Mathematics - Ph.D. / Sc.D.
http://hdl.handle.net/1721.1/7680
2016-04-03T13:33:57ZPseudoholomorphic quilts with figure eight singularity
http://hdl.handle.net/1721.1/101823
Pseudoholomorphic quilts with figure eight singularity
Bottman, Nathaniel Sandsmark
In this thesis, I prove several results toward constructing a machine that turns Lagrangian correspondences into A[infinity],-functors between Fukaya categories. The core of this construction is pseudoholomorphic quilts with figure eight singularity. In the first part, I propose a blueprint for constructing an algebraic object that binds together the Fukaya categories of many different symplectic manifolds. I call this object the "symplectic A[infinity]-2-category Symp". The key to defining the structure maps of Symp is the figure eight bubble. In the second part, I establish a collection of strip-width-independent elliptic estimates. The key is function spaces which augment the Sobolev norm with another term, so that the norm of a product can be bounded by the product of the norms in a manner which is independent of the strip-width. Next, I prove a removable singularity theorem for the figure eight singularity. Using the Gromov compactness theorem mentioned in the following paragraph, I adapt an argument of Abbas-Hofer to uniformly bound the norm of the gradient of the maps in cylindrical coordinates centered at the singularity. I conclude by proving a "quilted" isoperimetric inequality. In the third part, which is joint with Katrin Wehrheim, I use my collection of estimates to prove a Gromov compactness theorem for quilts with a strip of (possibly non-constant) width shrinking to zero. This features local C[infinity]-convergence away from the points where energy concentrates. At such points, we produce a nonconstant quilted sphere.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015.; Cataloged from PDF version of thesis.; Includes bibliographical references (pages 107-109).
2015-01-01T00:00:00ZA-infinity algebras for Lagrangians via polyfold theory for Morse trees with holomorphic disks
http://hdl.handle.net/1721.1/101822
A-infinity algebras for Lagrangians via polyfold theory for Morse trees with holomorphic disks
Li, Jiayong, Ph. D. Massachusetts Institute of Technology
For a Lagrangian submanifold, we define a moduli space of trees of holomorphic disk maps with Morse flow lines as edges, and construct an ambient space around it which we call the quotient space of disk trees. We show that this ambient space is an M-polyfold with boundary and corners by combining the infinite dimensional analysis in sc-Banach space with the finite dimensional analysis in Deligne-Mumford space. We then show that the Cauchy-Riemann section is sc-Fredholm, and by applying the polyfold perturbation we construct an A[infinity]. algebra over Z₂ coefficients. Under certain assumptions, we prove the invariance of this algebra with respect to choices of almost-complex structures.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015.; Cataloged from PDF version of thesis.; Includes bibliographical references (pages 253-254).
2015-01-01T00:00:00ZPreserving patient privacy in biomedical data analysis
http://hdl.handle.net/1721.1/101821
Preserving patient privacy in biomedical data analysis
Simmons, Sean Kenneth
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.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015.; Cataloged from PDF version of thesis.; Includes bibliographical references (pages 147-154).
2015-01-01T00:00:00ZEconomic behavior from an evolutionary perspective
http://hdl.handle.net/1721.1/101820
Economic behavior from an evolutionary perspective
Zhang, Ruixun, Ph. D. Massachusetts Institute of Technology
The conflict between rational models of economic behavior and their systematic deviations, often referred to as behavioral economics, is one of the most hotly debated issues in social sciences. This thesis reconciles the two opposing perspectives by applying evolutionary principles to economic behavior and deriving implications that cut across species, physiology, and genetic origins. In the context of a binary-choice model, we first show that risk aversion emerges via natural selection if reproductive risk is "systematic", i.e., correlated across individuals in a given generation. The degree of risk aversion is determined by the stochastic nature of reproductive rates, and different statistical properties lead to different utility functions. More generally, irrational behaviors are not just mere divergence from rationality, but seeds necessary for successfully coping with environmental transformations. Furthermore, there is an optimal degree of irrationality in the population depending on the degree of environmental stochasticity. When applied to evolutionary biology, we show that what appears to be group selection may, in fact, simply be the consequence of natural selection occurring in stochastic environments with "systematic" risks. Those individuals with highly correlated risks will appear to form "groups", even if their actions are totally autonomous, mindless, and, prior to selection, uniformly randomly distributed in the population. Evolutionary principles can also be used to model the dynamics of financial markets. In a multiperiod model of the contagion of investment ideas, we show that heterogeneous investment styles can coexist in the long run, implying a wider variation of diverse strategies compared to traditional theories. These results may provide new insights to the survival of a wide range of hedge funds. In a model that investors maximize their relative wealth, the initial wealth plays a critical role in determining how the optimal behavior deviates from the Kelly Criterion, regardless of whether the investor is myopic or maximizing the infinite-horizon wealth.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015.; Cataloged from PDF version of thesis.; Includes bibliographical references (pages 155-174).
2015-01-01T00:00:00Z