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#### MAT 531: Topology, Geometry II

Course Instructor: Mark McLean

Class coordinates: Tuesday, Thursday 3:00pm-4:20pm in Physics P130.

Material Covered: Foundations of differentiable manifolds: differentiable maps, vector fields and flows, and differential forms and integration on manifolds. Stokes' theorem. Froebenius theorem. Lie derivatives. Immersions and submersions. DeRham chomology, cochain complexes, degree of a map, Mayer-Vietoris Theorem.

Spring, 3 credits, Letter graded (A, A-, B+, etc.)

Textbook: Introduction to Smooth Manifolds (2nd Ed.), by Lee, Springer 2012.

Office Hours: Tuesday, Thursday 1pm-2pm, in Math 3-112.

Exams: There are two exams, a midterm and a final exam. Both are worth 25% of the final grade.
• The midterm exam will be on Thursday March 10th in P130 from 3:00pm to 4:20pm (I.e. in class).
• The final exam will be on Tuesday May 10th from 2:15pm-5:00pm.
Homework: Homework is a fundamental part of this course, and you will have to work hard on the assigned problems in order to succeed. Assignments will be posted on the course website at the beginning of each week, and will be due on Wednesday of the following week at 3pm. A box or envelope will be placed outside the office of the course assistant Mohamad Rabah, where you may turn in your homework. Late homework will not be accepted. Homework will account for 50% of the total grade.

The exercises will be from the course textbook. Starred questions are for extra credit.

Number. Due Date. Exercises.
1. Wednesday Feb 9th at 3pm. 1-1, 1-3, 1-5, 1-6, 1-7, 1-9.
2. Wednesday Feb 16th at 3pm. 2-5, 2-7, 2-14, 3-1, 3-2 (for extra credit: show that the vector space of smooth functions on a positive dimensional manifold cannot have a countable basis).
3. Wednesday Feb 23rd at 3pm. 3-3, 3-4, 3-5, 3-6, 3-7, 4-4.
4. Wednesday Mar 2nd at 3pm. 4-5, 4-10 , 4-13, 5-6, 5-7.
5. Wednesday Mar 30th at 3pm. 6-1, 8-3, 8-4, 8-11, 8-13, 8-16.
6. Wednesday Apr 6th at 3pm. 9-2, 9-3, 9-4, 9-5, ☆9-8, 9-17 & ☆9-18.
7. Wednesday Apr 13th at 3pm. 10-6, 10-7, 10-10, 10-12, 10-18
8. Wednesday Apr 20th at 3pm. 11-7, 11-9, & ☆11-10, 12-8, 14-1, 14-3.
9. Wednesday Apr 27th at 3pm. 14-5, 14-6, 14-7, 14-9, 15-1.
10. Wednesday May 4th at 3pm. 15-3, 15-5, ☆15-8, 16-3 (a), 16-4 (you may assume the retraction is smooth here), 16-5, & ☆16-6
Practice Problems. (Not officially graded, but I can informally grade if you wish). 17-1, 17-5, 17-6, 17-7, 17-10, 17-13, 19-4, 19-5, 19-6

1. Jan 25th and 27th. Chapters 1 and 2.
2. Feb 1st and 3rd. Chapter 3.
3. Feb 8th and 10th. Chapter 4.
4. Feb 15th and 17th. Chapter 5.
5. Feb 22nd and 24th. Chapter 6.
6. March 1st and 3rd. Chapter 8.
7. March 8th and 10th. Review and Midterm.
Spring Break.
8. March 22nd and 24th. Chapter 9.
9. March 29th and 31st. Chapters 10 and 11.
10. April 5th and 7th. Chapters 13 and 14.
11. April 12th and 14th. Chapters 15 and 16.
12. April 19th and 21st. Chapter 17.
13. April 16th and 28th. Chapter 19.
13. May 3rd and 5th. Review.

Disability Support Services (DSS) Statement: If you have a physical, psychological, medical, or learning disability that may impact your course work, please contact the Student Accessibility Support Center, Stony Brook Union Suite 107, (631) 632-6748, or at sasc@stonybrook.edu. They will determine with you what accommodations are necessary and appropriate. All information and documentation is confidential. Students who require assistance during emergency evacuation are encouraged to discuss their needs with their professors and the Student Accessibility Support Center. For procedures and information go to the following website: https://ehs.stonybrook.edu//programs/fire-safety/emergency-evacuation/evacuation-guide-disabilities and search Fire Safety and Evacuation and Disabilities.