This course is the second part of a sequence of two courses dedicated to the study of the notion of differentiable manifold. In the first course we have seen the basic definitions (smooth manifold, submanifold, smooth map, immersion, embedding, foliation, etc.), some examples (spheres, projective spaces, Lie groups, etc.) and some fundamental results (constant rank theorem, Witney’s embedding theorem, Frobenius theorem, Stokes Formula, etc.). In this course we will proceed with this study, introducing more advanced notions (e.g., vector bundles, principal bundles, connections, etc.) and aiming at a deeper understanding of smooth manifolds. If you have not taken Math 518 the past semester, you should make yourself familiar with the topics study in that course (please see the web page of Math 518).

**Lecturer:**Rui Loja Fernandes**Email:** ruiloja (at) illinois.edu**Office:** 346 Illini Hall**Office Hours:** Wed 11:00 AM (or by appointment);**Class meets:** MWF 9:00-9:50 PM, 441 Altgeld Hall;**Prerequisites:** Math 518 or equivalent.

**In this page:**

- Announcements
- Syllabus
- Textbooks
- Grading Policy and Exams
- Homework Assignments
- Emergency information for students in Mathematics courses

**Announcements:**

- Class will meet for the first time on Wednesday, January 22.
- I have made available the final papers as links below under the titles of the presentations.
- No more announcements will be posted.
- Schedule of presentations:
- Wednesday, April 30:
- James Wratten: de Rham’s Theorem via simplicial methods
- Joel Villatoro:i Symmetries of differential equations
- Shiyu Shen: Foliations and Reeb stability
- Edward J. Sanchez: Cobordism and Hopf’s theorem

- Friday, May 2:
- Matthew Romney: The Hodge star theorem
- Nima Rasekh: Orbifolds and Euler characteristic
- Malik Obeidin: Exotic smooth structures on S^7
- Georgios Kydonakis: Bochner’s linearization theorem

- Monday, May 5:
- Daan Michiels: The orbit type stratification
- Marissa Loving: de Rham’s Theorem via simplicial methods
- Feng Liang: Darboux’s theorem and symplectic geometry
- Melinda Lanius: Morse Theory

- Wednesday, May 7:
- Xinghua Gao: Symmetric spaces
- Eliana Duarte: de Rham’s theorem (using sheaf cohomology)
- Erin Compaan: Topologies on spaces of smooth functions
- Yiwang Chen: Triangulations of surfaces

- Wednesday, April 30:

**Syllabus:**

**Differential Forms.**Review of differential forms. De Rham Cohomology. Properties of de Rham cohomology: homotopy invariance and Mayer-Vietoris sequence. The de Rham Theorem. Computations in cohomology: triangulations and Euler’s formula, PoincarĂ© Duality, degree of a map, index of a vector field.**Fiber Bundles.**Vector bundles, principal bundles, connections, parallel transport, curvature, Chern-Weyl theory, PoincarĂ©-Hopf Theorem.**Morse Theory.**(this topic will be covered only if time permits).

**Textbooks:**

The first two topics are covered in my lecture notes, but the following textbooks are highly recommended.

**Recommended Textbooks:**

- R. Bott and L.W. Tu,
,Springer-Verlag, GTM vol 82, 1982.**Differential Forms in Algebraic Topology** - John M. Lee,
, Springer-Verlag, GTM vol 218, 2003. See the contents and first chapter here.**Introduction to Smooth Manifolds** - Michael Spivak,
, (3rd edition) Publish or Perish, 1999.*A Comprehensive Introduction to Differential Geometry, Vol. 1*

**Grading Policy**

**Expository Paper**Students will be required to write (in LaTeX) and present a paper. The papers will be due by the last week of classes. Presentations will be in class, during the last week of classes.**Problem Sessions**There will be problem sessions every other week. Students are not required to turn in homework, but homework problems will be discussed in these sessions.

**Homework Assignments and Sections covered so far:**

**Homework #1**Read Lectures 18 and 19 and solve exercises 18.2, 18.3, 19.1, 19.2, 19.3, 19.4.**Homework #2**Read Lectures 20 and 21 and solve exercises 20.3, 20.7, 20.8, 20.9, 21.1, 21.4, 21.5, 21.6, 21.7, 21.8.**Homework #3**Read Lectures 13 and 22 and solve exercises 13.1, 13.4, 13.6, 13.7, 13.8, 22.1, 22.3, 22.6, 22.8.**Homework #4**Read Lectures 14 and 23 and solve exercises 14.1, 14.3, 14.6, 14.7, 14.8, 14.10, 14.11, 23.2, 23.3, 23.7, 23.8.**Homework #5**Read Lectures 24-26 and solve exercises 24.3, 24.4, 25.1, 25.3, 25.6, 26.1, 26.2, 26.3, 26.6, 26.8.

**Sections of the Lecture Notes covered:** Lectures 13-14, 18-30.

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

**Emergency information for students in Mathematics courses**

For important emergency information related to fires, tornados or active threats, please look at the following leaflet.

*Last updated May 8, 2014.*