University of Illinois at Urbana-Champaign

Math 518 - Differentiable Manifolds I - Fall 2020

(section XXX)

The notion of differentiable manifold makes precise the concept of a space which locally looks like the usual euclidean space Rn. Hence, it generalizes the usual notions of curve (locally looks like R1) and surface (locally looks like R2). This course consists of a precise study of this fundamental concept of Mathematics and some of the constructions associated with it: for example, much of the infinitesimal analysis (i.e., calculus) extends from euclidean space to smooth manifolds. On the other hand, the global analysis of smooth manifolds requires new techniques and even the most elementary questions quickly lead to open questions. If you would like to have a preview of this course and experience a taste of it, you may wish to watch an old recording of 3 lectures by Fields medalist John Milnor.

Lecturer: Rui Loja Fernandes
Email: ruiloja (at)
Office: 346 Illini Hall
Office Hours: TBD
Class meets: TBD
Prerequisites: Restricted to Graduate Students; Undergraduate students may register with approval.

In this page:




You can find my lecture notes here, but the following two textbooks are highly recommended.

Recommended Textbooks:

Grading Policy and Exams

There will be weekly homework, 1 midterm and a final exam. All exams/midterms will be closed book.

Homework Assignments and Sections covered so far:

Sections of the Lecture Notes covered:

(PDF files can be viewed using Adobe Acrobat Reader which can be downloaded for free from Adobe Systems for all operating systems.)

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Last updated March 1, 2020