English and L A TEX for Mathematicians

Transcription

1 English and L A TEX for Mathematicians Melchior Grützmann melchiorgfreehostingcom/english Department of Mathematics 12 th April, 2012

2 Outline Last week: Structure of an article Prehistoric and antique phase Renaissance and Enlightenment Modern mathematics

3 Prehistoric phase Figure: Ishango Bone, , NE Congo comparing and counting notion of natural numbers (equivalence relation)

4 Beginnings of Geometry oldest preserved pictures of geometric shapes (Egypt) Stonehenge (England), astronomic calendar?

5 Antique usage of mathematics around the Mediterranean book-keeping arithmetic (multiplication tables, division), construction works (eg pyramids in Egypt) geometry, engineering, mechanics plane geometry (Thales, Pythagoras, Archimedes, Euclid) logic and reason to give first proofs (Aristotle)

6 Antique mathematics in China -330 Mohist canon physics and geometry -300 Chou Pei Suan Ching decimal position system -212 Qin Shi Huang: burning of all non-official books -200 Jiǔzhāng Suànshù mathematical problem solving 200 suan pan (Chinese abacus)

7 Hindu Arabic mathematics 500 Aryabhata, trigonometric functions 800 numeral positional system 825 algorithm and algebra 1000 inductive proof

8 Renaissance (Europe) rediscovery of the ancient Greek and Arabic contributions solution formula for cubic (Ferro, Tartaglia, 1525) and quartic equations (Ferrari, 1535)

9 Scientific Revolution, 17th century laws of kinematics (Galileo), laws of planetary motions (T Brahe, J Kepler), natural logarithms (Napier, Bürgi), Carthesian coordinate systems (Descartes) laws of motion (Newton), differential and integral calculus (Newton, Leibniz), elementary probability theory (Fermat, Pascal),

10 Scientific Revolution, 18th century Leonhard Euler (by E Handmann) graph theory, complex analysis, fluid dynamics (Euler s equations) JL Lagrange: number theory, calculus of variations PS Laplace: celestial mechanics, statistics, Laplace pde, Laplace transform (sa O Heaviside)

11 Modern mathematics, 19th century I CF Gauß: science; functions of complex variables, geometry, convergence of series NI Lobachevsky, J Bolyai, CF Gauß: hyperbolic geometry alternative to Euclidean geom B Riemann: spherical geometry, manifold, Riemannian geometry H Grassmann: vector spaces, WR Hamilton: noncommutative algebra B Bolzano, AL Cauchy, K Weierstrass: more rigorous calculus, ϵ-δ-definition of limit

12 Modern mathematics, 19th century II NH Abel, E Galois: limits of algebra, finite group theory G Cantor: systematic set theory, G Peano, LEJ Brouwer, D Hilbert, B Russell, AN Whitehead: logic as science (foundations of math) founding of: London MS, SM de France, Edinburgh MS, AMS, Quaternion Society

13 plane geometry (ancient Greeks, Arabia, China, Japan) analysis situs (topology)

14 19th century hyperbolic geometry (Lobachevsky, Bolyai, Gauß) alternative to Euclidean geometry construction of regular n-gons (Gauß, P Wantzel) with ruler and compass/ impossibility of the classical Greek problems: doubling the cube, squaring the disk, trisecting the angle Riemannian geometry (B Riemann), abstract curves and surfaces axiomatization of geometry (D Hilbert, F Klein) analytical mechanics (JL Lagrange, CG Jacobi, WR Hamilton) Poisson and symplectic geometry

15 Last week: Structure of an article Idea of a manifold A manifold is glued together from open subsets U Rn that overlap in open subsets Uα Uβ Rn 1 The geometric quantities are those expressions that remain invariant under coordinate changes2 1 2 H Whitney G Ricci-Curbastro, T Levi-Civita

16 20th century areas of differential geometry 1 Riemannian and semi-riemannian geometry metric tensor g Γ(S 2 T M), symmetric non-degenerate bilinearform; 2 Finsler geometry smooth Banach norm on TM, 3 symplectic geometry ω Ω 2 (M), skew-symmetric non-degenerate bilinearform, dω = 0 (even dimensional manifolds); 4 contact geometry α Ω 1 (M), dα maximally non-degenerate (odd-dimensional manifolds); 5 complex and Kähler geometry Ũ Cn instead of R n ; 6 Cauchy Riemann geometry boundaries of complex manifolds; 7 Lie groups continuous groups parametrized by manifolds; 8 differential topology without a metric / sympl str

17 Other modern geometry algebraic geometry replace R by an algebraic field / commutative ring; tropical geometry 3 replace R by a semi-ring; non-commutative geometry (A Connes, J Lott) replace R by a non-commutative topological ring 3 named after a Brazilian

18 Further literature wiki History of Mathematics, Differential geometry VJ Katz History of Mathematics An Introduction, Addison-Wesley (1998), ISBN G G Joseph The Crest of the Peacock The Non-European Roots of Mathematics, Princeton University Press (2000), ISBN

19 Next week: typical mathematical expressions examples and correct usage what to avoid