Recent Progress in Mathematics

This book consists of five chapters that present problems of current research in mathematics, with its history and development, current state, and possible future direction. The chapters are expository in nature and deal with important areas of mathematics such as algebraic geometry, topology, partial differential equations, Riemannian geometry, and harmonic analysis. The book is addressed to researchers who are interested in those subject areas.

Format: Hardback
Length: 200 pages
Publication date: 01 October 2022
Publisher: Springer Verlag, Singapore

This comprehensive book delves into the realm of contemporary mathematics, encompassing five chapters that present diverse and cutting-edge research problems. Each chapter explores the historical background, current state, and potential future directions of the respective field, making it an invaluable resource for scholars and researchers alike.

The first chapter delves into the captivating world of classical enumerative geometry, shedding light on its evolution before the advent of string theory and the remarkable advancements achieved after its integration. The author highlights the significant contributions made by renowned mathematicians, such as Donaldson and Thomas, in the study of Calabi-Yau 4-folds, while also discussing the latest developments in quantum singularity theory. Furthermore, the book explores the intriguing realm of Vafa-Witten invariants and their profound implications in the study of quantum gravity.

The second chapter focuses on the finite-time singularity problem for three-dimensional incompressible Euler equations. The author presents a comprehensive analysis, encompassing Kato's classic local well-posedness results, Beale-Kato-Majda's blow-up criterion, and recent breakthroughs in understanding the singularity problem for the 2D Boussinesq equations. This chapter provides a deep understanding of the complex dynamics and mathematical challenges associated with these equations.

The third chapter delves into the realm of singularity models in three-dimensional Riemannian manifolds. The author discusses recent developments that have led to a complete classification of all singularity models within this context. By presenting an alternative proof of the classification of noncollapsed steady gradient Ricci solitons in dimension 3, the author showcases the power of mathematical analysis and its ability to unravel the mysteries of complex geometrical structures.

The fourth chapter explores the fascinating world of the Neumann–Poincare operator (NPO). The author reviews some of the most significant advancements in this field, including the study of visibility and invisibility via polarization tensors, the decay rate of eigenvalues and surface localization of plasmon, singular geometry and the essential spectrum, analysis of stress, and the structure of the elastic NPO. This chapter offers a comprehensive overview of the latest research developments and their profound implications in various mathematical and physical applications.

The fifth chapter presents an explicit description of the shift locus as a complex of spaces over a contractible building. The author employs intricate mathematical techniques to elucidate the underlying structure and properties of this complex, providing a clear understanding of its geometric and topological aspects. This chapter offers valuable insights into the study of differential geometry and its applications in various fields, including physics and engineering.

In conclusion, this book serves as a treasure trove of knowledge and insights into the diverse fields of contemporary mathematics. It provides a comprehensive overview of classical enumerative geometry, the finite-time singularity problem, singularity models, the Neumann–Poincare operator, and the shift locus. Each chapter is written by renowned experts in their respective fields, ensuring that the content is both authoritative and up-to-date. Whether you are a mathematician, physicist, or engineer, this book will undoubtedly enhance your understanding of these complex and fascinating subjects.

Weight: 483g
Dimension: 235 x 155 (mm)
ISBN-13: 9789811937071
Edition number: 1st ed. 2022