丘成桐中国大学生数学竞赛大纲.doc

2010年中国大学生数学竞赛（丘成桐教授发起）竞赛大纲一．Syllabuses for Geometry and Topology ?????? Geometry: Curves and surfaces 1)????? Plane curves and space curves 2)????? The fundamental theorem of curves 3)????? Concept and examples of surfaces 4)????? The first and second fundamental forms 5)????? Normal curvature, principal curvature and the Gauss curvature 6)????? Orthogonal moving frames and structure equations of surfaces 7)????? Existence and uniqueness of surfaces 8)????? Isometric transformation of surfaces 9)????? Covariant derivatives on surfaces 10)?? Geodesic curvatures and geodesics, Geodesic coordinates 11)?? The Gauss-Bonnet formula 12)?? Laplacian operator on surfaces Geometry on manifolds 1)?Manifolds 2)?Vector fields and differentials 3)?Tensors and differential forms 4)?Stokes formula 5)?De Rham theorem 6)?Lie derivatives 7)?Lie algebras 8)?Maurer-Cartan equations 9)?Vector bundles 10)Connection and curvatures 11) Structure equations 12) Riemannian metrics 13) The Hodge star operator and Laplacian operator 14) The Hodge theorem References: M. Do Carmo, Differential geometry of curves and surfaces. S S Chern and Chen Weihuan, Lectures on differential geometry Q. Chen and CK Peng, Differential geometry T. Frenkel: Geometry from physics J. Milnor, Morse theory ? Topology Point Set Topology 1)????? Open set and closed set 2)????? Continuous maps 3)????? Haudorff space, seperability and countable axioms 4)????? Compactness and Heine-Borel theorem 5)????? Connectivity and path connectivity 6)????? Quotient space and quotient topology? Fundamental groups ? 1)?Definition of fundamental groups, homotopic maps 2)?Computation of fundamental groups: Van Kampen theorem 3)?Covering maps and covering spaces 4)?Applications: Brouwer fixed point theorem, Lefschetz fixed point theorem ? Complexes and homology groups ? 1)?Simplex, complexes and polyhedron 2)?Barycentric subdivision and simplex approximation 3)?Computation of fundamental groups of complexes 4)?Classification of surfaces