MAT 402  Seminar in Mathematics: Vector Bundles on Manifolds  

Spring 2018  TuTh 10:00-11:20am Earth and Space 177

Instructor: Ljudmila Kamenova

Office: Math Tower 3-115
Office hours: Wednesdays 11am-12 in the MLC, 12-2pm in Math Tower 3-115

Syllabus: The seminar topic this semester will be "Vector bundles on manifolds". We are going to review the basics of manifolds, differential forms and de Rham cohomology. During most of the semester we are going to work with the definitions and properties of vector bundles. We would also introduce general fiber bundles, parallel transport, connections, geodesics. We are going to explore examples of vector bundles on projective spaces. Towards the end of the semester we would talk about the splitting principle and characteristic classes.

This seminar course is aimed primarily at third- and fourth-year students majoring in mathematics.

I will start by giving a few initial lectures, but after a week or two will assign topics to students to present in class.

Prerequisites: Basic understanding of analysis in several dimensions (as in MAT 322) and topology and geometry (as in MAT364), or an equivalent course. If you would like to take MAT402, but you are not sure about the requirements, please contact me.

Required Text: There is no required textbook. We are going to use Chris Wendl's notes on bundles and connections which are available for non-commercial use on his web-page. A very well-written text on manifolds and vector bundles is Aleksey Zinger's notes on manifolds and vector bundles. Another source on differential geometry and bundles is Ivan Avramidi's lecture notes on topics in differential geometry.

Here are some further reading materials:

Grading: The grading will be based primarily on class participation (40%), the quality of the student's in-class presentations (30%), and a final term paper consisting of the written version of one of the presentations (30%). The term paper should be typed of length 5 to 10 pages. Not attending regularly will have a negative effect on the grade. Regular attendance is therefore required.

The final term papers are due in the last week of classes.

Homework: Here are some practice problems.

HW 1: Click here for the problems.

HW 2: Click here for the problems.

HW 3: Click here for the problems.

HW 4: Click here for the problems.

HW 5: Click here for the problems.

HW 6: Click here for the problems.

DSS Statement. 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.

Academic Integrity Statement. Each student must pursue his or her academic goals honestly and be personally accountable for all submitted work. 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 tofollow their school-specific procedures. For more comprehensive information on academic integrity, including categories of academic dishonesty, please refer to the academic judiciary website.

Critical Incident Management Statement. 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.