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Syllabus for Oral Examination \\
Ren Guo (November 30, 2005)
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    Algebraic Topology\\



    The Fundamental Group

    The Seifert-Van Kampen Theorem

    Covering Spaces

    Lifting properties

    Classification of covering spaces

    Deck Transformations and group actions\\

    Simplicial  homology

    Singular homology

    Homotopy Invariance

    Exact Sequence and Excision

    Cellular Homology

    Mayer-Vietoris Sequence\\

    Cohomology ring

    K\"unneth formula

    Cup and Cap Products

    Hopf invariant

    Poincare Duality\\

   Reference

   Allen Hatcher, Algebraic Topology

   James Vick, Homology Theory: An Introduction to Algebraic Topology

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  Riemannian Geometry\\

   Riemannian metrics

   Levi-Civita connection

   Parallel translation

   Curvature tensor

   Sectional curvature, Ricci curvature, Scalar curvature\\

   Geodesic, Exponential map, Gauss lemma

   Reimannian manifolds as metric spaces

   Hopf-Rinow theorem

   First and second variations of arc length

   Jacobi fields\\

   Manifolds with constant sectional curvature

   Cartan-Hadamard theorem

   Bonnet-Myers theorem

   Synge theorem\\

  Reference

Karsten Grove, Riemannian Geometry: A Metric Entrance Peter

Petersen, Riemannian Geometry

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