Geometry, Lie Theory and Applications: The Abel Symposium 2019

The book "Symmetry and Invariance: Applications in Differential Geometry and Lie Theory" summarizes the presentations and discussions from the Abel Symposium 2019, focusing on symmetry and invariance ideas in differential geometry and Lie theory. It includes survey and research articles written by leading international researchers, covering topics such as geometric structures, differential invariants, integrability analysis, and parabolic geometry.

Format: Paperback / softback
Length: 330 pages
Publication date: 08 February 2023
Publisher: Springer Nature Switzerland AG

This remarkable book is a compilation of insightful contributions from esteemed participants of the Abel Symposium 2019, which took place in the picturesque town of Ålesund, Norway. The symposium focused on the diverse applications of symmetry and invariance principles, encompassing areas such as equivalence and deformation theory of geometric structures, the classification of differential invariants and invariant differential operators, the integrability analysis of equations of mathematical physics, the advancements in parabolic geometry, and the mathematical aspects of general relativity. The chapters in this book are authored by renowned international researchers, presenting a blend of survey and research articles. This comprehensive volume offers readers a deep dive into the cutting-edge research in differential geometry and Lie theory, while also providing valuable insights into their practical applications, particularly in the realms of general relativity and string theory.

Introduction:
The Abel Symposium 2019, held in the enchanting town of Ålesund, Norway, was a pivotal event where renowned scholars from around the world gathered to explore the profound implications of symmetry and invariance concepts. This symposium aimed to unravel the mysteries of geometric structures, differential equations, and mathematical physics, shedding light on their applications in various fields.

Symmetry and Invariance:
The central theme of the Abel Symposium 2019 revolved around the applications of symmetry and invariance principles. Symmetry is a fundamental concept in mathematics that describes the underlying structure of objects or systems. It allows for the classification and analysis of objects based on their invariant properties, which remain unchanged under certain transformations. Invariance, on the other hand, refers to the preservation of certain properties under specific transformations, such as translations, rotations, and dilations.

Equivalence and Deformation Theory:
One of the key areas of focus during the symposium was the equivalence and deformation theory of geometric structures. This field explores the relationship between different geometric structures and their deformations, allowing for the study of geometric transformations and their effects on objects. The participants discussed various topics, including the classification of symmetry groups, the study of differential equations and their invariants, and the development of deformation theories for different types of geometric structures.

Classification of Differential Invariants and Invariant Differential Operators:
Another important aspect of the symposium was the classification of differential invariants and invariant differential operators. Differential invariants are mathematical quantities that are conserved under certain transformations, and they play a crucial role in the study of differential equations and their solutions. Invariant differential operators, on the other hand, are operators that preserve the invariants of differential equations and are used to study their behavior and solutions.

Integrability Analysis of Equations of Mathematical Physics:
The participants also delved into the integrability analysis of equations of mathematical physics. This field involves the study of systems of differential equations that describe physical phenomena, such as the motion of particles, the behavior of fluids, and the evolution of complex systems. The symposium discussed various methods and techniques for analyzing these equations, including the use of symmetry and invariance principles, perturbation analysis, and numerical methods.

Progress in Parabolic Geometry:
Parabolic geometry is a branch of mathematics that studies geometric objects that are modeled by parabolic equations. It plays a crucial role in the study of general relativity and other areas of physics. The symposium featured presentations on recent developments in parabolic geometry, including the study of hyperbolic surfaces, the geometry of black holes, and the connections between parabolic geometry and quantum mechanics.

Mathematical Aspects of General Relativity:
General relativity is a fundamental theory of physics that describes the nature of space, time, and gravity. It is one of the most complex and challenging theories in modern physics. The symposium included presentations on various mathematical aspects of general relativity, such as the study of black holes, gravitational waves, and the geometric interpretation of spacetime.

Conclusion:
The Abel Symposium 2019 was a remarkable event that brought together experts from diverse fields to explore the frontiers of symmetry and invariance. The contributions presented in this book provide a comprehensive overview of the current research in differential geometry and Lie theory, as well as their applications in general relativity and string theory. The chapters are written by leading international researchers, and they offer valuable insights into the latest developments in these fields.

This book will be a valuable resource for researchers, students, and scholars interested in differential geometry, Lie theory, and their applications in physics. It will serve as a stepping stone for further exploration and discovery in these exciting and rapidly evolving areas of mathematics.

Weight: 528g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030812980
Edition number: 1st ed. 2022