable bundles for C general, and hence obtaining a self-contained proof of the resulting formula for the degree of V₂. / Using limit linear series and a result controlling degeneration from separable maps to inseparable maps, we give a formula for the number of self-maps of P¹ with ramification to order e[sub]i at general points P[sub]i the case that all e[sub]i are less than the characteristic. We also develop a new, more functorial construction for the basic theory of limit linear series, which works transparently in positive and mixed characteristics, yielding a result on lifting linear series from characteristic p to characteristic 0, and even showing promise for generalization to higher-dimensional varieties. Now, let C be a curve of genus 2 over a field k of positive characteristic, and V₂ the Verschiebung rational map induced by pullback under Frobenius on moduli spaces of semistable vector bundles of rank two and trivial determinant. We show that if the Frobenius-unstable vector bundles are deformation-free in a suitable sense, then they are precisely the undefined points of V₂, and may each be resolved by a single blow-up; in this setting, we are able to calculate the degree of V₂ in terms of the number of Frobenius-unstable bundles, and describe the image of the exceptional divisors. We finally examine the Frobenius-unstable bundles on C by studying connections with vanishing p-curvature on certain unstable bundles on C. Using explicit formulas for p-curvature, we completely describe the Frobenius-unstable bundles in characteristics 3, 5, 7. We classify logarithmic connections with vanishing p-curvature on vector bundles of rank 2 on P¹ in terms of self-maps of P¹ with prescribed ramification. Using our knowledge of such maps, we then glue the connections to a nodal curve and deform to a smooth curve to / by Brian Osserman. / Ph.D.

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.