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Triangles, Rotation, a Theorem and the Jackpot
by Dave Auckly - arXiv , 2013
This paper introduced undergraduates to the Atiyah-Singer index theorem. It includes a statement of the theorem, an outline of the easy part of the heat equation proof. It includes counting lattice points and knot concordance as applications.
Lectures on Minimal Surface Theory
by Brian White - arXiv , 2013
The goal was to give beginning graduate students an introduction to some of the most important basic facts and ideas in minimal surface theory. Prerequisites: the reader should know basic complex analysis and elementary differential geometry.
Noncompact Harmonic Manifolds
by Gerhard Knieper, Norbert Peyerimhoff - arXiv , 2013
We provide a survey on recent results on noncompact simply connected harmonic manifolds, and we also prove many new results, both for general noncompact harmonic manifolds and for noncompact harmonic manifolds with purely exponential volume growth.
Notes on Symmetric Spaces
by Jonathan Holland, Bogdan Ion - arXiv , 2012
Contents: Affine connections and transformations; Symmetric spaces; Orthogonal symmetric Lie algebras; Examples; Noncompact symmetric spaces; Compact semisimple Lie groups; Hermitian symmetric spaces; Classification of real simple Lie algebras.
Geometric Wave Equations
by Stefan Waldmann - arXiv , 2012
We discuss the solution theory of geometric wave equations as they arise in Lorentzian geometry: for a normally hyperbolic differential operator the existence and uniqueness properties of Green functions and Green operators is discussed.
Introduction to Evolution Equations in Geometry
by Bianca Santoro - arXiv , 2012
The author aimed at providing a first introduction to the main general ideas on the study of the Ricci flow, as well as guiding the reader through the steps of Kaehler geometry for the understanding of the complex version of the Ricci flow.
Gauge Theory for Fiber Bundles
by Peter W. Michor - Universitaet Wien , 1991
Gauge theory usually investigates the space of principal connections on a principal fiber bundle (P,p,M,G) and its orbit space under the action of the gauge group (called the moduli space), which is the group of all principal bundle automorphisms...
Orthonormal Basis in Minkowski Space
by Aleks Kleyn, Alexandre Laugier - arXiv , 2012
In this paper, we considered the definition of orthonormal basis in Minkowski space, the structure of metric tensor relative to orthonormal basis, procedure of orthogonalization. Contents: Preface; Minkowski Space; Examples of Minkowski Space.
Lectures on Fibre Bundles and Differential Geometry
by J.L. Koszul - Tata Institute of Fundamental Research , 1960
From the table of contents: Differential Calculus; Differentiable Bundles; Connections on Principal Bundles; Holonomy Groups; Vector Bundles and Derivation Laws; Holomorphic Connections (Complex vector bundles, Almost complex manifolds, etc.).
Synthetic Geometry of Manifolds
by Anders Kock - University of Aarhus , 2009
This textbook can be used as a non-technical and geometric gateway to many aspects of differential geometry. The audience of the book is anybody with a reasonable mathematical maturity, who wants to learn some differential geometry.
Ricci-Hamilton Flow on Surfaces
by Li Ma - Tsinghua University , 2003
Contents: Ricci-Hamilton flow on surfaces; Bartz-Struwe-Ye estimate; Hamilton's another proof on S2; Perelman's W-functional and its applications; Ricci-Hamilton flow on Riemannian manifolds; Maximum principles; Curve shortening flow on manifolds.
Lectures on Exterior Differential Systems
by M. Kuranishi - Tata Institute of Fundamental Research , 1962
Contents: Parametrization of sets of integral submanifolds (Regular linear maps, Germs of submanifolds of a manifold); Exterior differential systems (Differential systems with independent variables); Prolongation of Exterior Differential Systems.
Discrete Differential Geometry: An Applied Introduction
by M. Desbrun, P. Schroeder, M. Wardetzky - Columbia University , 2008
This new and elegant area of mathematics has exciting applications, as this text demonstrates by presenting practical examples in geometry processing (surface fairing, parameterization, and remeshing) and simulation (of cloth, shells, rods, fluids).
Global Theory Of Minimal Surfaces
by David Hoffman - American Mathematical Society , 2005
The wide variety of topics covered make this volume suitable for graduate students and researchers interested in differential geometry. The subjects covered include minimal and constant-mean-curvature submanifolds, Lagrangian geometry, and more.
Ricci Flow and the Poincare Conjecture
by John Morgan, Gang Tian - American Mathematical Society , 2007
This book provides full details of a complete proof of the Poincare Conjecture following Grigory Perelman's preprints. The book is suitable for all mathematicians from advanced graduate students to specialists in geometry and topology.
Lectures on Calabi-Yau and Special Lagrangian Geometry
by Dominic Joyce - arXiv , 2002
An introduction to Calabi-Yau manifolds and special Lagrangian submanifolds from the differential geometric point of view, followed by recent results on singularities of special Lagrangian submanifolds, and their application to the SYZ Conjecture.
Notes on the Atiyah-Singer Index Theorem
by Liviu I. Nicolaescu - University of Notre Dame , 2005
This is arguably one of the deepest and most beautiful results in modern geometry, and it is surely a must know for any geometer / topologist. It has to do with elliptic partial differential operators on a compact manifold.
An introductory course in differential geometry and the Atiyah-Singer index theorem
by Paul Loya - Binghamton University , 2005
This is a lecture-based class on the Atiyah-Singer index theorem, proved in the 60's by Sir Michael Atiyah and Isadore Singer. Their work on this theorem lead to a joint Abel prize in 2004. Requirements: Knowledge of topology and manifolds.
An Introduction to Gaussian Geometry
by Sigmundur Gudmundsson - Lund University , 2009
These notes introduce the beautiful theory of Gaussian geometry i.e. the theory of curves and surfaces in three dimensional Euclidean space. The text is written for students with a good understanding of linear algebra and real analysis.
Introduction to Homological Geometry
by Martin A. Guest - arXiv , 2001
This is an introduction to some of the analytic aspects of quantum cohomology. The small quantum cohomology algebra, regarded as an example of a Frobenius manifold, is described without going into the technicalities of a rigorous definition.
Functional Differential Geometry
by Gerald Jay Sussman, Jack Wisdom - MIT , 2005
Differential geometry is deceptively simple. It is surprisingly easy to get the right answer with informal symbol manipulation. We use computer programs to communicate a precise understanding of the computations in differential geometry.
Combinatorial Geometry with Application to Field Theory
by Linfan Mao - InfoQuest , 2009
Topics covered in this book include fundamental of mathematical combinatorics, differential Smarandache n-manifolds, combinatorial or differentiable manifolds and submanifolds, Lie multi-groups, combinatorial principal fiber bundles, etc.
by Karsten Grove, Peter Petersen - Cambridge University Press , 1997
This volume is an up-to-date panorama of Comparison Geometry, featuring surveys and new research. Surveys present classical and recent results, and often include complete proofs, in some cases involving a new and unified approach.
Tight and Taut Submanifolds
by Thomas E. Cecil, Shiing-shen Chern - Cambridge University Press , 1997
Tight and taut submanifolds form an important class of manifolds with special curvature properties, one that has been studied intensively by differential geometers since the 1950's. This book contains six articles by leading experts in the field.
A Geometric Approach to Differential Forms
by David Bachman - arXiv , 2003
This is a textbook on differential forms. The primary target audience is sophomore level undergraduates enrolled in a course in vector calculus. Later chapters will be of interest to advanced undergraduate and beginning graduate students.
Probability, Geometry and Integrable Systems
by Mark Pinsky, Bjorn Birnir - Cambridge University Press , 2007
The three main themes of this book are probability theory, differential geometry, and the theory of integrable systems. The papers included here demonstrate a wide variety of techniques that have been developed to solve various mathematical problems.
Projective and Polar Spaces
by Peter J. Cameron - Queen Mary College , 1991
The author is concerned with the geometry of incidence of points and lines, over an arbitrary field, and unencumbered by metrics or continuity (or even betweenness). The treatment of these themes blends the descriptive with the axiomatic.
Exterior Differential Systems and Euler-Lagrange Partial Differential Equations
by R. Bryant, P. Griffiths, D. Grossman - University Of Chicago Press , 2008
The authors present the results of their development of a theory of the geometry of differential equations, focusing especially on Lagrangians and Poincare-Cartan forms. They also cover certain aspects of the theory of exterior differential systems.
Algebraic geometry and projective differential geometry
by Joseph M. Landsberg - arXiv , 1998
Homogeneous varieties, Topology and consequences Projective differential invariants, Varieties with degenerate Gauss images, Dual varieties, Linear systems of bounded and constant rank, Secant and tangential varieties, and more.
The Convenient Setting of Global Analysis
by Andreas Kriegl, Peter W. Michor - American Mathematical Society , 1997
This book lays the foundations of differential calculus in infinite dimensions and discusses those applications in infinite dimensional differential geometry and global analysis not involving Sobolev completions and fixed point theory.
Cusps of Gauss Mappings
by Thomas Banchoff, Terence Gaffney, Clint McCrory - Pitman Advanced Pub. Program , 1982
Gauss mappings of plane curves, Gauss mappings of surfaces, characterizations of Gaussian cusps, singularities of families of mappings, projections to lines, focal and parallel surfaces, projections to planes, singularities and extrinsic geometry.
Natural Operations in Differential Geometry
by Ivan Kolar, Peter W. Michor, Jan Slovak - Springer , 1993
A comprehensive textbook on all basic structures from the theory of jets. It begins with an introduction to differential geometry. After reduction each problem to a finite order setting, the remaining discussion is based on properties of jet spaces.
Projective Differential Geometry Old and New
by V. Ovsienko, S. Tabachnikov - Cambridge University Press , 2004
This book provides a route for graduate students and researchers to contemplate the frontiers of contemporary research in projective geometry. The authors include exercises and historical comments relating the basic ideas to a broader context.
Synthetic Differential Geometry
by Anders Kock - Cambridge University Press , 2006
Synthetic differential geometry is a method of reasoning in differential geometry and calculus. This book is the second edition of Anders Kock's classical text, many notes have been included commenting on new developments.