MATH 432/532 - Basic Analysis II
Spring 2013

Presentation Topics

 The following is a list of possible topics for student presentations and final papers, but I also encourage you to propose other topics that relate to the content of our course. Be sure to discuss your choice with me early so I can make sure there is no duplication and we can agree on the scope of expectations for your project. I will help you locate resources to begin reading and working through the mathematics. As you make progress, you should meet with me regularly to make sure everything is on track. Lebesgue's Theorem (necessary and sufficient conditions for integrability), see pp. 203-207 A nonintegrable derivative, see pp. 207-210 The generalized Riemann Integral, see pp. 213-221 Fourier Series, see pp. 228-243 Taylor Series for complex functions Double series The Mean Value Theorem for functions f : Rn→R and T : Rn→Rn Possible sets of discontinuities for functions, see pp. 29-34, 75-78, 94-97, 125-128 The historical development of the function concept The covariant derivative and geodesics Orbital mechanics The covariant derivative and Foucault pendulum Taylor Series Solutions to Schrödinger's Equation The mechanics of spinning tops General relativity Hamiltonian Mechanics (using the theory of differential forms to model and solve classical mechanics problems) Whitney Embedding Theorem (What size Euclidean space Rn is needed for embedding a k-dimensional manifold?) Intersection theory (invariants of submanifolds derived from their intersections) The Lefschetz fixed-point theorem (invariants of manifolds derived from fixed points of smooth maps on manifolds) The Poincaré-Hopf index theorem (invariants of manifolds derived from vector fields) The curvature of surfaces Green's Theorem, the Divergence Theorem, and Stokes' Theorem for vector fields DeRham cohomology (invariants of manifolds derived from differential forms) The Degree Formula (the transformation of an integral under a smooth map) The Gauss-Bonnet Theorem (relating the integral of curvature to the Euler characteristic) The Cauchy Integral Formula (relating line integrals to winding numbers)