Course Descriptions

16:640:548 - Differential Topology

Paul Feehan

Course Description:

Differential Topology of central importance in Mathematics and required background for every research mathematician and theoretical physicist. Differential Topology has core applications in all areas of Complex Analysis and Geometry, Differential Geometry, Geometric Analysis, Geometric Topology, Global Analysis, Mathematical Physics, Partial Differential Equations, and Theoretical Physics. This course will give a broad introduction to Differential Topology, with prerequisites that we shall try to keep to a minimum in order to introduce students to the field while also providing guidance for more advanced students. Topics may vary depending on the audience and their interests but should include:

I. Smooth manifolds and smooth maps

II. Submersions, immersions, and embeddings

III. Transversality and Sard’s theorem

IV. Intersection theory

V. Vector fields, Lie derivatives and brackets, distributions, tensor fields, and vector bundles

VI. Riemannian metrics

VII. Differential forms, orientations, integration on manifolds, and de Rham cohomology

VIII. Morse theory

IX. Symplectic manifolds

X. Banach manifolds

Text:

Primary References:

 · J. Lee, Introduction to smooth manifolds, Springer, 2013

· V. Guillemin and A. Pollack, Differential topology, AMS Chelsea, 201

Secondary References:

The following references may be useful for more advanced students with specific interests:

· R. Abraham, J. Marsden, T. Ratiu, Manifolds, tensor analysis, and applications, Springer, 1988

· M. Hirsch, Differential topology, Springer, 1994

· J. W. Milnor, Morse theory, Princeton University Press, 1963

· S. Lang, Introduction to differentiable manifolds, Springer, 2002

· J. Margalef Roig and E. Outerelo Dominguez, Differential topology, North Holland, 1992

· F. Warner, Foundations of differentiable manifolds and Lie groups, Springer, 1983

· E. Zeidler, Nonlinear functional analysis and its applications, volume I, Springer, 1986

Prequisites:

Analysis, Linear Algebra, and Elementary Topology or permission of the instructor.