3 Main Branches of Modern Mathematics Analysis n

3 Main Branches of Modern Mathematics Analysis n Algebra n Geometry n

A Brief History of Geometry Maggie & Michael ILG 131 2008/6/2 16: 00 伍業輝 Ng Ip Fai

What is Geometry? n A study of spaces and transformations ¨ Spaces: the geometric spaces ¨ Transformations: translations, rotations, symmetries, etc.

Ancient Period The concept of n Pythagorean Theorem n Pareto’s Five Regular Polyhedrons n Euclid’s Elements n Archimedes’s Volume Formula of Sphere n Zu Chong-Zi’s (祖沖之) Principle n

Middle / Towards Modern Stage n n n n Cartesian coordinate system Newton & Leibniz: Calculus Gaussian Elegant Theorem and Gauss-Bonnet’s Theorem Non-Euclidean Geometry and Riemannian Geometry Lagrange: Calculus of Variation Laplace: Celestial Mechanics Euler’s Characteristic & Wave Equation Klein’s Program

Modern Time Poincaré’s plane & fundamental groups n Hilbert’s Foundations of Geometry n Einstein’s General Relativity n de Rham’s Cohomology, Hodge’s Theory, Cartan’s Differential Form n Chen’s (陳省身) characteristic class, Chen-Weil and Chen-Simon’s Theories n

Contemporary Era Rauch’s Theorem n Atiyah-Singer’s Index Theorem n Yau (丘成桐): Geometric Analysis n Donaldson, Seiberg-Witten’s Theories n M. Gromov: Symplectic Geometry n Mandelbrot: Fractal Geometry and Chaos n Computational Geometry n (Wu & Chen 2004: 24 -25)

Geometry as an Experiential Science Ancient Period I

Beginning of Geometry n Ancient Civilizations ¨ Chinese, Babylonian, Egyptian ¨ Greek n “Geometry” is derived from Greek roots ¨ geo: earth ¨ metry: measurement n To measure the areas of lands

Pythagorean Theorem Pythagoras (About 6 th Century BC) 2 2 2 n a + b = c in a right triangle n Concept of Directions n Trigonometry n 5 3 4

Geometry as a Logical Science Ancient Period II (3 rd century BC onwards)

Euclidean Geometry Euclid (About 300 – 260 BC) n The beginning of Axiomatic Geometry n ¨ Breakthrough n of Mathematics Elements: Words and graphs only; no symbol

Common Notions x = a, y = a n x = y, a = b n Coincide n The whole n imply implies > x=y x+a=y+b x–a=y–b Equal The part

Postulates I ~ IV A straight line can be drawn from any point to any other point. n A finite straight line can be produced continuously in a line. n A circle may be described with any center and distance. n All right angles are equal to one another. n

Postulate V: Parallel Postulate n If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, then the two straight lines, if produced indefinitely meet on that side on which are the angles less than two right angles. n Given a line and a point not on the line, exactly one line can be drawn through the point parallel to the line.

Any Triangle is Isosceles? n n n n Let the perpendicular bisector of BC meet the bisector of A at E. Let EF AB, EG AC. BC = EC, EF = EG. BEF CEG, AEF AEG. BF = CG, AF = AG. AB = AF + FB = AG + GC = AC. ABC is an isosceles triangle. (Li & Zhou 1995: 351 -52)

Geometry as a Quantitative Science Towards Modern Stage (17 th century onwards)

Cartesian coordinate system n Descartes (1596 -1650) on Plane Pairs of Numbers ¨ Plane Curves Equations with 2 Variables ¨ Points Algebra Geometry 2 2 n n n E R ; or generally, E R n

Calculus Newton (1642 -1727) & Leibniz (1647 -1716) n Calculus is NOT geometry; but a tool for it n Finding tangent lines and areas n Analytical Geometry n

Geometry as an Invariant in a Transformation Group Modern Time

Concept of Group n A group is a set G, together with a binary operation * on G which satisfies: G, g, h G ¨ f *(g*h) = (f *g)*h, f, g, h G ¨ !e G such that g*e = g = e*g, g G ¨ g G h G such that g*h = e = h*g ¨ g*h n Compare it with a vector space yourselves.

Erlangen’s Guiding Principle Given a set V and a transformation group G on the elements in V. n Then V is called a space, its elements is called points, and the subspaces of V is called graphs. n And then the study of graphs about the invariants in the group G is called the geometry of V corresponding to G. n n To Classify geometry

Non-Euclidean Geometry Gauss, Bolyai, and Lobatchewsky n The Parallel Postulate can’t be proved by the first 4 postulates. n Both the geometries with and without the Parallel Postulate are self-consistent. n

Differential Geometry Gauss (1777 -1855), Riemann (1826 -1866) n Riemann’s “On the Hypotheses which lie at the Bases of Geometry” n ¨ Über die Hypothesen, welche der Geometrie zu Grunde liegen n Riemannian Geometry ¨ Einstein’s General Theory of Relativity

Gaussian Elegant Theorem Gauss Theorema Egregium n Gauss Curvature is intrinsic. n

Global Inner Geometry Cartan (1869 -1951) & Chen Xing-Shen (1911 -2004) n Localization vs. Globalization n Viewpoint of Dual n

Algebraic Geometry Use algebraic methods to study geometry n Deep and Difficult n ¨ Not only the background of geometry and analysis are required, but a deep understanding of algebra is also needed.

Calendar of Geometry (Wu & Chen 2004: 32) “Date” Events Year 01/01 Dawn of Ancient Greek Civilization 1650 B. C. 04/15 Pythagorean Theorem 600 B. C. 05/15 Euclid’s Elements 300 B. C. 05/20 Archimedes’s Volume of sphere 250 B. C. 06/15 (Year 0) 0 AD 11/25 Cartesian coordinate system 1630 11/29 Newton & Leibniz: Calculus 1670 12/14 -17 Gauss & Riemann: non-Euclidean & inner geometry 1820 -1850 12/22 Poincaré: Fundamental Groups; Ricci: Tensor analysis 1900 12/24 Einstein’s General Relativity 1910 -1920 12/27 Cartan & Chen: Global Inner Geometry 1950 12/29 Mandelbrot: Fractal Geometry 1970 12/31 (Year 2000) 2000

Error in the “Proof” Suppose AB > AC. n Let the bisector of A meet BC at D. n BD/ DC = AB/ AC > 1. n BD > DC. n M is on [BD]. n (Li & Zhou 1995: 352)

Bibliography n Li, C. -M. & Zhou H. -S. (1995). Research of Elementary Mathematics. Beijing: Higher Education Press. n Burton, D. (2007). The History of Mathematics: An Introduction, 6 th ed. New York: Mc. Graw-Hill. n Au, K. -K. , Cheung K. -L. & Cheung L. -F. (2007). Towards Differential Geometry: Lecture notes of EPYMT 2007. Hong Kong: The Chinese University of Hong Kong. n Beardon, A. (2005). Algebra and Geometry. Cambridge: Cambridge University Press. n Garding, L. (1977). Encounter with Mathematics. New York: Springer-Verlag

On-line References n Wu, Z. -Y. & Chen, W. -H. (2004). “A Short History of Development of Geometry”. Mathmedia 28(1). http: //www. math. sinica. edu. tw/math_media/d 281/ 28103. pdf n Wikipedia: History of Geometry (and others). http: //en. wikipedia. org/wiki/History_of_geometry n The History of Geometry. http: //math. rice. edu/~lanius/Geom/his. html

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