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This volume presents lectures given at the Wisła 19 Summer School: Differential Geometry, Differential Equations, and Mathematical Physics, which took place from August 19 - 29th, 2019 in Wisła, Poland, and was organized by the Baltic Institute of Mathematics. The lectures were dedicated to symplectic and Poisson geometry, tractor calculus, and the integration of ordinary differential equations, and are included here as lecture notes comprising the first three chapters. Following this, chapters combine theoretical and applied perspectives to explore topics at the intersection of differential geometry, differential equations, and mathematical physics. Specific topics covered include: Parabolic geometry Geometric methods for solving PDEs in physics, mathematical biology, and mathematical finance Darcy and Euler flows of real gases Differential invariants for fluid and gas flow Differential Geometry, Differential Equations, and Mathematical Physics is ideal for graduate students and researchers working in these areas. A basic understanding of differential geometry is assumed.
An emerging field of discrete differential geometry aims at the development of discrete equivalents of notions and methods of classical differential geometry. The latter appears as a limit of a refinement of the discretization. Current interest in discrete differential geometry derives not only from its importance in pure mathematics but also from its applications in computer graphics, theoretical physics, architecture, and numerics. Rather unexpectedly, the very basic structures of discrete differential geometry turn out to be related to the theory of integrable systems. One of the main goals of this book is to reveal this integrable structure of discrete differential geometry. For a given smooth geometry one can suggest many different discretizations. Which one is the best? This book answers this question by providing fundamental discretization principles and applying them to numerous concrete problems. It turns out that intelligent theoretical discretizations are distinguished also by their good performance in applications. The intended audience of this book is threefold. It is a textbook on discrete differential geometry and integrable systems suitable for a one semester graduate course. On the other hand, it is addressed to specialists in geometry and mathematical physics. It reflects the recent progress in discrete differential geometry and contains many original results. The third group of readers at which this book is targeted is formed by specialists in geometry processing, computer graphics, architectural design, numerical simulations, and animation. They may find here answers to the question “How do we discretize differential geometry?” arising in their specific field. Prerequisites for reading this book include standard undergraduate background (calculus and linear algebra). No knowledge of differential geometry is expected, although some familiarity with curves and surfaces can be helpful.
This is a textbook on differential geometry well-suited to a variety of courses on this topic. For readers seeking an elementary text, the prerequisites are minimal and include plenty of examples and intermediate steps within proofs, while providing an invitation to more excursive applications and advanced topics. For readers bound for graduate school in math or physics, this is a clear, concise, rigorous development of the topic including the deep global theorems. For the benefit of all readers, the author employs various techniques to render the difficult abstract ideas herein more understandable and engaging. Over 300 color illustrations bring the mathematics to life, instantly clarifying concepts in ways that grayscale could not. Green-boxed definitions and purple-boxed theorems help to visually organize the mathematical content. Color is even used within the text to highlight logical relationships. Applications abound! The study of conformal and equiareal functions is grounded in its application to cartography. Evolutes, involutes and cycloids are introduced through Christiaan Huygens' fascinating story: in attempting to solve the famous longitude problem with a mathematically-improved pendulum clock, he invented mathematics that would later be applied to optics and gears. Clairaut’s Theorem is presented as a conservation law for angular momentum. Green’s Theorem makes possible a drafting tool called a planimeter. Foucault’s Pendulum helps one visualize a parallel vector field along a latitude of the earth. Even better, a south-pointing chariot helps one visualize a parallel vector field along any curve in any surface. In truth, the most profound application of differential geometry is to modern physics, which is beyond the scope of this book. The GPS in any car wouldn’t work without general relativity, formalized through the language of differential geometry. Throughout this book, applications, metaphors and visualizations are tools that motivate and clarify the rigorous mathematical content, but never replace it.
This volume presents lectures given at the Summer School Wisła 18: Nonlinear PDEs, Their Geometry, and Applications, which took place from August 20 - 30th, 2018 in Wisła, Poland, and was organized by the Baltic Institute of Mathematics. The lectures in the first part of this volume were delivered by experts in nonlinear differential equations and their applications to physics. Original research articles from members of the school comprise the second part of this volume. Much of the latter half of the volume complements the methods expounded in the first half by illustrating additional applications of geometric theory of differential equations. Various subjects are covered, providing readers a glimpse of current research. Other topics covered include thermodynamics, meteorology, and the Monge–Ampère equations. Researchers interested in the applications of nonlinear differential equations to physics will find this volume particularly useful. A knowledge of differential geometry is recommended for the first portion of the book, as well as a familiarity with basic concepts in physics.
The 2019 'Australian-German Workshop on Differential Geometry in the Large' represented an extraordinary cross section of topics across differential geometry, geometric analysis and differential topology. The two-week programme featured talks from prominent keynote speakers from across the globe, treating geometric evolution equations, structures on manifolds, non-negative curvature and Alexandrov geometry, and topics in differential topology. A joy to the expert and novice alike, this proceedings volume touches on topics as diverse as Ricci and mean curvature flow, geometric invariant theory, Alexandrov spaces, almost formality, prescribed Ricci curvature, and Kähler and Sasaki geometry.
Discusses the differential geometric aspects of complex manifolds. This work contains standard materials from general topology, differentiable manifolds, and basic Riemannian geometry. It discusses complex manifolds and analytic varieties, sheaves and holomorphic vector bundles. It also gives a brief account of the surface classification theory.
The Nordic Summer School 1985 presented to young researchers the mathematical aspects of the ongoing research stemming from the study of field theories in physics and the differential geometry of fibre bundles in mathematics. The volume includes papers, often with original lines of attack, on twistor methods for harmonic maps, the differential geometric aspects of Yang-Mills theory, complex differential geometry, metric differential geometry and partial differential equations in differential geometry. Most of the papers are of lasting value and provide a good introduction to their subject.
Starting in the middle of the 80s, there has been a growing and fruitful interaction between algebraic geometry and certain areas of theoretical high-energy physics, especially the various versions of string theory. Physical heuristics have provided inspiration for new mathematical definitions (such as that of Gromov-Witten invariants) leading in turn to the solution of problems in enumerative geometry. Conversely, the availability of mathematically rigorous definitions and theorems has benefited the physics research by providing the required evidence in fields where experimental testing seems problematic. The aim of this volume, a result of the CIME Summer School held in Cetraro, Italy, in 2005, is to cover part of the most recent and interesting findings in this subject.
This lecture notes volume presents significant contributions from the “Algebraic Geometry and Number Theory” Summer School, held at Galatasaray University, Istanbul, June 2-13, 2014. It addresses subjects ranging from Arakelov geometry and Iwasawa theory to classical projective geometry, birational geometry and equivariant cohomology. Its main aim is to introduce these contemporary research topics to graduate students who plan to specialize in the area of algebraic geometry and/or number theory. All contributions combine main concepts and techniques with motivating examples and illustrative problems for the covered subjects. Naturally, the book will also be of interest to researchers working in algebraic geometry, number theory and related fields.