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MAT 531: Topology and Geometry II

### Spring 2018

###
Department of Mathematics

SUNY at Stony Brook

**Course Description:** This course is an introduction to the basic structures and concepts of smooth manifolds.
Topics to be covered include: foundations of differentiable manifolds, differentiable maps, vector fields and flows, differential forms and
integration on manifolds, Stokes' theorem, Froebenius' theorem, Lie derivatives, immersions and submersions, de Rham cohomology, cochain complexes,
degree of a map, Mayer-Vietoris Theorem.
**Textbook:**
*Introduction to Smooth Manifolds,* by John M. Lee, 2nd Edition, Springer, 2012.

**Instructor:** Marcus Khuri, Math Tower 3-122. Office Hours: Tuesdays and Thursdays from 2:30-4:00pm.
Email: khuri@math.sunysb.edu

**Course Assistants:**
Aleksander Doan, Math Tower 2-106. Office Hours: Wednesdays from 12:00-1:00pm in office, and Wednesdays from 10:00am-12:00pm
in MLC. Email: aleksander.doan@stonybrook.edu

**Class schedule:** TuTh 10:00-11:20am, Mathematics 4-130

**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 envelop will be placed
outside the office of the course assistant, where you may turn in your homework.
Late homework will not be accepted. Homework will account for 50% of
the total grade.

Homework 1 (Due 1/31) - §1: 5, 6, 7; §2: 5, 7, 10, 14;

Homework 2 (Due 2/7) - §3: 3, 4, 6; §4: 5, 6, 8;

Homework 3 (Due 2/14) - §4: 10, 13; §5: 4, 6, 7, 10, 11;

Homework 4 (Due 2/21) - §8: 4, 10, 11, 12, 13, 14, 16;

Homework 5 (Due 3/7) - §9: 2, 5, 17, 18; §19: 4, 5, 12;

Midterm Examination in class on 3/1 -- it covers sections 1-5, 8, 9, 19.

Homework 6 (Due 3/21) - §12: 2, 9, 10, 11, 12;

Homework 7 (Due 3/28) - §14: 1, 2, 5, 6, 7, 9;

Homework 8 (Due 4/4) - §15: 3, 5, 6, 10, 12, 16;

Homework 9 (Due 4/11) - §16: 2, 4, 6, 12, 14, 15, 17;

Homework 10 (Due 4/18) - §17: 1, 2, 8, 10, 11, 12, 13;

Homework 11 (Due 4/25) - §17: 6, 7; §18: 1, 2, 7, 8, 9;

Homework 12 (Due 5/2) - Problems.

**Exams:** There will be an in class midterm as well as a final,
each respectively accounting for 20% and 30% of the total grade. **The midterm exam will
be in class on Thursday, March 1, and the final exam will take place on Friday, May 11, 11:15am-1:45pm
in our classroom Mathematics 4-130.** By enrolling in this course, you are attesting to the fact
that you will be available for the exams at these times.

**Disability Support Services (DSS)**

**If you have a physical, psychological, medical or learning disability that may impact
your course work, please contact Disability Support Services, ECC (Educational Communications Center)
Building, room 128, (631) 632-6748. They will determine with you what accommodations, if any, 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 Disability Support Services.
For procedures and information go to the following website:
http://www.stonybrook.edu/ehs/fire/disabilities**

**Academic Integrity**

**Representing another person's work as your own is always wrong. Faculty are required to report any suspected instances
of academic dishonesty to the Academic Judiciary. Faculty in the Health Sciences Center (School of Health Technology & Management,
Nursing, Social Welfare, Dental Medicine) and School of Medicine are required to follow their school-specific procedures. For more
comprehensive information on academic integrity, including categories of academic dishonesty, please refer to the academic judiciary
website at
http://www.stonybrook.edu/commcms/academic_integrity/index.html**

**Critical Incident Management**

**Stony Brook University expects students to respect the rights, privileges, and property of other people. Faculty are required to
report to the Office of Judicial Affairs any disruptive behavior that interrupts their ability to teach, compromises the safety of
the learning environment, or inhibits students' ability to learn. Faculty in the HSC Schools and the School of Medicine are required
to follow their school-specific procedures.**