Thesis Defense

from Thursday
June 01, 2017 to Sunday
December 31, 2017
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Tuesday
June 06, 2017

2:00 PM - 3:30 PM
Math Tower 5-127
Dingxin Zhang, Stony Brook University
Degeneration of slopes

A. Grothendieck proves that the Newton polygons of a family of smooth projective algebraic varieties defined on a field of characteristic p > 0 go up under a smooth specialization. When the family acquires semistable singular members, we prove the smallest slope of the Newton polygons attached to the rigid cohomology groups cannot become smaller upon degeneration. This is achieved by constructing the “generic higher direct images” for a singular morphism, using the convergent topoi. Our result generalizes the theorem of Grothendieck, and partially answers a question of H. Esanult. As geometric applications, we prove that several families of algebraic varieties must have “ordinary” members. Special cases of these applications recover older theorems of L. Illusie and Wolf-Riedl.


Tuesday
December 05, 2017

1:00 PM
Math Tower 5-127
Robert Abramovic, Stony Brook University
The Positive Mass Theorem with Charge Outside Horizon(s)

A result of M. Herzlich proves the positive mass theorem for a Riemannian three-manifold containing a boundary whose mean curvature is bounded from above. The purpose of this thesis is to add charge to Herzlich's result. In particular, the positive mass theorem with charge is proved for time-symmetric initial data for the Einstein-Maxwell equations whose initial data contains either one or multiple boundary components. In the case of multiple boundary components, a condition is imposed on the electric field. The techniques used to prove the charged positive mass theorem for initial data with multiple components involves a nontrivial solution of the Dirac equation. Nontrivial solutions of the Dirac equation are then applied to prove the charged positive mass theorem for charged initial data consisting of the Riemannian 3-manifold with finitely many cylindrical ends. Further, a charged positive mass theorem for a Riemannian three-manifold with corners is proven in the non-time symmetric, extending a result by Shi and Tam.


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