MA 598 Ralph M. Kaufmann

Ralph Martin Kaufmann
Purdue University; Department of Mathematics
150 N. University Street; West Lafayette, IN
Phone: (765) 494-1205; Fax: (765) 494-0548
e-mail: rkaufman@math.purdue.edu URL: http://www.math.purdue.edu/~rkaufman

MA 59800 - Fall 2028
Topology and its Applications

Instructor: Ralph Kaufmann

Office: Math 730
Contact via e-mail

Date and Time: MWF 9:30-10:20 Rec 113

Syllabus

Office hours: M 3:00- 3:50 and by appointment

References: There will be not be based on an existing book. General references will be given, but all material will be developed in class. Some references are
Feynman Lectures on Physics
do Carmo: Differential Geometry of Curves and Surfaces
Munkres: Topology, Algebraic Topology.
Bott-Tu: Differential Forms in Algebraic Topology
Others will work as well.

Wikipedia, Wolfram Mathworld and n-lab (use with care).

First Week: I will do a short survey and assess the different needs and levels of the participants. The following table will be used to keep track of the subjects and references.

Number	Topics	Reference
1	Overview. Review of Vector Calculus Theorems and their topological content, 2d and 3d examples. Start of deRham	Any vector calculus book. Bott-Tu, Lecture notes
2	More deRham, beginning of homotopy theory	Bott-Tu, Munkres, Lecture notes
3	Potentials and Maxwell equations. deRham differential.	Lecture notes, Bott-Tu, basic resources.
4	Vector fields and forms. Hodge *, grad, curl, div.	Lecture notes, notes.
5	Hodge *, orientations, closed and exact forms, Poincaré Lemma, deRham cohomology for $,R^n$	Lecture notes, notes, Bott-Tu or other sources.
6	Application to Maxwell equation, Helmholz Theorem, Winding number	Feynman lectures, do Carmo
7	Lifts, winding number, relation to log, cuts & residues, turning number, curvature of curves as index theorem	do Carmo, lecture notes.
8	Gauss curvature, Theorema egregium, Gauss-Bonnet as another index theorem	do Carmo

Suggested Problems. There may be suggested problem sets,

Presentation Topics: There may be presentations.

NEWS: Any news will be here. Welcome to the semester.