Department of Mathematics
http://hdl.handle.net/1721.1/7841
2016-10-28T02:42:10ZRevealing and analyzing imperceptible deviations in images and videos
http://hdl.handle.net/1721.1/105088
Revealing and analyzing imperceptible deviations in images and videos
Wadhwa, Neal
The world is filled with objects that appear to follow some perfect model. A sleeping baby might look still and a house's roof .should be straight. However, both the baby and the roof can deviate subtly from their ideal models of perfect stillness and perfect straightness. These deviations can reveal important information like whether the baby is breathing normally or whether the house's roof is sagging. In this dissertation, we make the observation that these subtle deviations produce a visual signal that while invisible to the naked eye can be extracted from ordinary and ubiquitous images and videos. We propose new computational techniques to reveal these subtle deviations by producing new images and videos, in which the tiny deviations have been magnified. We focus on magnifying deviations from two ideal models: perfect stillness and perfect geometries in space. In the first case, we leverage the complex steerable pyramid, a localized version of the Fourier transform, whose notion of local phase can be used to process and manipulate small motions or changes from stillness in videos. In the second case, we find hidden geometric deformations in images by localizing edges to sub-pixel precision. In both cases, we experimentally validate that the tiny deviations we magnify are indeed real, comparing them to alternative ways of measuring tiny motions and subtle geometric deformations in the world. We also give a careful analysis of how noise in videos impacts our ability to see tiny motions. Additionally, we show the utility of revealing hidden deviations in a wide variety of fields, such as biology, physics and structural analysis.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016.; Cataloged from PDF version of thesis.; Includes bibliographical references (pages 191-197).
2016-01-01T00:00:00ZSpecial gradient trajectories counted by simplex straightening
http://hdl.handle.net/1721.1/104608
Special gradient trajectories counted by simplex straightening
Alpert, Hannah (Hannah Chang)
We prove three theorems based on lemmas of Gromov involving the simplicial norm on stratified spaces. First, the Gromov singular fiber theorem (with proof originally sketched by Gromov) relates the simplicial norm to the number of maximum multiplicity critical points of a smooth map of manifolds that drops in dimension by 1. Second, the multitangent trajectory theorem (proved with Gabriel Katz) relates the simplicial norm to the number of maximum-multiplicity tangent trajectories of a nowhere-vanishing gradient-like vector field on a manifold with boundary. And third, the Morse broken trajectory theorem relates the simplicial volume to the number of maximally broken trajectories of the gradient flow of a Morse--Smale function. Corollary: a Morse function on a closed hyperbolic manifold must have a critical point of every Morse index.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016.; Cataloged from PDF version of thesis.; Includes bibliographical references (pages 65-67).
2016-01-01T00:00:00ZYang-Mills replacement
http://hdl.handle.net/1721.1/104607
Yang-Mills replacement
Berchenko-Kogan, Yakov
We develop an analog of the harmonic replacement technique of Colding and Minicozzi in the gauge theory context. The idea behind harmonic replacement dates back to Schwarz and Perron, and the technique involves taking a function v: ... defined on a surface ... and replacing its values on a small ball B2 ... with a harmonic function u that has the same values as v on the boundary &B2 . The resulting function on ... has lower energy, and repeating this process on balls covering ..., one can obtain a global harmonic map in the limit. We develop the analogous procedure in the gauge theory context. We take a connection B on a bundle over a four-manifold X, and replace it on a small ball ... with a Yang-Mills connection A that has the same restriction to the boundary [alpha]B4 as B, and we obtain bounds on the difference ... in terms of the drop in energy. Throughout, we work with connections of the lowest possible regularity ... (X), the natural choice for this context, and so our gauge transformations are in ... (X) and therefore almost but not quite continuous, leading to more delicate arguments than are available in higher regularity.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016.; Cataloged from PDF version of thesis.; Includes bibliographical references (pages 87-88).
2016-01-01T00:00:00ZFields of rationality of cuspidal automorphic representations
http://hdl.handle.net/1721.1/104606
Fields of rationality of cuspidal automorphic representations
Binder, John (John Robert)
This thesis examines questions related to the growth of fields of rationality of cuspidal automorphic representations in families. Specifically, if F is a family of cuspidal automorphic representations with fixed central character, prescribed behavior at the Archimedean places, and such that the finite component [pi] [infinity] has a [Gamma]-fixed vector, we expect the proportion of [pi] [epsilon] F with bounded field of rationality to be close to zero if [Gamma] is small enough. This question was first asked, and proved partially, by Serre for families of classical cusp forms of increasing level. In this thesis, we will answer Serre's question affirmatively by converting the question to a question about fields of rationality in families of cuspidal automorphic GL2(A) representations. We will consider the analogous question for certain sequences of open compact subgroups F in UE/F(n). A key intermediate result is an equidistribution theorem for the local components of families of cuspidal automorphic representations.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016.; Cataloged from PDF version of thesis.; Includes bibliographical references (pages 115-120).
2016-01-01T00:00:00Z