Mathematics - Ph.D. / Sc.D.
http://hdl.handle.net/1721.1/7843
Sat, 01 Oct 2016 18:58:16 GMT2016-10-01T18:58:16ZSpecial 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).
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/1721.1/1046082016-01-01T00:00:00ZYang-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).
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/1721.1/1046072016-01-01T00:00:00ZFields 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).
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/1721.1/1046062016-01-01T00:00:00ZParabolic Springer resolution
http://hdl.handle.net/1721.1/104605
Parabolic Springer resolution
Boger, D. (Dorin)
Let G be a reductive group over a field k = k. Let P be a parabolic subgroup. We construct a functor Groupoid ... is a connected space, which induces an action of generalizing a classical result. It is also a part of a study of natural equivalences between ... for P, Q associated parabolic subgroups.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016.; Cataloged from PDF version of thesis.; Includes bibliographical references (pages 73-75).
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/1721.1/1046052016-01-01T00:00:00Z