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Geometric Analysis
This book covers recent advances in several important areas of geometric analysis including extremal eigenvalue problems, mini-max methods in minimal surfaces, CR geometry in dimension three, and the Ricci flow and Ricci limit spaces. An output of the CIME Summer School "Geometric Analysis" held in Cetraro in 2018, it offers a collection of lecture notes prepared by Ailana Fraser (UBC), André Neves (Chicago), Peter M. Topping (Warwick), and Paul C. Yang (Princeton). These notes will be a valuable asset for researchers and advanced graduate students in geometric analysis.
On a Free Boundary Problem for Embedded Minimal Surfaces and Instability Theorems for Manifolds with Positive Isotropic Curvature
In this thesis, we describe a min-max construction of embedded minimal surfaces satisfying the free boundary condition in any compact 3-manifolds with boundary. We also prove the instability of minimal surfaces of certain conformal type in 4- manifolds with positive isotropic curvature. Given a compact 3-manifold M with boundary [d̳]M, consider the problem of find- ing an embedded minimal surface [Sigma] which meets [d̳]M orthogonally along [d̳][Sigma]. These surfaces are critical points to the area functional with respect to variations preserving [delta]M. We will use a min-max construction to construct such a free boundary solution and prove the regularity of such solution up to the fre...
Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations
This volume presents the proceedings of the Southeast Geometry Seminar for the meetings that took place bi-annually between the fall of 2009 and the fall of 2011, at Emory University, Georgia Institute of Technology, University of Alabama Birmingham, and the University of Tennessee. Talks at the seminar are devoted to various aspects of geometric analysis and related fields, in particular, nonlinear partial differential equations, general relativity, and geometric topology. Articles in this volume cover the following topics: a new set of axioms for General Relativity, CR manifolds, the Mane Conjecture, minimal surfaces, maximal measures, pendant drops, the Funk-Radon-Helgason method, ADM-mass and capacity, and extrinsic curvature in metric spaces.
Finite Fields, with Applications to Combinatorics
This book uses finite field theory as a hook to introduce the reader to a range of ideas from algebra and number theory. It constructs all finite fields from scratch and shows that they are unique up to isomorphism. As a payoff, several combinatorial applications of finite fields are given: Sidon sets and perfect difference sets, de Bruijn sequences and a magic trick of Persi Diaconis, and the polynomial time algorithm for primality testing due to Agrawal, Kayal and Saxena. The book forms the basis for a one term intensive course with students meeting weekly for multiple lectures and a discussion session. Readers can expect to develop familiarity with ideas in algebra (groups, rings and fields), and elementary number theory, which would help with later classes where these are developed in greater detail. And they will enjoy seeing the AKS primality test application tying together the many disparate topics from the book. The pre-requisites for reading this book are minimal: familiarity with proof writing, some linear algebra, and one variable calculus is assumed. This book is aimed at incoming undergraduate students with a strong interest in mathematics or computer science.
Motivic Euler Products and Motivic Height Zeta Functions
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On Singular Vortex Patches, I: Well-Posedness Issues
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Multiplicative Invariant Fields of Dimension $leq 6$
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Function Spaces of Logarithmic Smoothness: Embeddings and Characterizations
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Symbolic Extensions of Amenable Group Actions and the Comparison Property
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Asymptotic Spreading for General Heterogeneous Fisher-KPP Type Equations
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