Seminar in Mathematics:

Winding Numbers in Geometry and Topology

Last update: 1/21/02

**Lecturer:** Detlef Gromoll, Math 5-116

Phone: 632-8290,
email: *detlef@math.sunysb.edu*

Classes: TuTh 11:20-12:40, Math P-131

Office Hours: W 11-12 UG Office P-143, and by appointment

**About this course:** This course will begin with a discussion of global properties of closed curves - mostly in the plane. Winding numbers count how many essential times such a curve surrounds a given point. These numbers have deep and often unexpected applications. Topics to be covered include: The intersection number formula, rotation index, total curvature and the Theorem of Turning Tangents, convex curves and the isoperimetric inequality, Jordan's Separation Theorem, Kronecker's Existence Theorem and the Fundamental Theorem of Algebra, Brouwer's Fixed Point Theorem, the Theorem of Borsuk Ulam, and the Ham-Sandwich Problem. We will also touch on space curves - in particular, the total curvature of a knot.

**Text:** Detailed notes will be available at no cost; additional reading is recommended.

**Prerequisites:** Basically only calculus is needed, including a little multivariate calculus in the plane and in space; but more experience in mathematical thinking will definitely be a plus.

**Grading:** Students are expected to present a talk, prepare a paper, and work problems.

**Special Needs:** If you have a physical, psychiatric,
medical, or learning disability that could adversely affect
your ability to carry out assigned course work, we urge you to contact
the staff in the Disabled Student Services office (DSS),
Room 133 Humanities, 632-6748/TDD. DSS will review your
situation and determine, with you, what accomodations are
necessary and appropriate. All information and documentation
of disability is confidential.

**Problems & Projects:**

Project 1 (2/14/02)
TeX source file for Project 1

Project 2 (5/13/02)