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Mathematical Foundations · Genealogy

수리·논리 기초 계보

논리 · 집합론 · 대수 · 기하의 형식적 연결 · Frege (1879)에서 Open Logic Project (2024)까지
기호논리 → 공리적 집합론 → 추상대수 → 대수기하

왜 이 네 기둥인가

Frege의 Begriffsschrift (1879)는 정량화와 기호 체계를 통해 현대 일계논리의 출발점을 마련했습니다. 이어진 Grundgesetze (1893)의 논리주의 기획은 Russell의 역설 (1902)을 만나 한계를 드러냈고, 이는 ZFC·NBG 같은 공리적 집합론이 형성되는 계기가 되었습니다.

Pinter·Open Logic·Monk의 집합론은 수와 함수, 위상공간을 정의하는 공통 형식 언어를 제공합니다.

Cameron과 Dummit·Foote의 추상대수는 군·환·체·Galois 이론을 통해 서로 다른 수학 대상을 구조의 관점에서 비교합니다. 조합론은 군 작용과 생성함수로 계수 문제를 대수적 구조와 연결합니다.

Shafarevich의 선형대수·대수기하 저작은 방정식과 공간의 대응을 따라 고전 기하에서 다양체와 스킴으로 이어지는 발전을 보여줍니다.

Girard의 Linear Logic (1987)은 명제를 반복 가능한 정적 대상으로만 보지 않고, 사용되는 자원의 흐름까지 논리 안에 포함시킨 전환점입니다.

16
현재 연결 자료
3
Logic
3
Set Theory
3
Algebra
7
Geometry
6,206
연결 자료 페이지
2,427
추출 목차 항목

핵심 형식 전환

Frege Begriffsschrift (1879) → [자연어 → 기호] → Frege Grundgesetze (1893) → Russell Paradox (1902) → [모순 회피 · 공리화] → ZFC · Pinter · Open Logic · Monk ↓ [객체 일반화] Cameron · Dummit-Foote (Abstract Algebra) ↓ [방정식 → 공간] Shafarevich LAG → BAG 1 (varieties) → BAG 2 (schemes) → AG V (Fano) ↑ Del Centina (사영기하 통사 역행) ↑ ACF/AG (model theory 다리) ↓ 자원 논리로의 전환 Girard Linear Logic (1987) — 정적 논리 → 자원·동역학 논리

시간순 · 분야순 계보

LOGICSETALGEBRAGEOMETRYGottlob Frege1879Gottlob Frege1893Jean-Yves Girard1987Pinter1971J. Donald Monk (CU Boulder)2011Open Logic2024Peter J. Cameron2008Peter J. Cameron2017David S. Dummit2004Shafarevich LAG2013Shafarevich BAG 12013Shafarevich BAG 22013A. N. Parshin1999Del Centina · Gimigliano2024ACF·AG (Leonesi+)2024(Springer Calc·Econ)2025 Frege 자기 확장 — 산수의 logicist 환원 시도Russell paradox (1902) → ZFC 공리적 집합론 필연역사적 도입에서 Frege·Russell paradox 출발점open vs Dover — 같은 NBG/ZFC 수준입문 → 강의노트 (Forcing·LC)ZFC → Model Theory → Algebraic Geometry집합론 → 대수 구조 정의 토대단권 입문 → 북미 표준 확장대수 → 조합론 (군 작용·Pólya)추상대수 → 선형대수·기하 변환선형대수→사영공간→대수다양체Varieties → Schemes (Grothendieck 적층)Schemes → Fano Varieties 분류사영기하학의 역사 → 현대 대수기하 출발점ACF/Model Theory ↔ 대수다양체 (Tarski)Descartes 해석기하 → 사영기하 역사 줄기Frege 정적 논리 → Girard 자원·동역학 논리 (위상 이동)Coherent space·proof-net ↔ Set ↔ Topos 대안 EDGES  crisis (Russell's paradox)topology shift (Frege → Girard)historical rootextends / leads to / specializes
첫 번째 기둥 · 논리

🜂 Logic & Foundations

기호 발명 → 일계논리 → 자원·동역학 논리. 수학의 모든 닻이 박히는 자리.

현재 연결 자료 3건 · 770쪽 · 추출 목차 0항목
logic
Begriffsschrift: a Formula Language of Pure Thought (1879)
Gottlob Frege · Halle a.S.: Louis Nebert (1879) — van Heijenoort 영역본 · 1879
📄 82p 📑 3 entries frege
→ **창세** — 수리논리·일계논리·정량화 발명. Russell·Wittgenstein 의 출발점.
📂 목차 보기
  1. §§1–12 I. Definition of the symbols
  2. §§13–22 II. Representation and derivation of judgments by pure thought
  3. §§23–31 III. Some topics from a general theory of sequences
logic
Linear Logic
Jean-Yves Girard · Theoretical Computer Science 50 (1987) · 1987
📄 101p 📑 10 entries girard
→ **원전** — 자원·동역학 논리의 출발 (∼ 통상 논리의 위상이동)
📂 목차 보기
  1. §1. The connectives of linear logic
  2. §2. The sequent calculus of linear logic
  3. §3. The phase semantics
  4. §4. Coherent semantics — coherent spaces
  5. §5. Proof-nets
  6. §6. The syntactic structure of proof-nets
  7. §7. The exponential isomorphism
  8. §8. The translation of intuitionistic logic
  9. §9. Examples and applications
  10. §10. Final remarks — to what is linear logic?
logic
Grundgesetze der Arithmetik (1893/1903) — 두 권 영인
Gottlob Frege · Jena: Hermann Pohle (Vol I 1893, Vol II 1903) · 1893
📄 587p 📑 5 entries unidentified
→ **Begriffsschrift 의 산수적 완성** — Russell 이 paradox 발견한 자리. 논리주의 (Logicism) 정점·붕괴.
📂 목차 보기
  1. Vol I — Einleitung (Introduction)
  2. Vol I — I. Darlegung der Begriffsschrift (Exposition)
  3. Vol I — II. Beweise der Grundgesetze der Anzahl
  4. Vol II — III. Die reellen Zahlen (Real numbers)
  5. Vol II — Anhang (Appendix on Russell's paradox, 1902)
두 번째 기둥 · 집합론

∅ Set Theory

Cantor → Russell → ZFC → Forcing. 모든 수학적 객체가 환원되는 우주.

현재 연결 자료 3건 · 690쪽 · 추출 목차 168항목
set
Lectures on Set Theory
J. Donald Monk (CU Boulder) · Lecture notes · 2011
📄 204p 📑 8 entries set_theory
→ Pinter ↔ Jech 사이 — ZFC 표준 정리 노트
📂 목차 보기
  1. 1. Introduction & ZFC axioms
  2. 2. Ordinals & transfinite recursion
  3. 3. Cardinals, cofinality, GCH
  4. 4. The constructible universe L
  5. 5. Forcing & generic extensions
  6. 6. Independence: CH, AC
  7. 7. Large cardinals
  8. 8. Descriptive set theory
set
Set Theory
Charles C. Pinter · Dover (orig. Addison-Wesley 1971) · 1971
📄 224p 📑 12 entries set_theory
→ 입문 표준 — 0–11장 ZFC 통상 주제
📂 목차 보기
  1. Chapter 0 Historical Introduction p10
  2. Chapter 1 Classes and Sets p25
  3. Chapter 2 Functions p52
  4. Chapter 3 Relations p72
  5. Chapter 4 Partially Ordered Classes p86
  6. Chapter 5 The Axiom of Choice and Related Principles p108
  7. Chapter 6 The Natural Numbers p119
  8. Chapter 7 Finite and Infinite Sets p133
  9. Chapter 8 Arithmetic of Cardinal Numbers p143
  10. Chapter 9 Arithmetic of the Ordinal Numbers p158
  11. Chapter 10 Transfinite Recursion. Selected Topics in the Theory of Ordinals p177
  12. Chapter 11 Consistency and Independence in Set Theory p201
set
Set Theory: An Open Introduction
Tim Button · Open Logic Project · 2024
📄 262p 📑 22 entries set_theory
→ Open textbook — Cantor·ZF·iterative 개념 균형
📂 목차 보기
  1. I Prelude p14
  2. 1 History and Mythology p15
  3. II Naïve Set Theory p25
  4. 2 Getting Started p27
  5. 3 Relations p40
  6. 4 Functions p52
  7. 5 The Size of Sets p64
  8. 6 Arithmetization p85
  9. 7 Infinite Sets p106
  10. III The Iterative Conception p117
  11. 8 The Iterative Conception p121
  12. 9 Steps towards Z p131
  13. 10 Ordinals p144
  14. 11 Stages and Ranks p161
  15. 12 Replacement p173
  16. 13 Ordinal Arithmetic p189
  17. 14 Cardinals p200
  18. 15 Cardinal Arithmetic p211
  19. 16 Choice p224
  20. A Biographies p242
  21. Photo Credits p251
  22. About the Open Logic Project p262
세 번째 기둥 · 대수

⊕ Algebra · Combinatorics

구조 추상화. 대칭(군) → 환·체 → Galois → 조합론.

현재 연결 자료 3건 · 1,527쪽 · 추출 목차 945항목
algebra
Introduction to Algebra (2nd ed., 2008)
Peter J. Cameron · Oxford University Press · 2008
📄 353p 📑 10 entries cameron
→ Cameron 단권 입문 — Group / Ring / Field / Modules 통합
📂 목차 보기
  1. 1. Rings
  2. 2. Groups
  3. 3. Vector spaces
  4. 4. Modules
  5. 5. The number systems
  6. 6. Rings and modules — further topics
  7. 7. Group theory — further topics
  8. 8. Galois theory
  9. 9. Algebras
  10. 10. Coding theory
algebra
Notes on Counting: An Introduction to Enumerative Combinatorics
Peter J. Cameron · Cambridge University Press · 2017
📄 227p 📑 9 entries cameron
→ Cameron 조합론 — Pólya 이론 + 생성함수 + 분배·분할
📂 목차 보기
  1. 1. Introduction
  2. 2. Formal power series
  3. 3. Subsets, partitions, permutations
  4. 4. Recurrence relations and generating functions
  5. 5. The exponential formula
  6. 6. Possible growth rates
  7. 7. Permutation groups and group actions
  8. 8. Posets, lattices and Möbius inversion
  9. 9. Species
algebra
Abstract Algebra (3rd ed.)
David S. Dummit · Richard M. Foote · Wiley · 2004
📄 947p 📑 26 entries standard
→ 북미 표준 — Cameron 단권의 **확장된 정전**
📂 목차 보기
  1. Preliminaries
  2. Part I — Group Theory
  3. Ch.1 Introduction to Groups
  4. Ch.2 Subgroups
  5. Ch.3 Quotient Groups & Homomorphisms
  6. Ch.4 Group Actions
  7. Ch.5 Direct·Semidirect Products & Abelian Groups
  8. Ch.6 Further Topics in Group Theory
  9. Part II — Ring Theory
  10. Ch.7 Introduction to Rings
  11. Ch.8 Euclidean·Principal Ideal·UFD
  12. Ch.9 Polynomial Rings
  13. Part III — Modules and Vector Spaces
  14. Ch.10 Introduction to Module Theory
  15. Ch.11 Vector Spaces
  16. Ch.12 Modules over PIDs
  17. Part IV — Field Theory and Galois Theory
  18. Ch.13 Field Theory
  19. Ch.14 Galois Theory
  20. Part V — An Introduction to Commutative Rings, Algebraic Geometry, and Homological Algebra
  21. Ch.15 Commutative Rings & Algebraic Geometry
  22. Ch.16 Artinian, Discrete Valuation, Dedekind
  23. Ch.17 Introduction to Homological Algebra
  24. Part VI — Introduction to the Representation Theory of Finite Groups
  25. Ch.18 Representation Theory & Character Theory
  26. Ch.19 Examples & Applications
네 번째 기둥 · 기하

△ Geometry · Algebraic Geom.

Euclid → 사영 → 다양체 → Scheme. 공간을 식과 짝짓는 위상 이동.

현재 연결 자료 7건 · 3,219쪽 · 추출 목차 1,314항목
geometry
From Here to Infinity: Tracing the Origin and Development of Projective Geometry
Andrea Del Centina · Alessandro Gimigliano · Springer · SSHMPS · 2024
📄 820p 📑 96 entries algebraic_geometry
→ Greek 부터 1930 까지 사영기하 통사
📂 목차 보기
  1. Foreword p7
  2. Introduction p14
  3. An Overview of the History of Projective Geometry p14
  4. Acknowledgements p30
  5. Notes to the Reader p31
  6. Abbreviations and Conventions p31
  7. Drawings and Notation p32
  8. Biographies p32
  9. Quotations p32
  10. Use of Our Earlier Publications p32
  11. Pictures p33
  12. Acknowledgments p33
  13. Chapter 1: The Greek Legacy p40
  14. 1.1 The Transmission of Greek Knowledge p40
  15. 1.2 Euclid´s Optics p46
  16. 1.3 The First Four Books of Apollonius´s Conics p51
  17. 1.4 Menelaus´s Theorem p65
  18. 1.5 Pappus´s Collection p68
  19. Chapter 2: Perspective in the Renaissance p85
  20. 2.1 Perspective in Art p86
  21. 2.2 From Artists to Mathematicians p93
  22. 2.3 Conclusions p118
  23. Chapter 3: New Ways of Looking at Conics p119
  24. 3.1 Back to the Cone, Maurolico´s Spatial Method p120
  25. 3.2 Kepler´s Analogy and the Discovery of Points at Infinity p128
  26. 3.3 Conclusions p141
  27. Chapter 4: Desargues, the Dawn of Projective Geometry p142
  28. 4.1 An Overview of Desargues´s Life and Works p143
  29. 4.2 Desargues´s Motivations and First Achievements p147
  30. 4.3 The Brouillon Project p160
  31. 4.4 The Origin of the Concept of Involution p179
  32. 4.5 Theory of ``Polarity´´, and Conjugate Diameters p189
  33. 4.6 New Approaches to Classical Questions p194
  34. 4.7 Further Remarks and Conclusions p201
  35. Chapter 5: Pascal´s Geometrical Achievements p206
  36. 5.1 A Brief Account on Pascal´s Life and Works p206
  37. 5.2 The Essay on Conics p210
  38. 5.3 Pascal´s Address to the Academy p216
  39. 5.4 The Conicorum Opus Completum According to Leibniz p218
  40. 5.5 The Notes Left by Leibniz p228
  41. 5.6 The Mystic Hexagram and its Uses p232
  42. 5.7 Problems Solved by Pascal p243
  43. 5.8 Further Remarks and Conclusion p248
  44. Chapter 6: An Interlude a Century and a Half Long p251
  45. 6.1 Conic Sections 1640-1680, an Overview p253
  46. 6.2 Two Works of G. A. Borelli p257
  47. 6.3 Philippe de La Hire p263
  48. 6.4 Geometrical Results in Newton´s Principia p283
  49. 6.5 A Rising Interest in the Chords Theorem p301
  50. 6.6 New Impulse in the Study of Conic Sections p313
  51. 6.7 The Rise of the Principle of Continuity p326
  52. 6.8 Conclusions p334
  53. Chapter 7: Towards a New Geometry p336
  54. 7.1 Monge´s Seminal Work p337
  55. 7.2 Carnot and His Géométrie de position p344
  56. 7.3 Servois and the Geometry of the Ruler p362
  57. 7.4 Brianchon, a Real Step towards Projective Geometry p370
  58. 7.5 Brianchon´s Memoir on Curves of Second Order p400
  59. 7.6 Conclusion, with a Glimpse into the Future p421
  60. Chapter 8: Poncelet, the Projective Properties of Figures p424
  61. 8.1 Poncelet´s Life and Career p425
  62. 8.2 Poncelet´s Early Studies p428
  63. 8.3 The Notebooks of Saratov p431
  64. 8.4 The Research in the Years 1815-1820 p477
  65. 8.5 The Treatise of 1822 p507
  66. 8.6 Conclusions and Further Remarks p547
  67. Chapter 9: The Algebraic Way to Projective Geometry p552
  68. 9.1 Gergonne´s Duality and the Controversy with Poncelet p552
  69. 9.2 A New Algebraic Technique: Abridged Notation p561
  70. 9.3 Möbius and the Barycentric Calculus p568
  71. 9.4 Plücker´s New Analytic Geometry p584
  72. 9.5 Conclusions p602
  73. Chapter 10: The Synthetic Route: The Contributions of Steiner and Chasles p603
  74. 10.1 A Short Account of Steiner´s Life and Career p604
  75. 10.2 Steiner´s First Articles p607
  76. 10.3 The Systematische Entwicklung p612
  77. 10.4 Chasles, a Bridge from Ancient to Modern p640
  78. 10.5 Chasles´s Last Two Treatises p669
  79. 10.6 Further Remarks and Conclusions p681
  80. Chapter 11: Von Staudt´s Pure Synthetism p684
  81. 11.1 A Short Account on von Staudt´s Life and Works p684
  82. 11.2 The Geometrie der Lage p687
  83. 11.3 The Beiträge zur Geometrie der Lage p710
  84. 11.4 Reception of von Staudt´s Work p718
  85. 11.5 Further Remarks and Conclusions p723
  86. Chapter 12: Projective Geometry 1870-1930 and Beyond p726
  87. 12.1 From 3-Dimensional to n-Dimensions Spaces p727
  88. 12.2 Luigi Cremona and Alfred Clebsch p734
  89. 12.3 Klein´s ``Erlangen Programme´´ p739
  90. 12.4 Geometry in n-Dimensional Spaces p745
  91. 12.5 The Rise of the Italian School of Algebraic Geometry p750
  92. 12.6 Axiomatization, Proof of the Fundamental Theorem p754
  93. 12.7 An axiomatic Theory of Projective Space p770
  94. 12.8 The Italian School of Algebraic Geometry p773
  95. 12.9 A Glimpse Beyond the 1930s p783
  96. 12.10 Conclusions p786
geometry
Algebraically Closed Fields and Algebraic Geometry
(Stefano Leonesi 외) · Springer · 2024
📄 537p 📑 111 entries algebraic_geometry
→ Model Theory → ACF → AG 연결
📂 목차 보기
  1. Acknowledgments p10
  2. Preliminaries p11
  3. 1 Some Concrete Structures p24
  4. 1.1 Purpose of This Chapter p24
  5. 1.2 Choices p25
  6. 1.3 Permutations p29
  7. 1.4 Decompositions p33
  8. 1.5 Towards Model Theory p40
  9. 1.6 Groups, Fields, Vector Spaces p41
  10. 2 Languages and Structures p45
  11. 2.1 Terms p45
  12. 2.2 Formulae p50
  13. 2.3 Structures p55
  14. 2.4 Consequence p64
  15. 2.5 Induction on the Complexity of Formulae p68
  16. 2.6 Definability and Interpretability p73
  17. 2.7 The Size of a Language p83
  18. 3 Theories and Models p84
  19. 3.1 The Connection Between Theories and Models p84
  20. 3.2 Groups and Vector Spaces p86
  21. 3.3 Rings and Fields p91
  22. 3.4 Addendum: Quotient Rings and Noetherian Rings p94
  23. 4 Morphisms, Substructures and Extensions p98
  24. 4.1 Maps That Preserve Formulae p98
  25. 4.2 Elementary Maps, Substructures, Extensions p100
  26. 4.3 Diagrams p108
  27. 4.4 Addendum: Potential Truth p113
  28. 5 Ultraproducts p117
  29. 5.1 Product Structures and Voting p117
  30. 5.2 Ultraproducts p123
  31. 5.3 The Real Field R p130
  32. 5.4 A Touch of Nonstandard Analysis p136
  33. 5.5 The Compactness Theorem p142
  34. 5.6 Elementary Classes p144
  35. 5.7 Addendum: The Unity of the World p145
  36. 6 Essential Field Theory p151
  37. 6.1 Simple Field Extensions p151
  38. 6.2 Algebraic and Finite Extensions p154
  39. 6.3 Algebraically Closed Fields p158
  40. 6.4 The Characteristic of a Field p160
  41. 6.5 Field Embeddings p161
  42. 6.6 Addendum: The Primitive Element Theorem p164
  43. 7 Complete Theories p166
  44. 7.1 The Löwenheim-Skolem Theorems p166
  45. 7.2 Completeness p170
  46. 7.3 Infinite Vector Spaces and Divisible Groups p172
  47. 7.4 Dense Linear Orders p175
  48. 7.5 Transcendence Bases p179
  49. 7.6 Algebraic Closure p184
  50. 7.7 Completeness of ACF0 and ACFp p186
  51. 7.8 Model-Theoretic Algebraic Closure p189
  52. 8 Model-Completeness and Quantifier Elimination p196
  53. 8.1 Model-Completeness p197
  54. 8.2 Preservation and Axiomatisability p204
  55. 8.3 Chains and Lindström's Test p208
  56. 8.4 Quantifier Elimination p214
  57. 8.5 Finite Relational Languages p218
  58. 8.6 T-Closures p220
  59. 8.7 Algebraically Prime Models p223
  60. 8.8 Strongly Minimal Theories p227
  61. 8.9 o-Minimal Theories p234
  62. 9 Types p237
  63. 9.1 Key Definitions p237
  64. 9.2 Isolated and Algebraic Types p240
  65. 9.3 An Application to Measurement Theory p243
  66. 9.4 Types and Elementary Maps p247
  67. 9.5 Homogeneity p250
  68. 9.6 Atomic Structures p254
  69. 9.7 Saturated Structures p262
  70. 10 Algebraically Closed Fields and Algebraic Geometry p270
  71. 10.1 Specialisations p271
  72. 10.2 Elements of Algebraic Geometry p278
  73. 10.3 Morley Rank p286
  74. 10.4 Dimension of an Algebraic Variety p296
  75. 11 Real Closed Fields p303
  76. 11.1 Ordered Fields p303
  77. 11.2 Real Closed Fields p309
  78. 11.3 Quantifier Elimination of RCF< and Its First Consequences p314
  79. 11.4 Applications to Real Geometry and Algebra p327
  80. 12 Fraïssé Limits and Measurement Scales p336
  81. 12.1 Classes of Finitely Generated Substructures p336
  82. 12.2 Fraïssé Limits p339
  83. 12.3 Two Concrete Limits p343
  84. 12.4 Saturation and Quantifier Elimination p346
  85. 12.5 Measurement Scales over Q p348
  86. 12.6 Model-Theoretic Preliminaries p350
  87. 12.7 Group-Theoretic Preliminaries p352
  88. 12.8 A Theorem on Rational Scale Types p355
  89. Further Reading p360
  90. Further Reading p360
  91. A Set-Theoretic Background p362
  92. A.1 Well-Ordered Sets p362
  93. A.2 Classes and Sets p367
  94. A.3 Comparability and Towers p369
  95. A.4 Transfinite Recursion p373
  96. A.5 Cardinal Numbers p379
  97. A.6 The Axiom of Choice p387
  98. B Solutions to Exercises p392
  99. B.1 Chapter 1 p392
  100. B.2 Chapter 2 p399
  101. B.3 Chapter 3 p415
  102. B.4 Chapter 4 p418
  103. B.5 Chapter 5 p426
  104. B.6 Chapter 6 p437
  105. B.7 Chapter 7 p444
  106. B.8 Chapter 8 p459
  107. B.9 Chapter 9 p482
  108. B.10 Chapter 10 p497
  109. B.11 Chapter 11 p504
  110. B.12 Chapter 12 p515
  111. B.13 Appendix A p519
geometry
Calculus with Applications to Economics
(Springer 2025) · Springer · 2025
📄 480p 📑 100 entries algebraic_geometry
→ Descartes·Newton 기원에서 출발하는 미적분 신간
📂 목차 보기
  1. Chapter 1 Descartes’ Analytic Geometry p16
  2. 1.1 Lines p16
  3. 1.2 Quadratic Equations
  4. 1.3 Lines and Circles p26
  5. 1.4 Hyperbola p29
  6. 1.5 Parabola p30
  7. 1.6 Long Division of Polynomials and Horner’s Rule
  8. 1.7 Roots of Polynomials
  9. 1.8 Tangents to Polynomial Curves
  10. 1.9 Derivatives of Polynomials
  11. 1.10 Polynomial and Rational Curves p44
  12. 1.11 Descartes’ Rule of Signs
  13. 1.12 Tangents to the graphs of inverse and implicit functions p53
  14. Chapter 2 Functions and Graphs p56
  15. 2.1 Functions p56
  16. 2.2 Graphs
  17. 2.3 Implicit Functions p64
  18. 2.4 An Application of Conic Sections p69
  19. 2.5 Exponents and Logarithms p70
  20. 2.6 Cobb-Douglas Functions
  21. 2.7 Inverse Functions
  22. 2.8 Trigonometric Functions
  23. 2.9 Inverse Trigonometric Functions
  24. Chapter 3 Limits and Continuity: The " Method of Weierstrass p90
  25. 3.1 Instantaneous Rate of Change p90
  26. 3.2 Limits and Infinity
  27. 3.3 The Limit of a Sequence and Real Numbers
  28. 3.4 Continuous Functions
  29. 3.5 Demand and Supply Functions
  30. 3.6 Newton’s Method p114
  31. 3.7 Continuity of Implicit Functions p119
  32. 3.8 Classification of Points of Discontinuity
  33. 3.9 Three Theorems on Limits
  34. 3.10 Financial Mathematics and Euler’s number e = 2:71828
  35. 3.11 Remarkable Limits
  36. Chapter 4 Newton’s Method of Fluxions
  37. 4.1 Basic Rules
  38. 4.2 Derivatives of Inverse Functions
  39. 4.3 Tangents and Normals.
  40. 4.4 Linearization and Leibniz Di
  41. 4.5 Related Rates p162
  42. 4.6 Rolle’s Theorem
  43. 4.7 Lagrange’s Theorem
  44. 4.8 Darboux’ Theorem
  45. 4.9 Critical Points
  46. 4.10 Concavity and Inflection Points
  47. 4.11 The Shape of a Graph p206
  48. 4.12 The Shapes of Implicit Functions Graphs
  49. 4.13 Cost Function, Revenue, Profit p226
  50. 4.14 Lorenz Curves p235
  51. 4.15 Elasticity
  52. 4.16 L’Hospital’s Rule
  53. 4.17 Taylor’s Formula
  54. Chapter 5 Arithmetica Infinitorum:Wallis’ Theory p258
  55. 5.1 Areas below Graphs of Monotonic Functions p258
  56. 5.2 Areas below Parabolas and Hyperbolas
  57. 5.3 Areas below Exponentials and Logarithms
  58. 5.4 Riemann’s Theory
  59. 5.5 Cavalieri’s Principle
  60. 5.6 The Rectangle Rules
  61. 5.7 The Trapezoidal Rule p289
  62. 5.8 The Newton-Leibniz Formula
  63. 5.9 Wallis’ Infinite Product
  64. 5.10 Brouncker’s Continued Fraction p299
  65. 5.11 Evaluation of ˇ and Brouncker’s Continued Fraction p308
  66. 5.12 Lorenz Curves: Robin Hood and Gini Indexes p310
  67. 5.13 Consumer’s Surplus
  68. 5.14 Some Problems in Arithmetica Infinitorum p317
  69. Chapter 6 Antiderivatives and Indefinite Integrals: Newton’s Theory
  70. 6.1 Integration Rules
  71. 6.2 Integration by Parts
  72. 6.3 Partial Fractions p345
  73. 6.4 Trigonometric Integrals p356
  74. 6.5 Implicit Functions p360
  75. 6.6 Substitutions in Definite Intgrals
  76. 6.7 Areas between Curves p371
  77. 6.8 The Disk Method p376
  78. 6.9 The Washer Method
  79. 6.10 The Shell Method
  80. 6.11 Elasticity
  81. 6.12 Applications
  82. Chapter 7 Euler’s Theory of Differential Equations
  83. 7.1 Graphical Solution of Differential Equations
  84. 7.2 The Isochrone of Leibniz and Perrault’s Tractrix p402
  85. 7.3 Analytic Methods p406
  86. 7.4 Integrating Factors p409
  87. 7.5 Picard’s Iterative Method
  88. 7.6 Numerical Solutions: Euler’s Method p420
  89. 7.7 Exponential Decay
  90. 7.8 Bounded Growth p425
  91. 7.9 Unbounded and Logistic Growth p427
  92. 7.10 Subtangent, Logarithmic Convexity, and Elasticity of Demand p432
  93. 7.11 The Solow-Swan growth model p434
  94. Chapter 8 Optimization
  95. 8.1 Level Curves and Gradients
  96. 8.2 General Methods of Optimization p449
  97. 8.3 Classification of Critical Points p454
  98. 8.4 The Hessian’s Method p455
  99. 8.5 Classical Surfaces and their Critical Points p460
  100. 8.6 Constraint Optimization p466
geometry
Algebraic Geometry V — Fano Varieties
A. N. Parshin · I. R. Shafarevich (Eds.) · Springer · Encyclopaedia of Mathematical Sciences vol. 47 · 1999
📄 249p 📑 7 entries algebraic_geometry
→ Shafarevich BAG 1·2 의 **상급 응용** — Iskovskikh·Prokhorov 의 Fano 분류 개론
📂 목차 보기
  1. I. Iskovskikh & Prokhorov — Fano Varieties
  2. Ch.1 Basic Concepts
  3. Ch.2 Smooth Fano 3-folds
  4. Ch.3 Fano fibrations
  5. Ch.4 Mori theory & minimal model program
  6. Ch.5 Singular Fano varieties
  7. Ch.6 Birational rigidity & rationality questions
geometry
Basic Algebraic Geometry 1: Varieties in Projective Space
Igor R. Shafarevich · Springer (3rd ed.) · 2013
📄 326p 📑 16 entries shafarevich
→ Mumford 와 함께 표준 — Hartshorne 보다 부드러운 입구
📂 목차 보기
  1. Basic Algebraic Geometry 1 p3
  2. Part I: Book 1: Varieties in Projective Space p19
  3. Chapter 1: Basic Notions p20
  4. Chapter 2: Local Properties p99
  5. Chapter 3: Divisors and Differential Forms p163
  6. Chapter 4: Intersection Numbers p249
  7. Algebraic Appendix p299
  8. 1 Linear and Bilinear Algebra p299
  9. 2 Polynomials p301
  10. 3 Quasilinear Maps p301
  11. 4 Invariants p303
  12. 5 Fields p304
  13. 6 Commutative Rings p305
  14. 7 Unique Factorisation p308
  15. 8 Integral Elements p309
  16. 9 Length of a Module p310
geometry
Basic Algebraic Geometry 2: Schemes & Complex Manifolds
Igor R. Shafarevich · Springer (3rd ed.) · 2013
📄 271p 📑 20 entries shafarevich
→ Schemes 본격 + 복소다양체
📂 목차 보기
  1. Basic Algebraic Geometry 2 p3
  2. Preface to Books 2-3 p5
  3. Part I: Book 2: Schemes and Varieties p15
  4. Chapter 5: Schemes p16
  5. Chapter 6: Varieties p61
  6. Part II: Book 3: Complex Algebraic Varieties and Complex Manifolds p124
  7. Chapter 7: The Topology of Algebraic Varieties p125
  8. Chapter 8: Complex Manifolds p159
  9. Chapter 9: Uniformisation p210
  10. Historical Sketch p238
  11. 1 Elliptic Integrals p238
  12. 2 Elliptic Functions p240
  13. 3 Abelian Integrals p242
  14. 4 Riemann Surfaces p244
  15. 5 The Inversion of Abelian Integrals p246
  16. 6 The Geometry of Algebraic Curves p248
  17. 7 Higher Dimensional Geometry p250
  18. 8 The Analytic Theory of Complex Manifolds p252
  19. 9 Algebraic Varieties over Arbitrary Fields and Schemes p253
  20. References for the Historical Sketch p259
geometry
Linear Algebra and Geometry
I. R. Shafarevich · A. O. Remizov · Springer · 2013
📄 536p 📑 89 entries shafarevich
→ 선형대수 → 기하학 본격적 다리
📂 목차 보기
  1. Linear Algebra and Geometry p3
  2. Preliminaries p10
  3. Chapter 1: Linear Equations p21
  4. 1.1 Linear Equations and Functions p21
  5. 1.2 Gaussian Elimination p26
  6. 1.3 Examples* p35
  7. Chapter 2: Matrices and Determinants p44
  8. 2.1 Determinants of Orders 2 and 3 p44
  9. 2.2 Determinants of Arbitrary Order p49
  10. 2.3 Properties that Characterize Determinants p56
  11. 2.4 Expansion of a Determinant Along Its Columns p58
  12. 2.5 Cramer's Rule p61
  13. 2.6 Permutations, Symmetric and Antisymmetric Functions p63
  14. 2.7 Explicit Formula for the Determinant p69
  15. 2.8 The Rank of a Matrix p72
  16. 2.9 Operations on Matrices p79
  17. 2.10 Inverse Matrices p89
  18. Chapter 3: Vector Spaces p97
  19. 3.1 The Definition of a Vector Space p97
  20. 3.2 Dimension and Basis p104
  21. 3.3 Linear Transformations of Vector Spaces p119
  22. 3.4 Change of Coordinates p125
  23. 3.5 Isomorphisms of Vector Spaces p130
  24. 3.6 The Rank of a Linear Transformation p136
  25. 3.7 Dual Spaces p138
  26. 3.8 Forms and Polynomials in Vectors p145
  27. Chapter 4: Linear Transformations of a Vector Space to Itself p150
  28. 4.1 Eigenvectors and Invariant Subspaces p150
  29. 4.2 Complex and Real Vector Spaces p159
  30. 4.3 Complexification p166
  31. 4.4 Orientation of a Real Vector Space p171
  32. Chapter 5: Jordan Normal Form p178
  33. 5.1 Principal Vectors and Cyclic Subspaces p178
  34. 5.2 Jordan Normal Form (Decomposition) p182
  35. 5.3 Jordan Normal Form (Uniqueness) p186
  36. 5.4 Real Vector Spaces p190
  37. 5.5 Applications* p193
  38. Chapter 6: Quadratic and Bilinear Forms p207
  39. 6.1 Basic Definitions p207
  40. 6.2 Reduction to Canonical Form p214
  41. 6.3 Complex, Real, and Hermitian Forms p220
  42. Chapter 7: Euclidean Spaces p229
  43. 7.1 The Definition of a Euclidean Space p229
  44. 7.2 Orthogonal Transformations p239
  45. 7.3 Orientation of a Euclidean Space* p246
  46. 7.4 Examples* p249
  47. 7.5 Symmetric Transformations p261
  48. 7.6 Applications to Mechanics and Geometry* p271
  49. 7.7 Pseudo-Euclidean Spaces p281
  50. 7.8 Lorentz Transformations p291
  51. Chapter 8: Affine Spaces p305
  52. 8.1 The Definition of an Affine Space p305
  53. 8.2 Affine Spaces p310
  54. 8.3 Affine Transformations p317
  55. 8.4 Affine Euclidean Spaces and Motions p325
  56. Chapter 9: Projective Spaces p334
  57. 9.1 Definition of a Projective Space p334
  58. 9.2 Projective Transformations p343
  59. 9.3 The Cross Ratio p350
  60. 9.4 Topological Properties of Projective Spaces* p354
  61. Chapter 10: The Exterior Product and Exterior Algebras p363
  62. 10.1 Plücker Coordinates of a Subspace p363
  63. 10.2 The Plücker Relations and the Grassmannian p367
  64. 10.3 The Exterior Product p372
  65. 10.4 Exterior Algebras* p381
  66. 10.5 Appendix* p388
  67. Chapter 11: Quadrics p398
  68. 11.1 Quadrics in Projective Space p398
  69. 11.2 Quadrics in Complex Projective Space p407
  70. 11.3 Isotropic Subspaces p411
  71. 11.4 Quadrics in a Real Projective Space p423
  72. 11.5 Quadrics in a Real Affine Space p427
  73. 11.6 Quadrics in an Affine Euclidean Space p438
  74. 11.7 Quadrics in the Real Plane* p441
  75. Chapter 12: Hyperbolic Geometry p446
  76. 12.1 Hyperbolic Space* p447
  77. 12.2 The Axioms of Plane Geometry* p456
  78. 12.3 Some Formulas of Hyperbolic Geometry* p467
  79. Chapter 13: Groups, Rings, and Modules p479
  80. 13.1 Groups and Homomorphisms p479
  81. 13.2 Decomposition of Finite Abelian Groups p487
  82. 13.3 The Uniqueness of the Decomposition p493
  83. 13.4 Finitely Generated Torsion Modules over a Euclidean Ring* p496
  84. Chapter 14: Elements of Representation Theory p508
  85. 14.1 Basic Concepts of Representation Theory p508
  86. 14.2 Representations of Finite Groups p514
  87. 14.3 Irreducible Representations p519
  88. 14.4 Representations of Abelian Groups p522
  89. Historical Note p526
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