The Universe of Conics
This text presents the classical theory of conics in a modern form. It includes many novel results that are not easily accessible elsewhere. The approach combines synthetic and analytic methods to derive projective, affine and metrical properties, covering both Euclidean and non-Euclidean geometries.With more than two thousand years of history, conic sections play a fundamental role in numerous fields of mathematics and physics, with applications to mechanical engineering, architecture, astronomy, design and computer graphics.This text will be invaluable to undergraduate mathematics students, those in adjacent fields of study, and anyone with an interest in classical geometry.Augmented with more than three hundred fifty figures and photographs, this innovative text will enhance your understanding of projective geometry, linear algebra, mechanics, and differential geometry, with careful exposition and many illustrative exercises.
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.
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.
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.
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.
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.
Geometry Workbook for Dummies
Don't be a square! Strengthen your geometrical skills Lots of students need extra practice to master geometry. Thankfully, there's Geometry Workbook For Dummies. Packed with hundreds of practice problems and easy-to-understand concept explanations, this book takes a hands-on approach to showing you the geometric ropes. Inside, you'll find a helpful review of basic terms and concepts, so you can hit the ground running when you get to the more advanced stuff. In classic Dummies style, this workbook offers easy ways to understand theorems, proofs, and other geometry fundamentals. Figure out congruent triangles, wrap your mind around angle-arc theorems, connect radii and chords, and get smart about all the core concepts of geometry. Work through hundreds of practice problems to solidify your geometry know-how Clear up any confusion with easy-to-understand explanations of all key concepts Get tips for avoiding common mistakes and improving your test scores For students or parents looking for a hands-on approach to learning geometry, this is the perfect Dummies guide. It's great resource all on its own, or pair it with Geometry For Dummies for even more effective book learning.
Geometrodynamics of Gauge Fields
No detailed description available for "Geometrodynamics of Gauge Fields".
Topological Vector Spaces
With many new concrete examples and historical notes, Topological Vector Spaces, Second Edition provides one of the most thorough and up-to-date treatments of the Hahn-Banach theorem. This edition explores the theorem's connection with the axiom of choice, discusses the uniqueness of Hahn-Banach extensions, and includes an entirely new chapter on vector-valued Hahn-Banach theorems. It also considers different approaches to the Banach-Stone theorem as well as variations of the theorem.The book covers locally convex spaces; barreled, bornological, and webbed spaces; and reflexivity. It traces the development of various theorems from their earliest beginnings to present day, providing historical notes to place the results in context. The authors also chronicle the lives of key mathematicians, including Stefan Banach and Eduard Helly. Suitable for both beginners and experienced researchers, this book contains an abundance of examples, exercises of varying levels of difficulty with many hints, and an extensive bibliography and index.
Math Refresher Workbook for Adults Returning to School
Returning to school as an adult also creates a special learning environment. Adult learners bring their life experiences to the classroom, enriching discussions and broadening perspectives for everyone. Professors value the insights that adult students bring from their diverse backgrounds. This makes the educational experience better for all students.Modern education systems are flexible and support adult learners in ways that were not possible before. With online courses, evening classes, and part-time programs, it's easier for adults to balance school with other responsibilities. This flexibility allows them to continue their education without sacrificing their jobs or family time. Returning to school can also positively affect one's family. Children often see their parents as role models, and a parent going back to school can inspire a lifelong love of learning and perseverance. It shows that it's never too late to pursue your dreams and that education is a lifelong journey.Of course, going back to school as an adult has its challenges. Balancing work, family, and studies requires good time management and dedication. Money can also be an issue, but many companies offer scholarships and financial aid for adult learners. The long-term benefits often outweigh the initial costs, both in career advancement and personal growth.
Differential Forms
Get ready to twist your mind as you dive into The Beauty of Differential Forms. This mind-bending book takes you on a journey through the power and versatility of differential forms, from vectors and vector fields to integration, applications in fluid dynamics, electromagnetism, and general relativity. With a mix of humor and in-depth explanations, this book will have you hooked from start to finish. So, grab a seat, buckle up, and get ready to explore the twisted world of differential forms.
Lectures on Differential Geometry
This book presents a concise introduction to differential geometry. It is aimed at advanced undergraduate students and first year graduate students who wish to have a basic solid knowledge of the subject, and it can serve as a starting point for more advanced reading. The book is organized into lectures, so it can easily be used as a textbook for a beginning graduate-level course in differential geometry.
Lectures on Differential Geometry
This book presents a concise introduction to differential geometry. It is aimed at advanced undergraduate students and first year graduate students who wish to have a basic solid knowledge of the subject, and it can serve as a starting point for more advanced reading. The book is organized into lectures, so it can easily be used as a textbook for a beginning graduate-level course in differential geometry.
Euclid Transmogrified
This book tells the extraordinary story of how in the mid twentieth century, in a well-developed European country, a mathematically solid system of geometry was altered to such an extent that more than one million students were taught foundationally flawed geometry over more than a quarter of a century. The almost unbelievable situation came about in Ireland in the wake of early postcolonial developments that gave rise to an impressionable and unqualified civil service in a position of power over a nation's educational development. This is a multifaceted tale of a perfect storm within mathematics development in Ireland whose result was so destructive that its ramifications continue to be felt into the twenty-first century, forty years after a report of a Royal Irish Academy commission of professors of mathematics described the matter as a national scandal. Euclid Transmogrified: A National Scandal may be of interest to mathematics teachers, lecturers in mathematics, mathematical historians, lecturers in education, students of mathematical education, administrators of mathematical education, science administrators, science journalists and those educated in mathematics in Ireland, especially those born between 1954 and 2000. https: //sites.google.com/view/euclidtransmogrified1
Tutor Book’s
Need help with Geometry? Tutor Book's Geometry is extremely thorough. This book skillfully cover the many concepts that students need to master. Examples, including examples using numbers, are included for those theorems and definitions that skilled tutors and teachers know students find more difficult to understand. Many teaching tips and hints are included to support student learning. Memory tips are given where useful.
Geometry of Crystallographic Groups (Second Edition)
It is eleven years since the First Edition of Geometry of Crystallographic Groups appeared. This Second Edition expands on the first, providing details of a new result of automorphism of crystallographic groups, and on Hantzsche-Wendt groups/manifolds.Crystalographic groups are groups which act via isometries on some n-dimensional Euclidean space, so-named because in three dimensions they occur as the symmetry groups of a crystal. There are short introductions to the theme before every chapter, and a list of conjectures and open projects at the end of the book.Geometry of Crystallographic Groups is suitable as a textbook for students, containing basic theory of crystallographic groups. It is also suitable for researchers in the field, discussing in its second half more advanced and recent topics.
Differential Geometry
This textbook offers a different approach to classical textbooks in Differential Geometry. It includes practical examples and over 300 advanced problems designed for graduate students in various fields, such as fluid mechanics, gravitational fields, nuclear physics, electromagnetism, solid-state physics, and thermodynamics. Additionally, it contains problems tailored for students specializing in chemical, civil, and electrical engineering and electronics. The book provides fully detailed solutions to each problem and includes many illustrations to help visualize theoretical concepts. The book introduces Frenet equations for plane and space curves, presents the basic theory of surfaces, and introduces differentiable maps and differentials on the surface. It also provides the first and second fundamental forms of surfaces, minimal surfaces, and geodesics. Furthermore, it contains a detailed analysis of covariant derivatives and manifolds. The book covers many classical results, such as the Lancret Theorem, Shell Theorem, Joachimsthal Theorem, and Meusnier Theorem, as well as the fundamental theorems of plane curves, space curves, surfaces, and manifolds.
Weighted Morrey Spaces
This monograph is a testament to the potency of the method of singular integrals of layer potential type in solving boundary value problems for weakly elliptic systems in the setting of Muckenhoupt-weighted Morrey spaces and their pre-duals. A functional analytic framework for Muckenhoupt-weighted Morrey spaces in the rough setting of Ahlfors regular sets is built from the ground up and subsequently supports a Calder籀n-Zygmund theory on this brand of Morrey space in the optimal geometric environment of uniformly rectifiable sets. A thorough duality theory for such Morrey spaces is also developed and ushers in a never-before-seen Calder籀n-Zygmund theory for Muckenhoupt-weighted Block spaces. Both weighted Morrey and Block spaces are also considered through the lens of (generalized) Banach function spaces, and ultimately, a variety of boundary value problems are formulated and solved with boundary data arbitrarily prescribed from either scale of space. The fairly self-contained nature of this monograph ensures that graduate students, researchers, and professionals in a variety of fields, e.g., function space theory, harmonic analysis, and PDE, will find this monograph a welcome and valuable addition to the mathematical literature.
Algebraic Topology
This book is ideal as an introduction to algebraic topology and applied algebraic topology featuring a streamlined approach including coverage of basic categorical notions, simplicial, cellular, and singular homology, persistent homology, cohomology groups, cup products, Poincare Duality, homotopy theory, and spectral sequences. The focus is on examples and computations, and there are many end of chapter exercises and extensive student projects.
Multiplicative Differential Geometry
The author introduces the main conceptions for multiplicative surfaces: multiplicative first fundamental form, the main multiplicative rules for differentiations on multiplicative surfaces, and the main multiplicative regularity conditions for multiplicative surfaces. Many examples and problems are included.
Geometry and Its Applications
This unique textbook combines traditional geometry with current ideas to present a contemporary approach that is grounded in real-world applications. It balances introduces axiomatic, Euclidean geometry, non-Euclidean geometry, and transformational geometry. The text integrates applications and examples throughout and includes historical notes.
Dynamic Geometry on Time Scales
This book provides the first and second fundamental forms of surfaces on time scales. They are introduced minimal surfaces and geodesics on time scales. In the book are studied the covaraint derivatives on time scales, pseudo-spherical surfaces and \sigma_1, \sigma_2 manifolds on time scales.
The Geometry of Current
Delve into the hidden world that shapes our interconnected age with "The Geometry of Current: Unveiling the Hidden Structure of Networks." This book isn't about wires and cables; it's about the underlying mathematical beauty that governs how information flows through networks. Imagine social media platforms, power grids, or even the human brain as intricate geometric structures. "The Geometry of Current" explores how these seemingly complex systems can be understood through the lens of geometry and graph theory. By analyzing the connections within these networks, the book reveals hidden patterns and influences information flow. Readers will discover how network geometry can be used to predict everything from disease outbreaks to traffic congestion. "The Geometry of Current" isn't just about theory. It explores real-world applications, showcasing how understanding network geometry can help us design more efficient and resilient systems. It's a fascinating journey for anyone curious about the hidden forces that shape our interconnected world.
Topology - Recent Advances and Applications
Topology remains an active and fundamental area of research that plays a foundational role in many branches of mathematics and science, such as analysis, differential geometry, physics and even biology. It is hoped the papers in this book will contribute to stimulating research in this basic area of mathematics.
Surveys in Geometry II
The book is the second volume of a collection which consists of surveys that focus on important topics in geometry which are at the heart of current research. The topics in the present volume include the conformal and the metric geometry of surfaces, Teichm羹ller spaces, immersed surfaces of prescribed extrinsic curvature in 3-dimensional manifolds, symplectic geometry, the metric theory of Grassmann spaces, homogeneous metric spaces, polytopes, the higher-dimensional Gauss-Bonnet formula, isoperimetry in finitely generated groups and Coxeter groups.Each chapter is intended for graduate students and researchers. Several chapters are based on lectures given by their authors to middle-advanced level students and young researchers. The whole book is intended to be an introduction to important topics in geometry.
Mathematical Structures
This textbook is intended to be accessible to any second-year undergraduate in mathematics who has attended courses on basic real analysis and linear algebra. It is meant to help students to appreciate the diverse specialized mathematics courses offered at their universities. Special emphasis is on similarities between mathematical fields and ways to compare them. The organizing principle is the concept of a mathematical structure which plays an important role in all areas of mathematics. The mathematical content used to explain the structural ideas covers in particular material that is typically taught in algebra and geometry courses. The discussion of ways to compare mathematical fields also provides introductions to categories and sheaves, whose ever-increasing role in modern mathematics suggests a more prominent role in teaching. The book is the English translation of the second edition of "Mathematische Strukturen" (Springer, 2024) written in German. The translation was done with the help of artificial intelligence. A subsequent human revision was done primarily in terms of content.
Interaction between Science, Technology and Society vs Ethnomathematics
This book is the result of a didactic and, at the same time, exploratory study of the "ethno" and mathematical concepts involved in the construction of straws in Xai-Xai and the improvement of mathematical theories with this practice. The concepts covered focus on the interaction between science, technology and society in the context of ethnomathematical education. It involved the collaboration of students and selected builders from a community that is expert in building thatched roofs. The study was carried out from an interpretivist perspective, with a constructivist epistemology and a qualitative approach. The steps taken were: i) an exploration of the types of huts found in Xai-Xai and ii) the construction of two huts, one circular and the other rectangular, which culminated in an exploration of the "ethno" concepts and the maths involved. From this, it was concluded that, in addition to the mathematical concepts explored through the link between theory and practice, the act of building helped the students to get to know the "ethno" concepts involved and to gain more skills for building small huts that they would use to house their pets.
Theory of Functions on Complex Manifolds
No detailed description available for "Theory of Functions on Complex Manifolds".
A Primer on Smooth Manifolds
Differential Geometry is one of the major branches of current Mathematics, and it is an unavoidable language in modern Physics. The main characters in Differential Geometry are smooth manifolds: a class of geometric objects that locally behave like the standard Euclidean space.The book provides a first introduction to smooth manifolds, aimed at undergraduate students in Mathematics and Physics. The only prerequisites are the Linear Algebra and Calculus typically covered in the first two years. The presentation is as simple as possible, but it does not sacrifice the rigor.The lecture notes are divided into 10 chapters, with gradually increasing difficulty. The first chapters cover basic material, while the last ones present more sophisticated topics. The definitions, propositions, and proofs are complemented by examples and exercises. The exercises, which include part of the proofs, are designed to help the reader learn the language of Differential Geometry and develop their problem-solving skills in the area. The exercises are also aimed at promoting an active learning process. Finally, the book contains pictures which are useful aids for the visualization of abstract geometric situations. The lecture notes can be used by instructors as teaching material in a one-semester course on smooth manifolds.
Maths and Technology
This work aims to verify the feasibility of using the GeoGebra educational software in Vocational Education Integrated with Basic Education in the Youth and Adult Education Modality, surveying the possibilities of applying it in the classroom and showing how this resource involves the continuous and growing mastery of the technologies present in the school. This material is used as an instrument to carry out pedagogical work on the construction of knowledge, interpretation and application. The methodology used was bibliographical research and field research in which questionnaires were administered to the Course Coordinator, the Maths teacher and fourteen students, all from the Technical Nursing Course at an educational institution in the municipality of Corn矇lio Proc籀pio/PR. The field research shows that it is possible to use this software as long as both the teacher and the students are involved.
Lagrangian Floer Theory and Its Deformations
A-infinity structure was introduced by Stasheff in the 1960s in his homotopy characterization of based loop space, which was the culmination of earlier works of Sugawara's homotopy characterization of H-spaces and loop spaces. At the beginning of the 1990s, a similar structure was introduced by Fukaya in his categorification of Floer homology in symplectic topology. This structure plays a fundamental role in the celebrated homological mirror symmetry proposal by Kontsevich and in more recent developments of symplectic topology.A detailed construction of A-infinity algebra structure attached to a closed Lagrangian submanifold is given in Fukaya, Oh, Ohta, and Ono's two-volume monograph Lagrangian Intersection Floer Theory (AMS-IP series 46 I & II), using the theory of Kuranishi structures-a theory that has been regarded as being not easily accessible to researchers in general. The present lecture note is provided by one of the main contributors to the Lagrangian Floer theory and is intended to provide a quick, reader-friendly explanation of the geometric part of the construction. Discussion of the Kuranishi structures is minimized, with more focus on the calculations and applications emphasizing the relevant homological algebra in the filtered context.The book starts with a quick explanation of Stasheff polytopes and their two realizations-one by the rooted metric ribbon trees and the other by the genus-zero moduli space of open Riemann surfaces-and an explanation of the A-infinity structure on the motivating example of the based loop space. It then provides a description of the moduli space of genus-zero bordered stable maps and continues with the construction of the (curved) A-infinity structure and its canonical models. Included in the explanation are the (Landau-Ginzburg) potential functions associated with compact Lagrangian submanifolds constructed by Fukaya, Oh, Ohta, and Ono. The book explains calculations of potential functions for toric fibers in detail and reviews several explicit calculations in the literature of potential functions with bulk as well as their applications to problems in symplectic topology via the critical point theory thereof. In the Appendix, the book also provides rapid summaries of various background materials such as the stable map topology, Kuranishi structures, and orbifold Lagrangian Floer theory.
The Mathematics of Quantum Mechanics in Highly-Curved Spacetime
Unravel the intricate relationship between quantum mechanics and highly-curved spacetime in this groundbreaking book by Dr. Kuro Aksara. Through advanced mathematical analysis, Aksara explores the fusion of quantum mechanics and general relativity, and delves into the quantum dynamics of black holes, quantum cosmology, entanglement, quantum gravity, quantum technology, and more. Prepare to embark on a quantum voyage into the unknown, where the boundaries of our understanding are pushed to their limits.
The P-Adic Simpson Correspondence and Hodge-Tate Local Systems
This book delves into the p-adic Simpson correspondence, its construction, and development. Offering fresh and innovative perspectives on this important topic in algebraic geometry, the text serves a dual purpose: it describes an important tool in p-adic Hodge theory, which has recently attracted significant interest, and also provides a comprehensive resource for researchers. Unique among the books in the existing literature in this field, it combines theoretical advances, novel constructions, and connections to Hodge-Tate local systems.This exposition builds upon the foundation laid by Faltings, the collaborative efforts of the two authors with T. Tsuji, and contributions from other researchers. Faltings initiated in 2005 a p-adic analogue of the (complex) Simpson correspondence, whose construction has been taken up in several different ways. Following the approach they initiated with T. Tsuji, the authors develop new features of the p-adic Simpson correspondence, inspired by their construction of the relative Hodge-Tate spectral sequence. First, they address the connection to Hodge-Tate local systems. Then they establish the functoriality of the p-adic Simpson correspondence by proper direct image. Along the way, they expand the scope of their original construction. The book targets a specialist audience interested in the intricate world of p-adic Hodge theory and its applications, algebraic geometry and related areas. Graduate students can use it as a reference or for in-depth study. Mathematicians exploring connections between complex and p-adic geometry will also find it valuable.
A Full Axiomatic Development of High School Geometry
This textbook provides a full and complete axiomatic development of exactly that part of plane Euclidean geometry that forms the standard content of high school geometry. It begins with a set of points, a measure of distance between pairs of points and ten simple axioms. From there the notions of length, area and angle measure, along with congruence and similarity, are carefully defined and their properties proven as theorems. It concludes with a proof of the consistency of the axioms used and a full description of their models. It is provided in guided inquiry (inquiry-based) format with the intention that students will be active learners, proving the theorems and presenting their proofs to their class with the instructor as a mentor and a guide.The book is written for graduate and advanced undergraduate students interested in teaching secondary school mathematics, for pure math majors interested in learning about the foundations of geometry, for faculty preparing future secondary school teachers and as a reference for any professional mathematician. It is written with the hope of anchoring K-12 geometry in solid modern mathematics, thereby fortifying the teaching of secondary and tertiary geometry with a deep understanding of the subject.
Spin/Pin-Structures and Real Enumerative Geometry
Spin/Pin-structures on vector bundles have long featured prominently in differential geometry, in particular providing part of the foundation for the original proof of the renowned Atiyah-Singer Index Theory. More recently, they have underpinned the symplectic topology foundations of the so-called real sector of the mirror symmetry of string theory.This semi-expository three-part monograph provides an accessible introduction to Spin- and Pin-structures in general, demonstrates their role in the orientability considerations in symplectic topology, and presents their applications in enumerative geometry.Part I contains a systematic treatment of Spin/Pin-structures from different topological perspectives and may be suitable for an advanced undergraduate reading seminar. This leads to Part II, which systematically studies orientability problems for the determinants of real Cauchy-Riemann operators on vector bundles. Part III introduces enumerative geometry of curves in complex projective varieties and in symplectic manifolds, demonstrating some applications of the first two parts in the process. Two appendices review the Čech cohomology perspective on fiber bundles and Lie group covering spaces.
Metric Algebraic Geometry
Metric algebraic geometry combines concepts from algebraic geometry and differential geometry. Building on classical foundations, it offers practical tools for the 21st century. Many applied problems center around metric questions, such as optimization with respect to distances.After a short dive into 19th-century geometry of plane curves, we turn to problems expressed by polynomial equations over the real numbers. The solution sets are real algebraic varieties. Many of our metric problems arise in data science, optimization and statistics. These include minimizing Wasserstein distances in machine learning, maximum likelihood estimation, computing curvature, or minimizing the Euclidean distance to a variety.This book addresses a wide audience of researchers and students and can be used for a one-semester course at the graduate level. The key prerequisite is a solid foundation in undergraduate mathematics, especially in algebra and geometry. This is an openaccess book.
Metric Space Topology
This introductory book contains a rich collection of exercises and worked examples in Metric Spaces. Other than questions in the traditional setting, plenty of True-or-False type questions and open-ended questions are included. With detailed solutions, these are highly effective in helping students gain a bird's eye view and master the subject and pitfalls better. The presentation is clear in nurturing the mathematical insights and mathematical maturity of the readers.In this book, the pictorialization or visualization of abstract situations into simple pictures is very often crucially conducive to the understanding of the materials. This serves to give an insightful view of the intricate problems, as well as a clue or a direction to formulate rigorous arguments.The learning outcomes include:
