Non Additive Geometry
Non Additive Geometry introduces a groundbreaking approach to arithmetic geometry, replacing traditional structure of a commutative rings with Props and Bioperads -- algebraic systems that can handle matrix multiplication and block direct sums. These structures allow for a deeper exploration of algebraic geometry, where addition no longer holds as a universal operation, particularly at the critical 'Real prime.'The book presents an innovative and comprehensive study of this new geometric framework, discussing its implications for arithmetic geometry and its potential applications in physics. Chapters explore topics such as generalized schemes, sheaves, ideals and primes, localization, and higher K-theory, following Grothendieck's pioneering methods while extending them to accommodate the needs of arithmetic. The text also addresses future applications, leaving room for readers to explore new directions and potential breakthroughs.This monograph is essential reading for advanced graduate students, researchers, and professionals in mathematics and theoretical physics interested in the foundations of arithmetic geometry, the role of Props and Bioperads, and their applications to broaden our concept of geometry, and therefore have new geometrical data, such asSpec(�� ⊗ ��).
A Brief Account of the Historical Development of Pseudospherical Surfaces From 1827 to 1887
A Brief Account of the Historical Development of Pseudospherical Surfaces From 1827 to 1887
Geometric Deformations of Discriminants and Apparent Contours
Comp Formal Euclidean Geom (V1)
This book explores three computational formalisms for solving geometric problems. Part I introduces a trigonometric-based formalism, enabling calculations of distances, angles, and areas using basic trigonometry. Part II focuses on complex numbers, representing points in the plane to manipulate geometric properties like collinearity and concurrency, making it particularly useful for planar problems and rotations. Part III covers vector formalism, applying linear algebra to both plane and solid geometry. Vectors are effective for solving problems related to perpendicularity, collinearity, and the calculation of distances, areas, and volumes.Each formalism has its strengths and limitations, with complex numbers excelling in the plane and vectors being more versatile in three-dimensional space. This book equips readers to choose the best approach for various geometric challenges. This book, designed for math majors, especially future educators, is also valuable for gifted high school students and educators seeking diverse proofs and teaching inspiration.
Four-Dimensional Paper Constructions Mobius, Klein & Boy
Explore four-dimensional paper constructions inspired by the work of great mathematicians like M繹bius, Klein, Boy, Hopf, and others. These creations will help you visualize four-dimensional space and beyond, transporting you to higher-dimensional spaces. This book is designed to solidify your foundations in various areas of mathematics and physics, with a particular focus on topology.If you are familiar with higher-dimensional spaces from loving sci-fi stories, you may find the four-dimensional illustrations in this book especially intuitive. Imagine starting on Earth and traveling straight up into the universe -- where would you end up? Perhaps you would travel in one direction only to eventually return to your starting point. Can you imagine what happens during the course of this trip? By engaging with these four-dimensional paper constructions, you will gain a deeper understanding of this fascinating journey.
Geometry and Analysis on Finsler Spaces
Finsler geometry is a Riemannian geometry without quadratic restriction. It was originated from Riemann's ground-breaking 'Habilitation' address in the year 1854 and has many applications in many fields of the natural sciences including physics, psychology and ecology etc. The book is intended to provide basic materials on Finsler geometry for readers who are interested in Riemann-Finsler geometry, and to bring them into the frontiers of the active research on related topics.The book is comprised of three parts. The first part consists of Chapters 1-4, which cover the basic theory of Finsler geometry and important geometric invariants, including Riemannian quantities and non-Riemannian quantities. Chapters 5-6 present the theory of geodesics and comparison theorems, which are fundamental tools to investigate global Finsler geometry. The last part consisting of Chapters 7-9 presents the recent developments in the global analysis, including harmonic functions, the eigenvalue problem and heat flow etc., on Finsler manifolds although the problems discussed are classical.
Geometric Gems (V3)
Our physical world is embedded in a geometric environment. Plane geometry has many amazing wonders beyond those that are briefly touched on at school. The circle, one of the basic aspects of geometry, has a plethora of unexpected curiosities, which the authors present in an easily understandable fashion requiring nothing more than the very basics of school geometry to appreciate these curiosities and their justifications or proofs.The book is intended to be widely appreciated by a general readership, whose love for geometry should be greatly enhanced through exploring these many unexpected relationships. Geometric Gems is also suitable for mathematics teachers, to enhance the education of their students with these highly motivating circle properties.
Geometric Gems (V3)
Our physical world is embedded in a geometric environment. Plane geometry has many amazing wonders beyond those that are briefly touched on at school. The circle, one of the basic aspects of geometry, has a plethora of unexpected curiosities, which the authors present in an easily understandable fashion requiring nothing more than the very basics of school geometry to appreciate these curiosities and their justifications or proofs.The book is intended to be widely appreciated by a general readership, whose love for geometry should be greatly enhanced through exploring these many unexpected relationships. Geometric Gems is also suitable for mathematics teachers, to enhance the education of their students with these highly motivating circle properties.
Lectures on the H-Cobordism Theorem
Important lectures on differential topology by acclaimed mathematician John Milnor These are notes from lectures that John Milnor delivered as a seminar on differential topology in 1963 at Princeton University. These lectures give a new proof of the h-cobordism theorem that is different from the original proof presented by Stephen Smale. Milnor's goal was to provide a fully rigorous proof in terms of Morse functions. This book remains an important resource in the application of Morse theory.
Convexity and Its Applications in Discrete and Continuous Optimization
Using a pedagogical, unified approach, this book presents both the analytic and combinatorial aspects of convexity and its applications in optimization. On the structural side, this is done via an exposition of classical convex analysis and geometry, along with polyhedral theory and geometry of numbers. On the algorithmic/optimization side, this is done by the first ever exposition of the theory of general mixed-integer convex optimization in a textbook setting. Classical continuous convex optimization and pure integer convex optimization are presented as special cases, without compromising on the depth of either of these areas. For this purpose, several new developments from the past decade are presented for the first time outside technical research articles: discrete Helly numbers, new insights into sublinear functions, and best known bounds on the information and algorithmic complexity of mixed-integer convex optimization. Pedagogical explanations and more than 300 exercises make this book ideal for students and researchers.
Lectures on K瓣hler Groups
An introduction to the state of the art in the study of K瓣hler groups This book gives an authoritative and up-to-date introduction to the study of fundamental groups of compact K瓣hler manifolds, known as K瓣hler groups. Approaching the subject from the perspective of a geometric group theorist, Pierre Py equips readers with the necessary background in both geometric group theory and K瓣hler geometry, covering topics such as the actions of K瓣hler groups on spaces of nonpositive curvature, the large-scale geometry of infinite covering spaces of compact K瓣hler manifolds, and the topology of level sets of pluriharmonic functions. Presenting the most important results from the past three decades, the book provides graduate students and researchers with detailed original proofs of several central theorems, including Gromov and Schoen's description of K瓣hler group actions on trees; the study of solvable quotients of K瓣hler groups following the works of Arapura, Beauville, Campana, Delzant, and Nori; and Napier and Ramachandran's work characterizing covering spaces of compact K瓣hler manifolds having many ends. It also describes without proof many of the recent breakthroughs in the field. Lectures on K瓣hler Groups also gives, in eight appendixes, detailed introductions to such topics as the study of ends of groups and spaces, groups acting on trees and Hilbert spaces, potential theory, and L2 cohomology on Riemannian manifolds.
A Basic Course in Topology
This book serves as an introduction to topology, a branch of mathematics that studies the qualitative properties of geometric objects. It is designed as a bridge between elementary courses in analysis and linear algebra and more advanced classes in algebraic and geometric topology, making it particularly suitable for both undergraduate and graduate mathematics students. The authors employ the modern language of category theory to unify and clarify the concepts presented, with definitions supported by numerous examples and illustrations. The book includes over 170 exercises that reinforce and deepen the understanding of the material.
A Journey Through the Wonders of Plane Geometry
Geometry is often seen as one of the most beautiful aspects of mathematics. This beauty is probably a result of the fact that one can 'see' this aspect of mathematics. Most people are exposed to the very basic elements of geometry throughout their schooling, concentrated in the secondary school curriculum. High schools in the United States offer one year of concentrated geometry teaching, allowing students to observe how a mathematician functions, since everything that is accepted beyond the basic axioms must be proved. However, as the course is only one year long, a great amount of geometry remains to be exposed to the general audience. That is the challenge of this book, wherein we will present a plethora of amazing geometric relationships.We begin with the special relationship of the Golden Ratio, before considering unexpected concurrencies and collinearities. Next, we present some surprising results that arise when squares and similar triangles are placed on triangle sides, followed by a discussion of concyclic points and the relationship between circles and various linear figures. Moving on to more advanced aspects of linear geometry, we consider the geometric wonders of polygons. Finally, we address geometric surprises and fallacies, before concluding with a chapter on the useful concept of homothety, which is not included in the American year-long course in geometry.
Modern Approaches to Differential Geometry Related Fields
This volume consists of several papers written by the main participants of the 7th International Colloquium on Differential Geometry and its Related Fields (ICDG2023). Readers will find some papers that cover geometric structures on manifolds, such as quaternionic structures, Kaehler structures, Einstein structures, contact structures and so on, as well as other papers that deal with probability theory and discrete mathematics.In this volume, the authors present their recent research in differential geometry and related fields, offering a comprehensive overview for researchers not only within differential geometry but also across various areas of mathematics and theoretical physics. They aim for this volume to serve as a valuable guide for young scientists beginning their studies and research careers in the related fields. Together with previous proceedings, readers will gain insight into the progress of research on geometric structures in Riemannian manifolds.
Lectures on the H-Cobordism Theorem
Important lectures on differential topology by acclaimed mathematician John Milnor These are notes from lectures that John Milnor delivered as a seminar on differential topology in 1963 at Princeton University. These lectures give a new proof of the h-cobordism theorem that is different from the original proof presented by Stephen Smale. Milnor's goal was to provide a fully rigorous proof in terms of Morse functions. This book remains an important resource in the application of Morse theory.
Recent Topics on Topology - From Classical to Modern Applications
Topology plays a fundamental role in many branches of mathematics and science. In recent years, there has been a resurgence of interest in physics, particularly in condensed matter physics. This book provides an accessible yet rigorous overview of current research in topology, making it an invaluable resource for mathematicians, physicists, and researchers in related fields.
Foundations and Application of Graph Theory
Graph theory, an intriguing and vital area of mathematics, finds its roots in the 18th century, with Leonhard Euler's pioneering work on the Seven Bridges of K繹nigsberg problem. Since then, the field has burgeoned into a versatile and indispensable tool, influencing diverse areas like computer science, biology, sociology, and network theory. The foundational structures and profound applications of graph theory underscore its role as a linchpin in solving modern-day problems.This book, "Foundations and Applications of Graph Theory: From Basics to Advanced Concepts," is born out of a passion for unraveling the elegance and utility of graphs. It aims to bridge the gap between theoretical foundations and practical applications, catering to students, researchers, and professionals alike. With its structured progression from introductory topics to advanced themes, the book serves as both an educational guide and a reference for exploring the depths of graph theory.Through its chapters, readers will journey from the rudiments of graph definitions and properties to specialized topics like magic labeling, graph coloring, and isomorphism. The exploration is not merely theoretical; real-world applications across disciplines are woven throughout to demonstrate the relevance and adaptability of graph theory.-Dr. Rajpal Kosaliya (Author)
K-Theory and Representation Theory
Symmetry is one of the most important concepts in mathematics and physics. Emerging from the 2021 LMS-Bath Summer School, this book provides Ph.D. students and young researchers with some of the essential tools for the advanced study of symmetry. Illustrated with numerous examples, it explores some of the most exciting interactions between Dirac operators, K-theory and representation theory of real reductive groups. The final chapter provides a self-contained account of the representation theory of p-adic groups, from the very basics to an advanced perspective, with many arithmetic aspects.
Solving Problems in Point Geometry
For over two millennia, the complexities of elementary geometry have challenged learners, burdened by the intricacies of auxiliary graphics and cumbersome calculations. Inspired by Leibniz's query, this book introduces a groundbreaking method: point geometry. By operating directly on points, it integrates the strengths of coordinate, vector, and mass point methods, simplifying operations and problem-solving.Central to this method is the identity approach, which streamlines complex problems into concise equations, unlocking multiple propositions with ease. Through meticulously crafted examples, readers are invited to explore the joy of mathematical thinking.Beyond mathematics, point geometry holds promise for artificial intelligence, offering a simple yet rich knowledge representation and reasoning method. With most solutions generated by computer programs, the potential for simplifying reasoning methods is immense, paving the way for a brighter future in both education and AI advancement.In this ambitious endeavor, the authors seek to simplify knowledge representation and reasoning, reduce the burden of learning, and accelerate the progress of artificial intelligence. This book is not just a guide to geometry; it's a catalyst for transformative thinking and discovery.
Solving Problems in Point Geometry
For over two millennia, the complexities of elementary geometry have challenged learners, burdened by the intricacies of auxiliary graphics and cumbersome calculations. Inspired by Leibniz's query, this book introduces a groundbreaking method: point geometry. By operating directly on points, it integrates the strengths of coordinate, vector, and mass point methods, simplifying operations and problem-solving.Central to this method is the identity approach, which streamlines complex problems into concise equations, unlocking multiple propositions with ease. Through meticulously crafted examples, readers are invited to explore the joy of mathematical thinking.Beyond mathematics, point geometry holds promise for artificial intelligence, offering a simple yet rich knowledge representation and reasoning method. With most solutions generated by computer programs, the potential for simplifying reasoning methods is immense, paving the way for a brighter future in both education and AI advancement.In this ambitious endeavor, the authors seek to simplify knowledge representation and reasoning, reduce the burden of learning, and accelerate the progress of artificial intelligence. This book is not just a guide to geometry; it's a catalyst for transformative thinking and discovery.
Projective Geometry
The elements of geometry are points, lines and planes, together with Space and Nothing. In Euclidian geometry, we observe these as if we are point-centred beings concerned with extensive measurement.In non-Euclidian, or projective geometry, we are 'planar' beings observing spatial relations, free of measurement, as dynamic and transformational. This polar opposite vision of intensive space reveals remarkable secrets. In this fascinating and useful book, expert Lawrence Edwards invites the reader to discover these secrets through practical exercises in creative geometry. Topics include: the elements of geometry, duality, collineation, cross ratio, the circling and step measure, conics and the nature of a curve.
