FALL 2026
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:
J. Lee, Introduction to smooth manifolds, Springer, 2013
Prequisites:
Analysis, Linear Algebra, and Elementary Topology or permission of the instructor.
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FALL 2025
Hongbin Sun
Course Description:
Smooth manifolds are ubiquitous in the modern research of mathematics. This course is an introduction to smooth manifolds and their topology (differential topology).
In the second part, we will study differential topology, i.e. the topology of smooth manifolds. The topics include: Whitney immersion and embedding theorems, approximation theorem, Sard theorem, transversality, intersection numbers, Morse functions and Morse theory. If we have more time, I will talk about the h-cobordism theorem, which is the main part of Smale's proof of the Poincare conjecture of dimension at least 5.
Text:
John Lee, Introduction to smooth manifolds; Victor Guillemin & Alan Pollack, Differential topology
Prequisites:
Mathematical analysis, linear algebra, general topology (equivalent to Math 412, Math 350 and Math 441)
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FALL 2024
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.
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FALL 2023
Paul Feehan
Course Description:
Same as Fall 2024 above
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FALL 2022
Paul Feehan
Course Description:
Same as Fall 2023 above
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FALL 2021
Paul Feehan
Course Description:
Same as Fall 2022 above
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Schedule of Sections