Contemporary Mathematics, Volume: 570
2012; 185 pp; softcover
ISBN-13: 978-0-8218-4992-7
Expected publication date is June 15, 2012.
This volume contains research and expository articles from the courses and talks given at the RSME Lluis A. Santalo Summer School, "Geometric Analysis", held June 28-July 2, 2010, in Granada, Spain.
The goal of the Summer School was to present some of the many advances currently taking place in the interaction between partial differential equations and differential geometry, with special emphasis on the theory of minimal surfaces.
This volume includes expository articles about the current state of specific problems involving curvature and partial differential equations, with interactions to neighboring fields such as probability. An introductory, mostly self-contained course on constant mean curvature surfaces in Lie groups equipped with a left invariant metric is provided.
The volume will be of interest to researchers, post-docs, and advanced PhD students in the interface between partial differential equations and differential geometry.
Graduate students and research mathematicians interested in differential geometry and partial differential geometry.
J. A. Galvez and P. Mira -- Geometric PDEs in the presence of isolated singularities
W. H. Meeks III and J. Perez -- Constant mean curvature surfaces in metric Lie groups
R. W. Neel -- Stochastic methods for minimal surfaces
F. Pacard -- The role of minimal surfaces in the study of the Allen-Cahn equation
G. Tinaglia -- On curvature estimates for constant mean curvature surfaces
ISBN: 978-0-8218-5327-6
Series, Volume: Proceedings of Symposia in Applied Mathematics, Volume 70
Published: 15 July 2012; Copyright Year: 2012; Pages: 232; Hardcover
Applications
Geometry and Topology
Recommended for Bookstores
Applied Mathematics
Readership: Graduate students and research mathematicians interested in applied and computational topology.
What is the shape of data? How do we describe flows? Can we count by integrating? How do
we plan with uncertainty? What is the most compact representation? These questions, while unrelated,
become similar when recast into a computational setting. Our input is a set of finite, discrete, noisy samples
that describes an abstract space. Our goal is to compute qualitative features of the unknown space. It turns
out that topology is sufficiently tolerant to provide us with robust tools.
This volume is based on lectures delivered at the 2011 AMS Short Course on Computational Topology, held
January 4?5, 2011 in New Orleans, Louisiana.
The aim of the volume is to provide a broad introduction to recent techniques from applied and computational
topology. Afra Zomorodian focuses on topological data analysis via efficient construction of combinatorial
structures and recent theories of persistence. Marian Mrozek analyzes asymptotic behavior of
dynamical systems via efficient computation of cubical homology. Justin Curry, Robert Ghrist, and Michael
Robinson present Euler Calculus, an integral calculus based on the Euler characteristic, and apply it to sensor
and network data aggregation. Michael Erdmann explores the relationship of topology, planning, and
probability with the strategy complex. Jeff Erickson surveys algorithms and hardness results for topological
optimization problems.
A. Zomorodian -- Topological data analysis
M. Mrozek -- Topological dynamics: Rigorous numerics via cubical homology
J. Curry, R. Ghrist, and M. Robinson -- Euler calculus with applications to signals and sensing
M. Erdmann -- On the topology of discrete planning with uncertainty
J. Erickson -- Combinatorial optimization of cycles and bases
Index
ISBN: 978-0-8218-4923-1
Series, Proceedings of Symposia in Applied Mathematics, Volume 71
Published: 19 July 2012; Copyright Year: 2012; Pages: approximately 270; Hardcover;
Applications
Applied Mathematics
Readership: Graduate students and research mathematicians interested in computer science, classical and
quantum information theory, and the related mathematics and physics.
This volume is based on the 2008 Clifford Lectures on Information Flow in Physics, Geometry and
Logic and Computation, held March 12?15, 2008, at Tulane University in New Orleans, Louisiana.
The varying perspectives of the researchers are evident in the topics represented in the volume, including
mathematics, computer science, quantum physics and classical and quantum information. A number of
the articles address fundamental questions in quantum information and related topics in quantum physics,
using abstract categorical and domain-theoretic models for quantum physics to reason about such
systems and to model spacetime.
Readers can expect to gain added insight into the notion of information flow and how it can be understood
in many settings. They also can learn about new approaches to modeling quantum mechanics that
provide simpler and more accessible explanations of quantum phenomena, which donft require the arcane
aspects of Hilbert spaces and the cumbersome notation of bras and kets.
S. Abramsky and C. Heunen -- H*-algebras and nonunital Frobenius algebras: First steps in infinite-dimensional categorical quantum mechanics
H. Barnum, J. Barrett, M. Leifer, and A. Wilce -- Teleportation in general probabilistic theories
A. Brandenburger, A. Friedenberg, and H. J. Keisler -- Fixed points in epistemic game theory
B. Coecke and B. Edwards -- Spekken's toy theory as a category of processes
P. Hines and P. Scott -- Categorical traces from single-photon linear optics
K. H. Hofmann and M. Mislove -- Compact affine monoids, harmonic analysis and information theory
K. Martin -- The scope of a quantum channel
K. Martin and P. Panangaden -- Spacetime geometry from causal structure and a measurement
D. Pavlovic -- Geometry of abstraction in quantum computation
Ordinary Differential Equations
Qualitative Theory
ISBN: 978-0-8218-8749-3
Series: Graduate Studies in Mathematics, Volume 137
Bibliographic Information: Published: 11 July 2012; Copyright Year: 2012; Pages: 248; Hardcover;
Differential Equations
Textbook
Readership: Undergraduate and graduate students interested in ordinary differential equations, dynamical
systems, bifurcation theory, and Hamiltonian systems.
This textbook provides a comprehensive introduction to the qualitative theory of ordinary differential
equations. It includes a discussion of the existence and uniqueness of solutions, phase portraits,
linear equations, stability theory, hyperbolicity and equations in the plane. The emphasis is primarily on
results and methods that allow one to analyze qualitative properties of the solutions without solving the
equations explicitly. The text includes numerous examples that illustrate in detail the new concepts and
results as well as exercises at the end of each chapter. The book is also intended to serve as a bridge to
important topics that are often left out of a course on ordinary differential equations. In particular, it provides
brief introductions to bifurcation theory, center manifolds, normal forms and Hamiltonian systems.
Basic concepts and linear equations
Ordinary differential equations
Linear equations and conjugacies
Stability of hyperbolicity
Stability and Lyapunov functions
Hyperbolicity and topological conjugacies
Existence of invariant manifolds
Equations in the plane
Index theory
Poincare-Bendixson theory
Further topics
Bifurcations and center manifolds
Hamiltonian systems
Bibliography
Index