Geometry Books

Book SynopsisThe principle aim of this unique text is to illuminate the beauty of the subject both with abstractions like proofs and mathematical text, and with visuals, such as abundant illustrations and diagrams. With few mathematical prerequisites, geometry is presented through the lens of linear fractional transformations. The exposition is motivational and the well-placed examples and exercises give students ample opportunity to pause and digest the material. The subject builds from the fundamentals of Euclidean geometry, to inversive geometry, and, finally, to hyperbolic geometry at the end. Throughout, the author aims to express the underlying philosophy behind the definitions and mathematical reasoning. This text may be used as primary for an undergraduate geometry course or a freshman seminar in geometry, or as supplemental to instructors in their undergraduate courses in complex analysis, algebra, and number theory. There are elective courses that bring together seemingly disparate topics and this text would be a welcome accompaniment.Table of ContentsMotivation.- I Euclidean and Inversive Geometry.- Euclidean Isometries and Similarities.- Inversive Geometry.- Applications of Inversive Geometry.- II Non-Euclidean Geometry.- Spherical Geometry.- Appendix: Set Theory.

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Book SynopsisOlive Whicher's groundbreaking book presents an accessible - non-mathematician's - approach to projective geometry. Profusely illustrated, and written with fire and intuitive genius, this work will be of interest to anyone wishing to cultivate the power of inner visualization in a realm of structural beauty. Whicher explores the concepts of polarity and movement in modern projective geometry as a discipline of thought that transcends the limited and rigid space and forms of Euclid, and the corresponding material forces conceived in classical mechanics. Rudolf Steiner underlined the importance of projective geometry as, 'a method of training the imaginative faculties of thinking, so that they become an instrument of cognition no less conscious and exact than mathematical reasoning'. This seminal approach allows for precise scientific understanding of the concept of creative fields of formative (or etheric) forces at work in nature - in plants, animals and in the human being.

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Book SynopsisBased on a series of graduate lectures given by Vladimir Markovic at the University of Warwick in spring 2003, this book is accessible to those with a grounding in complex analysis looking for an introduction to the theory of quasiconformal maps and Teichmüller theory. Assuming some familiarity with Riemann surfaces and hyperbolic geometry, topics covered include the Grötzch argument, analytical properties of quasiconformal maps, the Beltrami differential equation, holomorphic motions and Teichmüller spaces. Where proofs are omitted, references to where they may be found are always given, and the text is clearly illustrated throughout with diagrams, examples, and exercises for the reader.Table of ContentsPreface ; 1. The Grotzch argument ; 2. Geometric definition of quasiconformal maps ; 3. Analytic properties of quasiconformal maps ; 4. Quasi-isometries and quasisymmetric maps ; 5. The Beltrami differential equation ; 6. Holomorphic motions and applications ; 7. Teichmuller spaces ; 8. Extremal quasiconformal mappings ; 9. Unique extremality ; 10. Isomorphisms of Teichmuller space ; 11. Local rigidity of Teichmuller spaces ; References ; Index

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The origami introduced in this book is based on simple techniques. Some were previously known by origami artists and some were discovered by the author. Curved-Folding Origami Design shows a way to explore new area of origami composed of curved folds. Each technique is introduced in a step-by-step fashion, followed by some beautiful artwork examples. A commentary explaining the theory behind the technique is placed at the end of each chapter.Features Explains the techniques for designing curved-folding origami in seven chapters Contains many illustrations and photos (over 140 figures), with simple instructions Contains photos of 24 beautiful origami artworks, as well as their crease patterns Some basic theories behind the techniques are introduced

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Book SynopsisKnot theory is a fascinating mathematical subject, with multiple links to theoretical physics. This enyclopedia is filled with valuable information on a rich and fascinating subject. Ed Witten, Recipient of the Fields MedalI spent a pleasant afternoon perusing the Encyclopedia of Knot Theory. It's a comprehensive compilation of clear introductions to both classical and very modern developments in the field. It will be a terrific resource for the accomplished researcher, and will also be an excellent way to lure students, both graduate and undergraduate, into the field. Abigail Thompson, Distinguished Professor of Mathematics at University of California, Davis Knot theory has proven to be a fascinating area of mathematical research, dating back about 150 years. Encyclopedia of Knot Theory provides short, interconnected articles on a variety of active areas in knot theory, and includes beautiful pictures, deeTrade Review"Knot theory is a fascinating mathematical subject, with multiple links to theoretical physics. This enyclopedia is filled with valuable information on a rich and fascinating subject."– Ed Witten, Recipient of the Fields Medal"I spent a pleasant afternoon perusing the Encyclopedia of Knot Theory. It’s a comprehensive compilation of clear introductions to both classical and very modern developments in the field. It will be a terrific resource for the accomplished researcher, and will also be an excellent way to lure students, both graduate and undergraduate, into the field." – Abigail Thompson, Distinguished Professor of Mathematics at University of California, Davis "An encyclopedia is expected to be comprehensive, and to include independent expository articles on many topics. The Encyclopedia of Knot Theory is all this. This book will be an excellent introduction to topics in the field of knot theory for advanced undergraduates, graduate students, and researchers interested in knots from many directions."– MAA Reviews"Knot theory is an area of mathematics that requires no introduction, and while this massive tome is certainly no introductory text, it does give a panoramic — and, well, encyclopaedic — view of this vast subject.[. . . ] A book with such an ambitious remit is bound to contain omissions and oddities. [. . .] But this is a small point compared to what has been achieved by this encyclopaedia, which would make a fine addition to any personal or departmental library, or to a departmental coffee table."– London Mathematical SocietyThe Encyclopedia of Knot Theory is close to 1000 pages, and every section, article, paragraph, and sentence inspires the reader to want to learn more knot theory. A wonderful attribute of this text is the reference section at the end of each article as opposed to the end of the book. This allows readers to highlight different sources that will allow them to dive deeper into the topic of that section. [. . .] And while it is nearly impossible to include discussions of every branch of the knot theorytree, the editors made a great choice to focus on current topics showing how the area is still a living subject. [. . .] As a knot theory enthusiast, I truly enjoyed reading about topics I was more familiar with while also exploring topics that were new to me. As an educator, I am excited to share this book with my students and encourage them to read more articles on the topics. Some of the articles in the book include thoughtful open questions for researchers in the field to enjoy, while also providing background for anyone new to knot theory research to use as a foundation. All in all, I loved this text.– American Mathematical MonthlyTable of ContentsI Introduction and History of Knots. 1. Introduction to Knots. II Standard and Nonstandard Representations of Knots. 2. Link Diagrams. 3. Gauss Diagrams. 4. DT Codes. 5. Knot Mosaics. 6. Arc Presentations of Knots and Links. 7. Diagrammatic Representations of Knots and Links as Closed Braids. 8. Knots in Flows. 9. Multi-Crossing Number of Knots and Links. 10. Complementary Regions of Knot and Link Diagrams. 11. Knot Tabulation. III Tangles. 12. What Is a Tangle? 13. Rational and Non-Rational Tangles. 14. Persistent Invariants of Tangles. IV Types of Knots. 15. Torus Knots. 16. Rational Knots and Their Generalizations. 17. Arborescent Knots and Links. 18. Satellite Knots. 19. Hyperbolic Knots and Links. 20. Alternating Knots. 21. Periodic Knots. V Knots and Surfaces. 22. Seifert Surfaces and Genus. 23. Non-Orientable Spanning Surfaces for Knots. 24. State Surfaces of Links. 25. Turaev Surfaces. VI Invariants Defined in Terms of Min and Max. 26. Crossing Numbers. 27. The Bridge Number of a Knot. 28. Alternating Distances of Knots. 29. Superinvariants of Knots and Links. VII Other Knotlike Objects. 30. Virtual Knot Theory. 31. Virtual Knots and Surfaces. 32. Virtual Knots and Parity. 33. Forbidden Moves,Welded Knots and Virtual Unknotting. 34. Virtual Strings and Free Knots. 35. Abstract and Twisted Links. 36. What Is a Knotoid? 37. What Is a Braidoid? 38. What Is a Singular Knot? 39. Pseudoknots and Singular Knots. 40. An Introduction to the World of Legendrian and Transverse Knots 41. Classical Invariants of Legendrian and Transverse Knots. 42. Ruling and Augmentation Invariants of Legendrian Knots. VIII Higher Dimensional Knot Theory. 43. Broken Surface Diagrams and Roseman Moves. 44. Movies and Movie Moves. 45. Surface Braids and Braid Charts. 46. Marked Graph Diagrams and Yoshikawa Moves. 47. Knot Groups. 48. Concordance Groups. IX Spatial Graph Theory. 49. Spatial Graphs. 50. A Brief Survey on Intrinsically Knotted and Linked Graphs. 51. Chirality in Graphs. 52. Symmetries of Graphs Embedded in Sᶟ and Other 3-Manifolds. 53. Invariants of Spatial Graphs. 54. Legendrian Spatial Graphs. 55. Linear Embeddings of Spatial Graphs. 56. Abstractly Planar Spatial Graphs. X Quantum Link Invariants. 57. Quantum Link Invariants. 58. Satellite and Quantum Invariants. 59. Quantum Link Invariants: From QYBE and Braided Tensor Categories. 60. Knot Theory and Statistical Mechanics. XI Polynomial Invariants. 61. What Is the Kauffman Bracket? 62. Span of the Kauffman Bracket and the Tait Conjectures. 63. Skein Modules of 3-Manifold. 64. The Conway Polynomial. 65. Twisted Alexander Polynomials. 66. The HOMFLYPT Polynomial. 67. The Kauffman Polynomials. 68. Kauffman Polynomial on Graphs. 69. Kauffman Bracket Skein Modules of 3-Manifolds. XII Homological Invariants. 70. Khovanov Link Homology. 71. A Short Survey on Knot Floer Homolog. 72. An Introduction to Grid Homology. 73. Categorification. 74. Khovanov Homology and the Jones Polynomial. 75. Virtual Khovanov Homology. XIII Algebraic and Combinatorial Invariants. 76. Knot Colorings. 77. Quandle Cocycle Invariants. 78. Kei and Symmetric Quandles. 79. Racks, Biquandles and Biracks. 80. Quantum Invariants via Hopf Algebras and Solutions to the Yang-Baxter Equation. 81. The Temperley-Lieb Algebra and Planar Algebras. 82. Vassiliev/Finite Type Invariants. 83. Linking Number and Milnor Invariants. XIV Physical Knot Theory. 84. Stick Number for Knots and Links. 85. Random Knots. 86. Open Knots. 87. Random and Polygonal Spatial Graphs. 88. Folded Ribbon Knots in the Plane. XV Knots and Science. 89. DNA Knots and Links. 90. Protein Knots, Links, and Non-Planar Graphs. 91. Synthetic Molecular Knots and Links.

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Book SynopsisIntuition.- I1: Geometry is about shapes.- I2: and more shapes.- I3: Polygons in the plane.- I4: Angles in the plane.- I5: Walking north, east, south, and west in the plane.- I6: Areas of rectangles.- I7: What is the area of the shaded triangle?.- I8: Adding the angles of a triangle.- I9: Pythagorean theorem.- I10: Side Side Side (SSS).- I11: Parallel lines.- I12: Rectangles between parallels and the Z-principle.- I13: Areas: The principle of parallel slices.- I14: If two lines in the plane do not intersect, they are parallel.- I15: The first magnification principle: preliminary form.- I16: The first magnification principle: final form.- I17: Area inside a circle of radius one.- I18: When are triangles congruent?.- I19: Magnifications preserve parallelism and angles.- I20: The principle of similarity.- I21: Proportionality of segments cut by parallels.- I22: Finding the center of a triangle.- I23: Concurrence theorem for altitudes of a triangle.- I24: Inscribing angles in circles.- I25: Fun facts about circles, and limiting cases.- I26: Degrees and radians.- I27: Trigonometry.- I28: Tangent a =(rise)/(run).- I29: Everything you always wanted to know about trigonometry but were afraid to ask.- I30: The law of sines and the law of cosines.- I31: Figuring areas.- I32: The second magnification principle.- I33: Volume of a pyramid.- I34: Of cones and collars.- I35: Sphereworld.- I36: Segments and angles in sphereworld.- I37: Of boxes, cylinders, and spheres.- I38: If it takes one can of paint to paint a square one widget on a side, how many cans does it take to paint a sphere with radius r widgets?.- I39: Excess angle formula for spherical triangles.- I40: Hyperbolic-land.- Construction.- C1: Copying triangles.- C2: Copying angles.- C3: Constructing perpendiculars.- C4:Constructing parallels.- C5: Constructing numbers as lengths.- C6 Given a number, construct its square root.- C7: Constructing parallelograms.- C8: Constructing a regular 3-gon and 4-gon.- C9: Constructing a regular 5-gon.- C10: Constructing a regular 6-gon.- C11: Constructing a regular 7-gon (almost).- C12: Constructing a regular tetrahedron.- C13: Constructing a cube and an octohedron.- C14: Constructing a dodecahedron and an icosahedron.- C15: Constructing the baricenter of a triangle.- C16: Constructing the altitudes of a triangle.- C17: Constructing a circle through three points.- C18: Bisecting a given angle.- C19: Putting circles inside angles.- C20: Inscribing circles in polygons.- C21: Circumscribing circles about polygons.- C22: Drawing triangles on the sphere.- C23: Constructing hyperbolic lines.- Proof.- P1: Distance on the line, motions of the line.- P2: Distance in the plane.- P3: Motions of the plane.- P4: A list of motions of the line.- P5: A complete list of motions of the line.- P6: Motions of the plane: Translations.- P7: Motions of the plane: Rotations.- P8: Motions of the plane: Vertical flip.- P9: Motions of the plane fixing (0,0) and (a,0).- P10: A complete list of motions of the plane.- P11: Distance in space.- P12: Motions of space.- P13: The triangle inequality.- P14: Co-ordinate geometry is about shapes and more shapes.- P15: The shortest path between two points.- P16: The unique line through two given points.- P17: Proving SSS.- Computer Programs.- CP1: Information you'll need about the CP-pages.- CP2: Given two points, construct the segment, ray, and line that pass through them.- CP3: Given a line and a point, construct the perpendicular to the line through the point, or the parallel to the line through the point.- CP4: Given asegment, construct its perpendicular bisector.- CP5: Given an angle, construct the bisector.- CP6: Given three vertices, construct the triangle and its medians.- CP7: Given three vertices, construct the triangle and its angle bisectors.- CP8: Given three vertices, construct the triangle and its altitudes.- CP9: Given a figure in the plane and a positive number R, magnify the figure by a factor of R.- CP10: Given a figure in the plane and two positive numbers R and S, magnify the figure by a factor of R in the horizontal direction and by a factor of S in the vertical direction.- CP11: Given the center and radius of a circle, and two positive numbers R and S, magnify the circle by a factor of R in the horizontal direction and by a factor of S in the vertical direction.- CP12: TRANSLATIONS: Given a figure in the plane and two numbers a and b, show the motion m(x,y) = (x + a, y + b).- CP13: ROTATIONS: Given a figure in the plane and two numbers c and s, so that c2 + s2 = 1, show the motion m(x,y) = (cx - sy, sx + cy).- CP14: FLIPS: Given a figure in the plane, show the motion m(x,y) = (x, -y).- CP15: Composing a set of two motions.- CP16: Composing a series of motions.- CP17: Given a point and a positive number R, construct the circle of radius R about the point.- CP18: Given three points in the plane, construct the unique circle that passes through all three points.- CP19: Given the center of a circle and a point on the circle, construct the tangent to the circle through the point.- CP20: Given a circle and a point outside the circle, construct the two lines tangent to the circle that pass through the point.- CP21: Given a point X inside or outside the circle of radius one and center O, construct the reciprocal point X'.- CP22: Given two points inside the circle ofradius one about (0,0), construct the hyperbolic line containing the two points.Table of ContentsIntuition.- I1: Geometry is about shapes.- I2:… and more shapes.- I3: Polygons in the plane.- I4: Angles in the plane.- I5: Walking north, east, south, and west in the plane.- I6: Areas of rectangles.- I7: What is the area of the shaded triangle?.- I8: Adding the angles of a triangle.- I9: Pythagorean theorem.- I10: Side Side Side (SSS).- I11: Parallel lines.- I12: Rectangles between parallels and the Z-principle.- I13: Areas: The principle of parallel slices.- I14: If two lines in the plane do not intersect, they are parallel.- I15: The first magnification principle: preliminary form.- I16: The first magnification principle: final form.- I17: Area inside a circle of radius one.- I18: When are triangles congruent?.- I19: Magnifications preserve parallelism and angles.- I20: The principle of similarity.- I21: Proportionality of segments cut by parallels.- I22: Finding the center of a triangle.- I23: Concurrence theorem for altitudes of a triangle.- I24: Inscribing angles in circles.- I25: Fun facts about circles, and limiting cases.- I26: Degrees and radians.- I27: Trigonometry.- I28: Tangent a =(rise)/(run).- I29: Everything you always wanted to know about trigonometry but were afraid to ask.- I30: The law of sines and the law of cosines.- I31: Figuring areas.- I32: The second magnification principle.- I33: Volume of a pyramid.- I34: Of cones and collars.- I35: Sphereworld.- I36: Segments and angles in sphereworld.- I37: Of boxes, cylinders, and spheres.- I38: If it takes one can of paint to paint a square one widget on a side, how many cans does it take to paint a sphere with radius r widgets?.- I39: Excess angle formula for spherical triangles.- I40: Hyperbolic-land.- Construction.- C1: Copying triangles.- C2: Copying angles.- C3: Constructing perpendiculars.- C4: Constructing parallels.- C5: Constructing numbers as lengths.- C6 Given a number, construct its square root.- C7: Constructing parallelograms.- C8: Constructing a regular 3-gon and 4-gon.- C9: Constructing a regular 5-gon.- C10: Constructing a regular 6-gon.- C11: Constructing a regular 7-gon (almost).- C12: Constructing a regular tetrahedron.- C13: Constructing a cube and an octohedron.- C14: Constructing a dodecahedron and an icosahedron.- C15: Constructing the baricenter of a triangle.- C16: Constructing the altitudes of a triangle.- C17: Constructing a circle through three points.- C18: Bisecting a given angle.- C19: Putting circles inside angles.- C20: Inscribing circles in polygons.- C21: Circumscribing circles about polygons.- C22: Drawing triangles on the sphere.- C23: Constructing hyperbolic lines.- Proof.- P1: Distance on the line, motions of the line.- P2: Distance in the plane.- P3: Motions of the plane.- P4: A list of motions of the line.- P5: A complete list of motions of the line.- P6: Motions of the plane: Translations.- P7: Motions of the plane: Rotations.- P8: Motions of the plane: Vertical flip.- P9: Motions of the plane fixing (0,0) and (a,0).- P10: A complete list of motions of the plane.- P11: Distance in space.- P12: Motions of space.- P13: The triangle inequality.- P14: Co-ordinate geometry is about shapes and more shapes.- P15: The shortest path between two points….- P16: The unique line through two given points.- P17: Proving SSS.- Computer Programs.- CP1: Information you’ll need about the CP-pages.- CP2: Given two points, construct the segment, ray, and line that pass through them.- CP3: Given a line and a point, construct the perpendicular to the line through the point, or the parallel to the line through the point.- CP4: Given a segment, construct its perpendicular bisector.- CP5: Given an angle, construct the bisector.- CP6: Given three vertices, construct the triangle and its medians.- CP7: Given three vertices, construct the triangle and its angle bisectors.- CP8: Given three vertices, construct the triangle and its altitudes.- CP9: Given a figure in the plane and a positive number R, magnify the figure by a factor of R.- CP10: Given a figure in the plane and two positive numbers R and S, magnify the figure by a factor of R in the horizontal direction and by a factor of S in the vertical direction.- CP11: Given the center and radius of a circle, and two positive numbers R and S, magnify the circle by a factor of R in the horizontal direction and by a factor of S in the vertical direction.- CP12: TRANSLATIONS: Given a figure in the plane and two numbers a and b, show the motion m(x,y) = (x + a, y + b).- CP13: ROTATIONS: Given a figure in the plane and two numbers c and s, so that c2 + s2 = 1, show the motion m(x,y) = (cx - sy, sx + cy).- CP14: FLIPS: Given a figure in the plane, show the motion m(x,y) = (x, -y).- CP15: Composing a set of two motions.- CP16: Composing a series of motions.- CP17: Given a point and a positive number R, construct the circle of radius R about the point.- CP18: Given three points in the plane, construct the unique circle that passes through all three points.- CP19: Given the center of a circle and a point on the circle, construct the tangent to the circle through the point.- CP20: Given a circle and a point outside the circle, construct the two lines tangent to the circle that pass through the point.- CP21: Given a point X inside or outside the circle of radius one and center O, construct the reciprocal point X’.- CP22: Given two points inside the circle of radius one about (0,0), construct the hyperbolic line containing the two points.

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Book SynopsisThese two volumes provide a self-contained account of research on algebraic cycles and motives. Twenty-two contributions from leading figures survey the key research strands, including: Abel-Jacobi/regulator maps and normal functions; Voevodsky's triangulated category of mixed motives; conjectures of Bloch-Beilinson and Murre on filtrations on Chow groups.Table of ContentsForeword; Part I. Survey Articles: 1. The motivic vanishing cycles and the conservation conjecture J. Ayoub; 2. On the theory of 1-motives L. Barbieri-Viale; 3. Motivic decomposition for resolutions of threefolds M. de Cataldo and L. Migliorini; 4. Correspondences and transfers F. D´eglise; 5. Algebraic cycles and singularities of normal functions M. Green and Ph. Griffiths; 6. Zero cycles on singular varieties A. Krishna and V. Srinivas; 7. Modular curves, modular surfaces and modular fourfolds D. Ramakrishnan.

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Finite Packing and Covering by Jr

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Book SynopsisThis book, first published in 2001, is a detailed exposition, in a single volume, of both the theory and applications of torsors to rational points. It is demonstrated that torsors provide a unified approach to several branches of the theory which were hitherto developing in parallel.Trade Review'… the book provides an excellent account of the subject for the non-expert.' T. Szamuely, Zentralblatt für Mathematik'The book is written in a clear and lucid manner with detailed examples that balance the abstract theory with concrete facts. It is reasonably self-contained and can therefore be recommended to newcomers to the recent development of the descent'. EMSTable of Contents1. Introduction; 2. Torsors: general theory; 3. Examples of torsors; 4. Abelian torsors; 5. Obstructions over number fields; 6. Abelian descent and Manin obstruction; 7. Conic bundle surfaces; 8. Bielliptic surfaces; 9. Homogenous spaces.

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a huge range and FREE tracked UK delivery on ALL orders.

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a huge range and FREE tracked UK delivery on ALL orders.

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Book SynopsisConsiders the so-called Unlikely Intersections, a topic that embraces well-known issues, such as Lang's and Manin-Mumford's, concerning torsion points in subvarieties of tori or abelian varieties. This book considers algebraic subgroups that meet a given subvariety in a set of unlikely dimension.Trade Review"Zannier's book is well written and a pleasure to read... [T]he author always makes an effort to point out key ideas and key steps, so a reader who wants to read and understand the complete proofs in this technically demanding field will find this monograph to be an extremely helpful entree into the subject... [T]he reviewer highly recommends Zannier's book as an excellent survey of and introduction to the important and hot topic of unlikely intersections in arithmetic geometry."--Joseph H. Silverman, Bulletin of the AMS "This book is indeed a great source of knowledge and inspiration for everybody interested in the unlikely intersection problems. The author must be commended for doing this job, and doing it so well."--Yuri Bilu, Mathematical Reviews ClippingsTable of Contents*FrontMatter, pg. i*Contents, pg. v*Preface, pg. ix*Notation and Conventions, pg. xi*Introduction: An Overview of Some Problems of Unlikely Intersections, pg. 1*Chapter 1: Unlikely Intersections in Multiplicative Groups and the Zilber Conjecture, pg. 15*Chapter 2: An Arithmetical Analogue, pg. 43*Chapter 3 Unlikely Intersections in Elliptic Surfaces and Problems of Masser, pg. 62*Chapter 4: About the Andre-Oort Conjecture, pg. 96*Appendix A: Distribution of Rational Points on Subanalytic Surfaces, pg. 128*Appendix B: Uniformity in Unlikely Intersections: An Example for Lines in Three Dimensions, pg. 136*Appendix C: Silverman's Bounded Height Theorem for Elliptic Curves: A Direct Proof, pg. 138*Appendix D: Lower Bounds for Degrees of Torsion Points: The Transcendence Approach, pg. 140*Appendix E: A Transcendence Measure for a Quotient of Periods, pg. 143*Appendix F: Counting Rational Points on Analytic Curves: A Transcendence Approach, pg. 145*Appendix G: Mixed Problems: Another Approach, pg. 147*Bibliography, pg. 149*Index, pg. 159

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Book SynopsisShaping everyday landscapes from cities to factories, the grid - an arrangement of individual elements along perpendicular lines - has been a basic structure of modern life. This book traces the relationship of these different yet intertwined methods of organizing the visual world and how we represent it in art.

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Book SynopsisThe Four Corners of Mathematics: A Brief History, from Pythagoras to Perelman describes the historical development of the âbig ideasâ in mathematics in an accessible and intuitive manner. In delivering this bird's-eye view of the history of mathematics, the author uses engaging diagrams and images to communicate complex concepts while also exploring the details of the main results and methods of high-level mathematics. As such, this book involves some equations and terminology, but the only assumption on the readersâ knowledge is A-level or high school mathematics.Features Divided into four parts, covering Geometry, Algebra, Calculus and Topology Presents high-level mathematics in a visual and accessible way with numerous examples and over 250 illustrations Includes several novel and intuitive proofs of big theorems, so even the nonexpert reader can appreciate them Sketches of the lives of important contributors, wi

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Book SynopsisZeta and L-functions have played a major part in the development of number theory. This book for graduate students and researchers presents a big picture of some key results and surrounding theory, whilst taking the reader on a journey through the history of their development.Trade Review'The book will be of interest to both young mathematicians and physicists as well as experienced scholars.' Nikolaj M. Glazunov, zbMATH OpenTable of ContentsIntroduction; 1. The Riemann zeta function; 2. The zeta function of a Z-scheme of finite type; 3. The Weil Conjectures; 4. L-functions from number theory; 5. L-functions from geometry; 6. Motives; Appendix A. Karoubian and monoidal categories; Appendix B. Triangulated categories, derived categories, and perfect complexes; Appendix C. List of exercises; Bibliography; Index.

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Book SynopsisIn knot theory, diagrams of a given canonical genus can be described by means of a finite number of patterns (generators). Diagram Genus, Generators and Applications presents a self-contained account of the canonical genus: the genus of knot diagrams. The author explores recent research on the combinatorial theory of knots and supplies proofs for a number of theorems.The book begins with an introduction to the origin of knot tables and the background details, including diagrams, surfaces, and invariants. It then derives a new description of generators using Hirasawa's algorithm and extends this description to push the compilation of knot generators one genus further to complete their classification for genus 4. Subsequent chapters cover applications of the genus 4 classification, including the braid index, polynomial invariants, hyperbolic volume, and Vassiliev invariants. The final chapter presents further research related to generators, which helps rTrade Review"Diagram Genus, Generators and Applications contains a systematical study of combinatorial properties of knot diagrams. It focuses on diagrams that represent the canonical genus of a knot, i.e., the minimal genus of all Seifert surfaces for a given knot that are obtained by applying Seifert’s algorithm to diagrams of the knot. The book contains the complete classification of knots up to canonical genus 4. This classification has lots of applications … The book … will certainly become a reference in this area. It is very clearly written and contains enough background material so that it can be used by graduate-level students to learn the subject and do work in this area on their own."—Thomas Fiedler, Institut de Mathématiques, Université Paul Sabatier, Toulouse"This book provides an essential resource for anyone currently doing research or interested in doing research on surfaces in knot complements and their applications. Enough background is included so non-experts can follow the exposition and appreciate the myriad results that ensue."—Professor Colin Adams, Williams College"This monograph is a systematic account of combinatorial knot theory, with a particular focus on spanning surfaces arising from Seifert’s construction. It includes a brief and nicely written introduction to knot theory, concentrating on the background needed for a diagrammatic treatment of knots, including the range of classical and modern knot polynomials.A strong feature of this book, and indeed much of the author’s work elsewhere, is the identification of diagrammatic examples with awkward or unexpected properties, and an analysis of the techniques that can be used effectively on them. This can provide examples that can’t possibly be tackled by certain procedures, and thus directs attention to places where the current repertoire of techniques is lacking.The main topic developed is the notion of diagram genus, or canonical genus, based on Seifert’s algorithm. The related graph theory leads to the selection of a class of alternating knot diagrams, termed generators, and a substantial account of these up to genus 4 is given.This is followed by the discussion of a number of combinatorial results and conjectures. In particular, some nice results for alternating or positive knots are given and their possible extension to the case when k of the knot crossings are switched is explored for small values of k. The earlier calculations are used to extend the knowledge of these results to cover knots with fewer restrictions on their genus or crossing number.There is a good account of the combinatorics for recognizing when a knot diagram actually represents the trivial knot. It is surprisingly easy to draw diagrams of the trivial knot with relatively few crossings that do not have an immediately obvious simplification, and some examples are included in the illustrations.A further section covers the question of finding the braid index for an alternating knot, and the conditions under which the Morton-Franks-Williams bound turn out to be sharp. The concluding section is intended as an appetizer for others and includes a variety of annotated questions and conjectures.The carefully written text is aimed at a graduate-level readership. It gives a comprehensive view of combinatorial questions, both in the monograph itself and in the well-annotated bibliography, and would serve both well as a reference and a source of new ideas.Features A comprehensive account of diagram-centered results in knot theory Focus on Seifert’s construction of oriented-spanning surfaces Analysis of diagrams representing the unknot and their reduction by Reidemeister moves Careful and persuasive writing An excellent reference text and source of ideas" —H.R. Morton, Department of Mathematical Sciences, University of LiverpoolTable of ContentsIntroduction. Preliminaries. The Maximal Number of Generator Crossings and ~-Equivalance Classes. Generators of Genus 4. Unknot Diagrams, Non-Trivial Polynomials, and Achiral Knots. The Signature. Braid Index of Alternating Knots. Minimal String Bennequin Surfaces. The Alexander Polynomial of Alternating Knots. Outlook.

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Book SynopsisIntended to introduce readers to the major geometrical topics taught at undergraduate level in a manner that is both accessible and rigorous, the author uses world measurement as a synonym for geometry - hence the importance of numbers, coordinates and their manipulation - and has included over 300 exercises, with answers to most of them.Table of Contents1. The Geometry of Numbers.- 1.1 Natural Numbers.- 1.2 Adding Natural Numbers.- 1.3 Multiplying Natural Numbers.- 1.4 Square and Triangular Numbers.- 1.5 Powers.- 1.6 Zero and Negative Numbers.- 1.7 Rational Numbers or Fractions.- 1.8 Powers of Rational Numbers.- 1.9 Rational Numbers as a Field.- 1.10 Real Numbers.- 1.11 Irrational Numbers.- 1.12 Four Famous Numbers: $$ \sqrt 2 $$ ?, ?, ?.- 2. Coordinate Geometry.- 2.1 Coordinates.- 2.2 ?nthe Space of Coordinates.- 2.3 The Line through Two Points.- 2.4 The Plane Containing Three Points.- 2.5 Distance and Angle.- 2.6 Polar Coordinates.- 2.7 Area.- 2.8 Hyperplanes.- 2.9 Angles between Hyperplanes and Nearest Points to Hyperplanes.- 3. The Geometry of the Euclidean Plane.- 3.1 The Life of Euclid “.- 3.2 The Euclidean Axioms for the Plane.- 3.3 Angles and Lines.- 3.4 Some Basic Facts about Triangles.- 3.5 General Polygons.- 3.6 Congruences and Similarities.- 3.7 Isosceles Triangles.- 3.8 Circles.- 3.9 Triangles and their Centres.- 3.10 Metric Properties of Triangles.- 3.11 Three Surprising (and Beautiful) Theorems.- 4. The Geometry of Complex Numbers.- 4.1 What is $$ \sqrt { - 1} $$.- 4.2 Modulus and Division.- 4.3 Unimodular Complex Numbers and the Unit Circle.- 4.4 Lines and Circles in the Complex Plane.- 4.5 Manipulating Complex Numbers.- 4.6 Infinity and the Riemann Sphere.- 4.7 Division and Inversion.- 4.8 Mobius Transformations.- 4.9 Cross Ratios.- 4.10 A Formula for the Cross Ratio.- 4.11 Roots of Unity.- 4.12 Formulre for the nth Roots of Unity.- 4.13 Solving Cubic and Biquadratic Polynomials.- 5. Solid Geometry.- 5.1 Points and Coordinates.- 5.2 Scalar Product.- 5.3 Cross Product.- 5.4 The Scalar Triple Product.- 5.5 The Vector Triple Product.- 5.6 Planes.- 5.7 Lines in Space.- 5.8 Isometries of Space.- 5.9 Projections.- 5.10 Polyhedra.- 6. Projective Geometry.- 6.1 The Projective Plane.- 6.2 Lines in the Projective Plane.- 6.3 Incidence and Duality.- 6.4 Desargues’ Theorem.- 6.5 Cross Ratios Again.- 6.6 Cross Ratios and Duality.- 6.7 Projectivities and Perspectivities.- 6.8 Quadrilaterals.- 6.9 Projective Transformations.- 6.10 Fixed Points and Eigenvectors.- 6.11 Pappus’ Theorem.- 6.12 Perspective Drawing: Tricks of the Trade.- 6.13 The Fano Plane.- 7. Conics and Quadric Surfaces.- 7.1 Conic Sections.- 7.2 The Conic as Quadratic Curve.- 7.3 Focal Properties of Conics.- 7.4 The Motion of the Planets.- 7.5 Quadric Surfaces.- 7.6 The General Quadric Surface.- 8. Spherical Geonnetry.- 8.1 Geodesics.- 8.2 Geodesic Triangles.- 8.3 Latitude and Longitude.- 8.4 Compass Bearings.- 8.5 The Celestial Sphere.- 8.6 Observer’s Coordinates.- 8.7 Time and Right Ascension.- 9. Quaternions and Octonions.- 9.1 Extended Complex Numbers.- 9.2 Multiplying Quaternions.- 9.3 Inverses of Quaternions.- 9.4 Real and Pure Parts of Quaternions.- 9.5 Multiplying Quaternions and Linear Transformations of ?4.- 9.6 Octonions.- 9.7 Vector Products in ?7.- 9.8 Octonions and Associativity.- 9.9 Hexadecanions?.

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Book SynopsisThis monograph could be used for a graduate course on symplectic geometry as well as for independent study.The monograph starts with an introduction of symplectic vector spaces, followed by symplectic manifolds and then Hamiltonian group actions and the Darboux theorem. After discussing moment maps and orbits of the coadjoint action, symplectic quotients are studied. The convexity theorem and toric manifolds come next and we give a comprehensive treatment of Equivariant cohomology. The monograph also contains detailed treatment of the Duistermaat-Heckman Theorem, geometric quantization, and flat connections on 2-manifolds. Finally, there is an appendix which provides background material on Lie groups. A course on differential topology is an essential prerequisite for this course. Some of the later material will be more accessible to readers who have had a basic course on algebraic topology. For some of the later chapters, it would be helpful to have some background on representation theory and complex geometry.Trade Review“The target audience is graduate students; ... this monograph could easily be used by researchers interested in learning the subject at a fast pace. It is a perfect text for a seminar course. ... the book's material is presented in a crisp and abridged manner. ... This makes the presentation short and highly valuable.” (Eduardo A. Gonzalez, Mathematical Reviews, December, 2020)Table of ContentsSymplectic vector spaces.- Hamiltonian group actions.- The Darboux-Weinstein Theorem.- Elementary properties of moment maps.- The symplectic structure on coadjoint orbits.- Symplectic Reduction.- Convexity.- Toric Manifolds.- Equivariant Cohomology.- The Duistermaat-Heckman Theorem.- Geometric Quantization.- Flat connections on 2-manifolds.

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Book SynopsisThis book presents, in a clear and structured way, the set function \mathcal{T} and how it evolved since its inception by Professor F. Burton Jones in the 1940s. It starts with a very solid introductory chapter, with all the prerequisite material for navigating through the rest of the book. It then gradually advances towards the main properties, Decomposition theorems, \mathcal{T}-closed sets, continuity and images, to modern applications.The set function \mathcal{T} has been used by many mathematicians as a tool to prove results about the semigroup structure of the continua, and about the existence of a metric continuum that cannot be mapped onto its cone or to characterize spheres. Nowadays, it has been used by topologists worldwide to investigate open problems in continuum theory.This book can be of interest to both advanced undergraduate and graduate students, and to experienced researchers as well. Its well-defined structure make this book suitable not only for self-study but also as support material to seminars on the subject. Its many open problems can potentially encourage mathematicians to contribute with further advancements in the field.Table of ContentsPreliminaries.- The Set Function T.- Decomposition Theorems.- T-Closed Sets.- Continuity of T.- Images of T.- Applications.- Questions.- References.- Index.

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Book SynopsisIn recent years cube complexes have become a cornerstone topic of geometric group theory and have proven to be a powerful tool in other areas, such as low dimensional topology, phylogenetic trees or in the context of optimization problems.This book covers a wide variety of algebraic and geometric properties of cube complexes and the groups acting on them. The content ranges from basic properties of metric spaces, notions of non-positive curvature, Gromov's link condition and the Švarc–Milnor theorem to advanced material such as the cubulation of half-space systems and the Roller boundary, the construction of cube complexes associated with Coxeter groups, and the Tits alternative for cubical groups.Being the first self-contained, comprehensive introduction to cube complexes this book serves as an entry point for researchers interested in the subject. The material is accessible to advanced undergraduate and graduate students. The text is illustrated with many figures and examples and comes with a large collection of exercises.Table of Contents- 1. Introduction. - 2. Metric Spaces Meet Groups. - 3. Non-positive Curvature. - 4. Cube Complexes and Gromov’s Link Condition. - 5. Hyperplanes and Half-Spaces. - 6. Cubulating Coxeter Groups. - 7. A Panoramic Tour.

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Book SynopsisLa géométrie rigide est devenue, au fil des ans, un outil indispensable dans un grand nombre de questions en géométrie arithmétique. Depuis ses premières fondations, posées par J. Tate en 1961, la théorie s'est développée dans des directions variées. Ce livre est le premier volume d'un traité qui expose un développement systématique de la géométrie rigide suivant l'approche de M. Raynaud, basée sur les schémas formels à éclatements admissibles près. Ce volume est consacré à la construction des espaces rigides dans une situation relative et à l'étude de leurs propriétés géométriques. L'accent est particulièrement mis sur l'étude de la topologie admissible d'un espace rigide cohérent, analogue de la topologie de Zariski d'un schéma. Parmi les sujets traités figurent l'étude des faisceaux cohérents et de leur cohomologie, le théorème de platification par éclatements admissibles qui généralise au cadre formel-rigide un théorème de Raynaud-Gruson dans le cadre algébrique, et le théorème de comparaison du type GAGA pour les faisceaux cohérents. Ce volume contient aussi de larges rappels et compléments de la théorie des schémas formels de Grothendieck. Ce traité est destiné tout autant aux étudiants ayant une bonne connaissance de la géométrie algébrique et souhaitant apprendre la géométrie rigide qu'aux experts en géométrie algébrique et en théorie des nombres comme source de références. Table of ContentsPréface par Michel Raynaud.- Avant-propos.- Introduction.- Chapitre 1. Préliminaires.- Chapitre 2. Géométrie formelle.- Chapitre 3. Éclatements admissibles.- Chapitre 4. Géométrie rigide.- Chapitre 5. Platitude.- Chapitre 6. Invariants différentiels. Morphismes lisses.- Chapitre 7. Espaces rigides quasi-séparés.- Bibliographie.- Index.

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Book SynopsisThe aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, BrasilWalter D. Neumann, Columbia University, New York, USAMarkus J. Pflaum, University of Colorado, Boulder, USADierk Schleicher, Aix-Marseille Université, FranceKatrin Wendland, Trinity College Dublin, Dublin, Ireland Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019)Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019)Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019)Volker Mayer, Mariusz Urbański, and Anna Zdunik, Random and Conformal Dynamical Systems (2021)Ioannis Diamantis, Boštjan Gabrovšek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021) Table of ContentsNotations and results from group theory; representations and representation-modules; simple and semisimple modules; orthogonality relations; the group algebra; characters of abelian groups; degrees of irreducible representations; characters of some small groups; products of representation and characters; on the number of solutions gm =1 in a group; a theorem of A. Hurwitz on multiplicative sums of squares ; permutation representations and characters; the class number; real characters and real representations; Coprime action; groups pa qb; Fronebius groups; induced characters; Brauer's permutation lemma and Glauberman's character correspondence; Clifford theory 1; projective representations; Clifford theory 2; extension of characters; Degree pattern and group structure; monomial groups; representation of wreath products; characters of p-groups; groups with a small number of character degrees; linear groups; the degree graph; groups all of whose character degrees are primes; two special degree problems; lengths of conjugacy classes; R. Brauer's theorem on the character ring; applications of Brauer's theorems; Artin's induction theorem; splitting fields; the Schur index; integral representations; three arithmetical applications; small kernels and faithful irreducible characters; TI-sets; involutions; groups whose Sylow-2-subgroups are generalized quaternion groups; perfect Fronebius complements. (Part contents).

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With emphasis on stochastic aspects of deterministic systems this short book introduces the reader to the basic facts and some special topics of applied ergodic theory. It adresses advanced undergraduate and graduate students from various disciplines, i.e. mathematicians, physicists, electrical and mechanical engineers. Based upon a sound (but non-technical) mathematical introduction, a number of typical examples from applications (mostly from mechanics) are thoroughly discussed. By studying both probabilistic and deterministic features of dynamical systems the reader will develop what might be considered a unified view on chaos and chance as two sides of the same thing.

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Book Synopsis„Shape optimization and spectral theory” is a survey book aiming to give an overview of recent results in spectral geometry and its links with shape optimization. It covers most of the issues which are important for people working in PDE and differential geometry interested in sharp inequalities and qualitative behaviour for eigenvalues of the Laplacian with different kind of boundary conditions (Dirichlet, Robin and Steklov). This includes: existence of optimal shapes, their regularity, the case of special domains like triangles, isospectrality, quantitative form of the isoperimetric inequalities, optimal partitions, universal inequalities and numerical results. Much progress has been made in these extremum problems during the last ten years and this edited volume presents a valuable update to a wide community interested in these topics. List of contributors Antunes Pedro R.S., Ashbaugh Mark, Bonnaillie-Noël Virginie, Brasco Lorenzo, Bucur Dorin, Buttazzo Giuseppe, De Philippis Guido, Freitas Pedro, Girouard Alexandre, Helffer Bernard, Kennedy James, Lamboley Jimmy, Laugesen Richard S., Oudet Edouard, Pierre Michel, Polterovich Iosif, Siudeja Bartłomiej A., Velichkov Bozhidar

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Book SynopsisTwo volume work containing a contemporary account on "Positivity in Algebraic Geometry". Both volumes also available as hardcover editions as Vols. 48 and 49 in the series "Ergebnisse der Mathematik und ihrer Grenzgebiete". A good deal of the material has not previously appeared in book form. Volume II is more at the research level and somewhat more specialized than Volume I. Volume II contains a survey of positivity for vector bundles, and moves on to a systematic development of the theory of multiplier ideals and their applications. Contains many concrete examples, applications, and pointers to further developmentsTrade ReviewFrom the reviews: "The main theme of this ... monograph is a comprehensive description of the fields of complex algebraic geometry connected with the notion of positivity. ... The book is written for mathematicians interested in the modern development of algebraic geometry." (EMS Newsletter, September, 2006)Table of ContentsNotation and Conventions.- Two: Positivity for Vector Bundles.- 6 Ample and Nef Vector Bundles.- 6.1 Classical Theory.- 6.1.A Definition and First Properties.- 6.1.B Cohomological Properties.- 6.1.C Criteria for Amplitude.- 6.1.D Metric Approaches to Positivity of Vector Bundles.- 6.2 Q-Twisted and Nef Bundles.- 6.2.A Twists by Q-Divisors.- 6.2.B Nef Bundles.- 6.3 Examples and Constructions.- 6.3.A Normal and Tangent Bundles.- 6.3.B Ample Cotangent Bundles and Hyperbolicity.- 6.3.C Picard Bundles.- 6.3.D The Bundle Associated to a Branched Covering.- 6.3.E Direct Images of Canonical Bundles.- 6.3.F Some Constructions of Positive Vector Bundles.- 6.4 Ample Vector Bundles on Curves.- 6.4.A Review of Semistability.- 6.4.B Semistability and Amplitude.- Notes.- 7 Geometric Properties of Ample Bundles.- 7.1 Topology.- 7.1.A Sommese’s Theorem.- 7.1.B Theorem of Bloch and Gieseker.- 7.1.C A Barth-Type Theorem for Branched Coverings.- 7.2 Degeneracy Loci.- 7.2.A Statements and First Examples.- 7.2.B Proof of Connectedness of Degeneracy Loci.- 7.2.C Some Applications.- 7.2.D Variants and Extensions.- 7.3 Vanishing Theorems.- 7.3.A Vanishing Theorems of Griffiths and Le Potier.- 7.3.B Generalizations.- Notes.- 8 Numerical Properties of Ample Bundles.- 8.1 Preliminaries from Intersection Theory.- 8.1.A Chern Classes for Q-Twisted Bundles.- 8.1.B Cone Classes.- 8.1.C Cone Classes for Q-Twists.- 8.2 Positivity Theorems.- 8.2.A Positivity of Chern Classes.- 8.2.B Positivity of Cone Classes.- 8.3 Positive Polynomials for Ample Bundles.- 8.4 Some Applications.- 8.4.A Positivity of Intersection Products.- 8.4.B Non-Emptiness of Degeneracy Loci.- 8.4.C Singularities of Hypersurfaces Along a Curve.- Notes.- Three: Multiplier Ideals and Their Applications.- 9 Multiplier Ideal Sheaves.- 9.1 Preliminaries.- 9.1.A Q-Divisors.- 9.1.B Normal Crossing Divisors and Log Resolutions.- 9.1.C The Kawamata—Viehweg Vanishing Theorem.- 9.2 Definition and First Properties.- 9.2.A Definition of Multiplier Ideals.- 9.2.B First Properties.- 9.3 Examples and Complements.- 9.3.A Multiplier Ideals and Multiplicity.- 9.3.B Invariants Arising from Multiplier Ideals.- 9.3.C Monomial Ideals.- 9.3.D Analytic Construction of Multiplier Ideals.- 9.3.E Adjoint Ideals.- 9.3.F Multiplier and Jacobian Ideals.- 9.3.G Multiplier Ideals on Singular Varieties.- 9.4 Vanishing Theorems for Multiplier Ideals.- 9.4.A Local Vanishing for Multiplier Ideals.- 9.4.B The Nadel Vanishing Theorem.- 9.4.C Vanishing on Singular Varieties.- 9.4.D Nadel’s Theorem in the Analytic Setting.- 9.4.E Non-Vanishing and Global Generation.- 9.5 Geometric Properties of Multiplier Ideals.- 9.5.A Restrictions of Multiplier Ideals.- 9.5.B Subadditivity.- 9.5.C The Summation Theorem.- 9.5.D Multiplier Ideals in Families.- 9.5.E Coverings.- 9.6 Skoda’s Theorem.- 9.6.A Integral Closure of Ideals.- 9.6.B Skoda’s Theorem: Statements.- 9.6.C Skoda’s Theorem: Proofs.- 9.6.D Variants.- Notes.- 10 Some Applications of Multiplier Ideals.- 10.1 Singularities.- 10.1.A Singularities of Projective Hypersurfaces.- 10.1.B Singularities of Theta Divisors.- 10.1.C A Criterion for Separation of Jets of Adjoint Series.- 10.2 Matsusaka’s Theorem.- 10.3 Nakamaye’s Theorem on Base Loci.- 10.4 Global Generation of Adjoint Linear Series.- 10.4.A Fujita Conjecture and Angehrn—Siu Theorem.- 10.4.B Loci of Log-Canonical Singularities.- 10.4.C Proof of the Theorem of Angehrn and Siu.- 10.5 The Effective Nullstellensatz.- Notes.- 11 Asymptotic Constructions.- 11.1 Construction of Asymptotic Multiplier Ideals.- 11.1.A Complete Linear Series.- 11.1.B Graded Systems of Ideals and Linear Series.- 11.2 Properties of Asymptotic Multiplier Ideals.- 11.2.A Local Statements.- 11.2.B Global Results.- 11.2.C Multiplicativity of Plurigenera.- 11.3 Growth of Graded Families and Symbolic Powers.- 11.4 Fujita’s Approximation Theorem.- 11.4.A Statement and First Consequences.- 11.4.B Proof of Fujita’s Theorem.- 11.4.C The Dual of the Pseudoeffective Cone.- 11.5.- Notes.- References.- Glossary of Notation.

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Book SynopsisIncidence geometry is a central part of modern mathematics that has an impressive tradition. The main topics of incidence geometry are projective and affine geometry and, in more recent times, the theory of buildings and polar spaces.Embedded into the modern view of diagram geometry, projective and affine geometry including the fundamental theorems, polar geometry including the Theorem of Buekenhout-Shult and the classification of quadratic sets are presented in this volume. Incidence geometry is developed along the lines of the fascinating work of Jacques Tits and Francis Buekenhout.The book is a clear and comprehensible introduction into a wonderful piece of mathematics. More than 200 figures make even complicated proofs accessible to the reader.Trade Review“This book provides an introduction into projective and affine spaces as well as polar spaces in the modern language of diagram geometry. … The book is well written and contains many enlightning pictures. It is mainly directed to students.” (Hans Cuypers, Nieuw Archief voor Wiskunde, Issue 4, December, 2015)“The book under review is a comprehensive monograph devoted to the foundations of incidence geometry … . a clear and good introduction for graduate students who want to learn about modern geometry and is also useful for lecturers who offer courses on this topic. The book may also be of interest for researchers because it contains several results which were previously only available in the original research articles. … At the end of the book the author gives some hints about further reading too.” (Gyӧrgy Kiss, Mathematical Reviews, November, 2013)“The book contains almost all classical results from projective geometry. It is written in a readable style. Especially the first two chapters can be recommended to those who are interested in a synthetic treatment of projective geometry.” (Boris Odehnal, Zentralblatt MATH, Vol. 1237, 2012)Table of ContentsI Projective and Affine Geometries.- 1. Introduction.- 2. Geometries and Pregeometries.- 3. Projective and Affine Planes.- 4. Projective Spaces.- 5. Affine Spaces.- 6. A Characterization of Affine Spaces.- 7. Residues and Diagrams.- 8. Finite geometries.- II Isomorphisms and Collineations.- 1. Introduction.- 2. Morphisms.- 3. Projections.- 4. Collineations of projective and affine spaces.- 5. Central Collineations.- 6. The Theorem of Desargues.- III Projective Geometry over a Vector Space.- 1. Introduction.- 2. The Projective Space P(V).- 3. Homogeneous Coordinates of Projective Spaces.- 4. Automorphisms of P(V).- 5. The Affine Space AG(W).- 6. Automorphisms of A(W).- 7. The First Fundamental Theorem.- 8. The Second Fundamental Theorem.- IV Polar Spaces and Polarities.- 1. Introduction.- 2. The Theorem of Buekenhout-Shult.- 3. The diagram of a polar space.- 4. Polarities.- 5. Sesquilinear Forms.- 6. Pseudo-quadrics.- 7. The Kleinian Polar Space.- 8. The Theorem of Buekenhout and Parmentier.- V Quadrics and Quadratic Sets.- 1. Introduction.- 2. Quadratic Sets.- 3. Quadrics.- 4. Quadratic Sets in PG(3, K).- 5. Perspective Quadratic Sets.- 6. Classification of the Quadratic Sets.- 7. The Kleinian Quadric.- 8. The Theorem of Segre.- 9. Further Reading.- References.- Index.

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Book SynopsisShiing-Shen Chern (1911-2004) was one of the leading differential geometers of the twentieth century. In 1946, he founded the Mathematical Institute of Academia Sinica in Shanghai, which was later moved to Nanking. In 1981, he founded the Mathematical Sciences Research Institute (MSRI) at Berkeley and acted as the director until 1984. In 1985, he founded the Nankai Institute of Mathematics in Tianjin. He was awarded the National Medal of Science in 1975; the Wolf Prize in mathematics in 1984; and the Shaw Prize in mathematical sciences in 2004.Chern's works span all the classic fields of differential geometry: the Chern-Simons theory; the Chern-Weil theory, linking curvature invariants to characteristic classes; Chern classes; and other areas such as projective differential geometry and webs that are mathematically rich but currently have a lower profile. He also published work in integral geometry, value distribution theory of holomorphic functions, and minimal submanifolds.Inspired by Chern and his work, former colleagues, students and friends — themselves highly regarded mathematicians in their own right — come together to honor and celebrate Chern's huge contributions. The volume, organized by Phillip Griffiths of the Institute for Advanced Study (Princeton), contains contributions by Michael Atiyah (University of Edinburgh), C-M Bai (Nankai), Robert Bryant (Duke University), Kung-Ching Chang (Peking University), Jeff Cheeger (New York University), Simon K Donaldson (Imperial College), Hélène Esnault (Universität Duisburg-Essen), Mo-Lin Ge (Nankai), Mark Green (University of California at Los Angeles), Phillip Griffiths (Institute for Advanced Study), F Reese Harvey (Rice University), Alain Hénaut (Université Bordeaux 1), Niky Kamran (McGill University), Bruce Kleiner (Yale), H Blaine Lawson, Jr (Suny at Stony Brook), Yiming Long (Nankai), Xiaonan Ma (UMR 7640 du CNRS), Luc Pirio (IRMAR, France), Graeme Segal (Oxford), Gang Tian (MIT), Jean-Marie Trepreau (Institut de Mathématiques de Jussieu), Jeff Viaclovsky (MIT), Wei Wang (Nankai), Wentsun Wu (Chinese Academy of Sciences), C N Yang (Tsinghua), Tan Zhang (Murray State University), Weiping Zhang (Nankai) and others.Table of ContentsIn Memory of Professor S S Chern (C N Yang); Twisted K-Theory and Cohomology (M Atiyah); Yangian and Its Applications (C-M Bai et al.); Geodesically Reversible Finsler 2-Spheres of Constant Curvature (R L Bryant); Multiple Solutions of the Prescribed Mean Curvature Equation (K C Chang & T Zhang); On the Differentiability of Lipschitz Maps from Metric Measure Spaces to Banach Spaces (J Cheeger & B Kleiner); Two-Forms on Four-Manifolds and Elliptic Equations (S K Donaldson); Partial Connection for p-Torsion Line Bundles in Characteristic p > 0 (H Esnault); Algebraic Cycles and Singularities of Normal Functions, II (M Green & P Griffiths); Planar Web Geometry Through Abelian Relations and Singularities (A Henaut); Transitive Analytic Lie Pseudo-Groups (N Kamran); Stability of Closed Characteristics on Compact Convex Hypersurfaces (Y Long & W Wang); -Invariant and Flat Vector Bundles II (X Ma & W Zhang); On Planar Webs with Infinitesimal Automorphisms (D Marin et al.); Projective Linking and Boundaries of Positive Holomorphic Chains in Projective Manifolds, Part II (F R Harvey & H B Lawson, Jr); Aspects of Metric Geometry of Four Manifolds (G Tian); Algebrisation Des Tissus De Codimension 1: La Generalisation D'un Theoreme De Bol (J-M Trepreau); Conformal Geometry and Fully Nonlinear Equations (J Viaclovsky); Memory of My First Research Teacher: The Great Geometer Chern Shiing-Shen (W Wu); Some Open Gromov-Witten Invariants of the Resolved Conifold (J Zhou).

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Book SynopsisThis book deals with the geometry of visual space in all its aspects. As in any branch of mathematics, the aim is to trace the hidden to the obvious; the peculiarity of geometry is that the obvious is sometimes literally before one's eyes.Starting from intuition, spatial concepts are embedded in the pre-existing mathematical framework of linear algebra and calculus. The path from visualization to mathematically exact language is itself the learning content of this book. This is intended to close an often lamented gap in understanding between descriptive preschool and school geometry and the abstract concepts of linear algebra and calculus. At the same time, descriptive geometric modes of argumentation are justified because their embedding in the strict mathematical language has been clarified.The concepts of geometry are of a very different nature; they denote, so to speak, different layers of geometric thinking: some arguments use only concepts such as point, straight line, and incidence, others require angles and distances, still others symmetry considerations. Each of these conceptual fields determines a separate subfield of geometry and a separate chapter of this book, with the exception of the last-mentioned conceptual field "symmetry", which runs through all the others: - Incidence: Projective geometry - Parallelism: Affine geometry - Angle: Conformal Geometry - Distance: Metric Geometry - Curvature: Differential Geometry - Angle as distance measure: Spherical and Hyperbolic Geometry - Symmetry: Mapping Geometry.The mathematical experience acquired in the visual space can be easily transferred to much more abstract situations with the help of the vector space notion. The generalizations beyond the visual dimension point in two directions: Extension of the number concept and transcending the three illustrative dimensions.This book is a translation of the original German 1st edition Geometrie – Anschauung und Begriffe by Jost-Hinrich Eschenburg, published by Springer Fachmedien Wiesbaden GmbH, part of Springer Nature in 2020. The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com). A subsequent human revision was done primarily in terms of content, so that the book will read stylistically differently from a conventional translation. Springer Nature works continuously to further the development of tools for the production of books and on the related technologies to support the authors.Table of ContentsWhat is geometry.- Parallelism: affine geometry.- From affine geometry to linear algebra.- Definition of affine space.- Parallelism and semiaffine mappings.- Parallel projections.- Affine coordinates and center of gravity.- Incidence: projective geometry.- Central perspective.- Far points and straight lines of projection.- Projective and affine space.-Semi-projective mappings and collineations.- Conic sections and quadrics; homogenization.- The theorems of Desargues and Brianchon.- Duality and polarity; Pascal's theorem.- The double ratio.- Distance: Euclidean geometry.- The Pythagorean theorem.- Isometries of Euclidean space.- Classification of isometries.- Platonic solids.- Symmetry groups of Platonic solids.- Finite rotation groups and crystal groups.- Metric properties of conic sections.- Curvature: differential geometry.- Smoothness.- Fundamental forms and curvatures.- Characterization of spheres and hyperplanes.- Orthogonal hyperface systems.- Angles: conformal geometry.- Conformal mappings.- Inversions.- Conformal and spherical mappings.- The stereographic projection.- The space of spheres.- Angular distance: spherical and hyperbolic geometry. The hyperbolic space. Distance on the sphere and in hyperbolic space. Models of hyperbolic geometry.- Exercises.- Solutions

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Book SynopsisA perennial bestseller by eminent mathematician G. Polya, How to Solve It will show anyone in any field how to think straight. In lucid and appealing prose, Polya reveals how the mathematical method of demonstrating a proof or finding an unknown can be of help in attacking any problem that can be "reasoned" out--from building a bridge to winning aTrade Review"Every prospective teacher should read it. In particular, graduate students will find it invaluable. The traditional mathematics professor who reads a paper before one of the Mathematical Societies might also learn something from the book: 'He writes a, he says b, he means c; but it should be d.' "--E. T. Bell, Mathematical Monthly "[This] elementary textbook on heuristic reasoning, shows anew how keen its author is on questions of method and the formulation of methodological principles. Exposition and illustrative material are of a disarmingly elementary character, but very carefully thought out and selected."--Herman Weyl, Mathematical Review "I recommend it highly to any person who is seriously interested in finding out methods of solving problems, and who does not object to being entertained while he does it."--Scientific Monthly "Any young person seeking a career in the sciences would do well to ponder this important contribution to the teacher's art."--A. C. Schaeffer, American Journal of Psychology "Every mathematics student should experience and live this book"--Mathematics Magazine "In an age that all solutions should be provided with the least possible effort, this book brings a very important message: mathematics and problem solving in general needs a lot of practice and experience obtained by challenging creative thinking, and certainly not by copying predefined recipes provided by others. Let's hope this classic will remain a source of inspiration for several generations to come."--A. Bultheel, European Mathematical Society

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Book SynopsisPolytheora; 3-dimensional regular solids assembled from regular polygons.

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Book SynopsisCollection of nearly 200 unusual problems dealing with congruence and parallelism, the Pythagorean theorem, circles, area relationships, Ptolemy and the cyclic quadrilateral, collinearity and concurrency and more. Arranged in order of difficulty. Detailed solutions.

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Book SynopsisThis book takes the reader on a journey through the world of college mathematics, focusing on some of the most important concepts and results in the theories of polynomials, linear algebra, real analysis, differential equations, coordinate geometry, trigonometry, elementary number theory, combinatorics, and probability. Preliminary material provides an overview of common methods of proof: argument by contradiction, mathematical induction, pigeonhole principle, ordered sets, and invariants. Each chapter systematically presents a single subject within which problems are clustered in each section according to the specific topic. The exposition is driven by nearly 1300 problems and examples chosen from numerous sources from around the world; many original contributions come from the authors. The source, author, and historical background are cited whenever possible. Complete solutions to all problems are given at the end of the book. This second edition includes new sections on quadratic polynomials, curves in the plane, quadratic fields, combinatorics of numbers, and graph theory, and added problems or theoretical expansion of sections on polynomials, matrices, abstract algebra, limits of sequences and functions, derivatives and their applications, Stokes' theorem, analytical geometry, combinatorial geometry, and counting strategies. Using the W.L. Putnam Mathematical Competition for undergraduates as an inspiring symbol to build an appropriate math background for graduate studies in pure or applied mathematics, the reader is eased into transitioning from problem-solving at the high school level to the university and beyond, that is, to mathematical research. This work may be used as a study guide for the Putnam exam, as a text for many different problem-solving courses, and as a source of problems for standard courses in undergraduate mathematics. Putnam and Beyond is organized for independent study by undergraduate and graduate students, as well as teachers and researchers in the physical sciences who wish to expand their mathematical horizons.Table of ContentsPreface to the Second Edition.- Preface to the First Edition.- A Study Guide.- 1. Methods of Proof.- 2. Algebra.- 3. Real Analysis.- 4. Geometry and Trigonometry.- 5. Number Theory.- 6. Combinatorics and Probability.- Solutions.- Index of Notation.- Index.

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Book SynopsisAn essential introduction to discrete and computational geometryDiscrete geometry is a relatively new development in pure mathematics, while computational geometry is an emerging area in applications-driven computer science. Their intermingling has yielded exciting advances in recent years, yet what has been lacking until now is an undergraduate textbook that bridges the gap between the two. Discrete and Computational Geometry offers a comprehensive yet accessible introduction to this cutting-edge frontier of mathematics and computer science.This book covers traditional topics such as convex hulls, triangulations, and Voronoi diagrams, as well as more recent subjects like pseudotriangulations, curve reconstruction, and locked chains. It also touches on more advanced material, including Dehn invariants, associahedra, quasigeodesics, Morse theory, and the recent resolution of the Poincaré conjecture. Connections to real-world applications are made throughout, and algorithms are presented independently of any programming language. This richly illustrated textbook also features numerous exercises and unsolved problems. The essential introduction to discrete and computational geometry Covers traditional topics as well as new and advanced material Features numerous full-color illustrations, exercises, and unsolved problems Suitable for sophomores in mathematics, computer science, engineering, or physics Rigorous but accessible An online solutions manual is available (for teachers only) Trade Review"Discrete and Computational Geometry meets an urgent need for an undergraduate text bridging the theoretical sides and the applied sides of the field. It is an excellent choice as a textbook for an undergraduate course in discrete and computational geometry! The presented material should be accessible for most mathematics or computer science majors in their second or third year in college. The book also is a valuable resource for graduate students and researchers."--Egon Schulte, Zentralblatt MATH "[W]e recommend this book for an undergraduate course on computational geometry. In fact, we hope to use this book ourselves when we teach such a class."--Brittany Terese Fasy and David L. Millman, SigAct News

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Book SynopsisDas Ziel des Buches ist eine umfassende Einführung sowohl in die geometrische wie die algebraische Topologie. Dabei werden lediglich gute Kenntnisse aus dem 1. Studienjahr in der Mathematik vorausgesetzt, die über die Analysis und lineare Algebra kaum hinausgehen; alle weiteren Hilfsmittel, wie die Grundbegriffe der mengentheoretischen Topologie, die Theorie der topologischen Gruppen und die algebraischen Grundlagen werden ebenfalls ausführlich dargestellt. Im Vordergrund stehen jedoch nicht die hieraus hervorgehenden technischen Apparate, sondern die geometrischen Fragestellungen, die erst den Anlass zu ihrer Entwicklung gaben.Table of ContentsEinführung - Allgemeine Topologie - Homotopie - Lie-Gruppen und homogene Räume - Homologie

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Book SynopsisThis introduction to the theory of rigid structures explains how to analyze the performance of built and natural structures under loads, paying special attention to the simplifying role of symmetry. Written for researchers and graduate students in structural engineering and mathematics, and of interest to computer scientists and physicists.Trade Review'Rigidity theory mathematicians and structural engineers are like two branches of a tribe that separated long ago. In the intervening time, the language and knowledge of each group has evolved to where concepts no longer align and common terms no longer have common meanings. As a result, when they interact today, confusion reigns. Frameworks, Tensegrities and Symmetry is a guide that both groups can use to understand the other.' William F. Baker, Skidmore, Owings & Merrill'The authors promise 'an attempt to build a bridge between two cultures' and they have done a remarkable job of this unenviable task. Requiring only a minimum of mathematical and engineering prerequisites the book develops intuitively, and rigorously, the rigidity theory of both bar frameworks and tensegrity frameworks and applies this theory to analyse built structures. Two masters of the field have carefully designed the book to move seamlessly between the analysis and synthesis of specific structures and providing the general, generic and symmetric theories.' Anthony Nixon, Lancaster UniversityTable of Contents1. Introduction; Part I. The General Case: 2. Frameworks and Rigidity; 3. First-Order Analysis of Frameworks; 4. Tensegrities; 5. Energy Functions and the Stress Matrix; 6. Prestress Stability; 7. Generic Frameworks; 8. Finite Mechanisms; Part II. Symmetric Structures: 9. Groups and Representation Theory; 10. First-Order Symmetry Analysis; 11. Generating Stable Symmetric Tensegrities; A. Useful Theorems and Proofs.

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Book SynopsisAfter the development of manifolds and algebraic varieties in the previous century, mathematicians and physicists have continued to advance concepts of space. This book and its companion explore various new notions of space, including both formal and conceptual points of view, as presented by leading experts at the New Spaces in Mathematics and Physics workshop held at the Institut Henri Poincaré in 2015. The chapters in this volume cover a broad range of topics in mathematics, including diffeologies, synthetic differential geometry, microlocal analysis, topos theory, infinity-groupoids, homotopy type theory, category-theoretic methods in geometry, stacks, derived geometry, and noncommutative geometry. It is addressed primarily to mathematicians and mathematical physicists, but also to historians and philosophers of these disciplines.Trade Review'The essays are self-contained, providing motivation to read selectively. Examples in each chapter illustrate the usefulness of these new notions of space … Recommended.' M. Clay, Choice MagazineTable of ContentsIntroduction Mathieu Anel and Gabriel Catren; Part I. Differential geometry: 1. An Introduction to diffeology Patrick Iglesias-Zemmour; 2. New methods for old spaces: synthetic differential geometry Anders Kock; 3. Microlocal analysis and beyond Pierre Schapira; Part II. Topology and algebraic topology: 4. Topo-logie Mathieu Anel and André Joyal; 5. Spaces as infinity-groupoids Timothy Porter; 6. Homotopy type theory: the logic of space Michael Shulman; Part III. Algebraic geometry: 7. Sheaves and functors of points Michel Vaquié; 8. Stacks Nicole Mestrano and Carlos Simpson; 9. The geometry of ambiguity: an introduction to the ideas of derived geometry Mathieu Anel; 10. Geometry in dg categories Maxim Kontsevich.

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a huge range and FREE tracked UK delivery on ALL orders.

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a huge range and FREE tracked UK delivery on ALL orders.

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a huge range and FREE tracked UK delivery on ALL orders.

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a huge range and FREE tracked UK delivery on ALL orders.

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