Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 83-85). / For each integer D >/= 5 with D =/- 0 or 1 mod 4, the Weierstrass curve WD is an algebraic curve and a finite volume hyperbolic orbifold which admits an algebraic and isometric immersion into the moduli space of genus two Riemann surfaces. The Weierstrass curves are the main examples of Teichmüller curves in genus two. The primary goal of this thesis is to determine the number and type of orbifold points on each component of WD. Our enumeration of the orbifold points, together with [Ba] and [Mc3], completes the determination of the homeomorphism type of WD and gives a formula for the genus of its components. We use our formula to give bounds on the genus of WD and determine the Weierstrass curves of genus zero. We will also give several explicit descriptions of each surface labeled by an orbifold point on WD. / by Ronen E. Mukamel. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 147-154). / The growing number of large biomedical databases and electronic health records promise to be an invaluable resource for biomedical researchers. Recent work, however, has shown that sharing this data- even when aggregated to produce p-values, regression coefficients, count queries, and minor allele frequencies (MAFs)- may compromise patient privacy. This raises a fundamental question: how do we protect patient privacy while still making the most out of their data? In this thesis, we develop various methods to perform privacy preserving analysis on biomedical data, with an eye towards genomic data. We begin by introducing a model based measure, PrivMAF, that allows us to decide when it is safe to release MAFs. We modify this measure to deal with perturbed data, and show that we are able to achieve privacy guarantees while adding less noise (and thus preserving more useful information) than previous methods. We also consider using differentially private methods to preserve patient privacy. Motivated by cohort selection in medical studies, we develop an improved method for releasing differentially private medical count queries. We then turn our eyes towards differentially private genome wide association studies (GWAS). We improve the runtime and utility of various privacy preserving methods for genome analysis, bringing these methods much closer to real world applicability. Building off this result, we develop differentially private versions of more powerful statistics based off linear mixed models. / by Sean Kenneth Simmons. / Ph. D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2000. / Includes bibliographical references (p. 140-148). / Crucial in spatial interpolation of stochastic processes is the determination of the underlying dependency of the data. The dependency can be represented by an underlying covariogram, variogram, or generalized covariogram. Estimating this function in a nonparametric way is the theme of this thesis. If the function can be found accurately, then kriging is the optimal linear interpolation technique. A nev,· technique for variogram model selection using the derivative of the empirical variogram and non-negative least squares is discussed. The eigenstructure of the spatial design matrix, the key matrix in Matheron's variogram estimator is determined. Then a nonparametric estimator of the variogram and covariogram of a spatial stochastic process is found. The optimal node selection is determined as well as conditions when the spectral coefficients can be found without a non-linear algorithm. A method of extending isotropic positive definite functions in ]Rd is determined in order to avoid a Gibbs effect on the Fourier-Bessel expansion. Finally, a nonparametric estimator of the generalized covariance is discussed. / by David John Gorsich. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2007. / This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections. / Includes bibliographical references (leaves 60-62). / We give nontrivial upper and lower bounds for the total number of distinct degree sequences among all simple, unlabeled graphs on n vertices (graphical partitions on n vertices). Our upper bound is ... for some constant C, and improvement of ... over the trivial upper bound which is asymptotic to ... Our lower bound is ..., and improvement of ... over the trivial lower bound which is asymptotic to ... / by Jason Matthew Burns. / Ph.D.

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2003. / Includes bibliographical references (leaf 61). / We develop and describe some of the basic tools of intersection theory in algebraic geometry. Some classical enumerative problems are then solved using these methods. In particular, we discuss the Fano variety of a cubic in surface in P3 , determinantal varieties, and the number of conics tangent to five conics in P2 . / by Astrid Adele Giugni. / S.M.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1967. / Vita. / Bibliography: p. 262-266. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018. / Cataloged from PDF version of thesis. / Includes bibliographical references (page 41). / Given a Galois cover of curves X --> Y with Galois group G which is totally ramified at a point x and unramified elsewhere, restriction to the punctured formal neighborhood of x induces a Galois extension of Laurent series rings k((u))/k((t)). If we fix a base curve Y, we can ask when a Galois extension of Laurent series fields comes from a global cover of Y in this way. Harbater proved that over a separably closed field, every Laurent series extension comes from a global cover for any base curve if G is a p-group, and he gave a condition for the uniqueness of such an extension. Using a generalization of Artin-Schreier theory to non-abelian p-groups, we characterize the curves Y for which this extension property holds and for which it is unique up to isomorphism, but over a more general ground field. / by Renee Hyunjeong Bell. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1995. / Includes bibliographical references (p. 65-66). / by Tom Charles Braden. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 49-51). / In this thesis we construct a functor from the perfect subcategory of the coherent version of the affine Hecke category in type A to the finite constructible Hecke category, partly categorifying a certain natural homomorphism of the corresponding Hecke algebras. This homomorphism sends generators of the Bernstein's commutative subalgebra inside the affine Hecke algebra to Jucys-Murphy elements in the finite Hecke algebra. Construction employs the general strategy devised by Bezrukavnikov to prove the equivalence of coherent and constructible variants of the affine Hecke category. Namely, we identify an action of the category Rep(GLn) on the finite Hecke category, and lift this action to a functor from the perfect derived category of the Steinberg variety, by equipping it with various additional data. / by Kostiantyn Tolmachov. / Ph. D.

Thesis: S.M., Massachusetts Institute of Technology, Department of Mathematics, 2018. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 27-29). / We first give an overview of methods used to study the decomposability of Jacobians of curves over the complex numbers. This involves studying the action of a finite group on an abelian variety in general. Next, we use methods for point counting properties of curves over finite fields to study the decomposability of Jacobians over number fields and finite fields. For example, we show that the genus of curves over number fields whose Jacobians are isomorphic to a product of elliptic curves satisfying certain reduction conditions is bounded and give restrictions on curves over number fields whose Jacobians are isomorphic to a product of elliptic curves. / by Soohyun Park. / S.M.