Math 441/442/443

Topology/Differential Geometry/Topics in Topology and Geometry

Class meets Mon-Wed-Fri: 1:30-2:20 in MEB 237

Instructor:	Daniel Pollack
Office:	C-550 Padelford Hall
Phone:	206-543-1809
E-mail:	pollack@math.washington.edu

The Mathematics sequence Math 441/2/3 is being revised as of Autumn 1999. The sequence will begin with Math 441, Topology. Math 442 is an Introduction to Differential Geometry, and Math 443 will continue with Topics in Geometry and Topology. The sequence as a whole will provide a solid introduction to a core area of modern mathematics. Topology and Differential Geometry form one of the most exciting areas of current mathematical research and play a major role in many other fields such as number theory, algebraic geometry, analysis and mathematical physics. In addition topology and geometry form the basis of numerous applications in fields outside of pure mathematics, such as physics, computer graphics, biology and engineering. Students who are considering pursuing a graduate degree in mathematics may find this course of particular interest. A more detailed description of each of the three courses follows.

Math 441 will be focus on the foundations of topology. In addition to being an important field in its own right topology provides the language and tools we need to explore differential geometry. The text for the course will be the book Foundations of Topology by C. Wayne Patty. The course will cover all of Chapters 1 through 4 of the text with the exception of sections 2.4, 2.5 and 4.4. A rough outline of the topics which this covers is thus

Topological Spaces: metric spaces, open sets and basis for a topology, closed sets, closures and interiors of sets, convergence and homeomorphisms.
New Spaces from Old Ones: subspaces, product topology, quotient spaces.
Connectedness: connected spaces, pathwise and local connectedness, totally disconnected spaces.
Compactness: compactness in metric spaces, compact spaces, local compactness, equicontinuity.

Other topics covered in the text will be treated in Math 442 and Math 443.

Math 442 will be in introduction to the differential geometry of curves and surfaces in Euclidean three space. The text for this course will be Differential Geometry of Curves and Surfaces by Manfredo P. Do Carmo. The first 3 chapters of the text will form the basic syllabus for the course. Again a rough outline is given by

Curves: parametrized curves, regular curves and arc length, curvature and torsion of curves, the fundamental theorem for the local theory of curves.
Regular Surfaces: continuity and differentiability in 3-space, inverse images of regular values, functions on surfaces, the tangent plane ad the differential of a smooth map, first fundamental form, orientation, area.
The Geometry of the Gauss Map: Definition of the Gauss map, second fundamental form, principal curvatures, Gauss and mean curvature, Gauss map in local coordinates.

Math 443 will build on the first two quarters and cover various topics in Geometry and Topology. We will begin by extending what was done in Math 441 to treat the intrinsic geometry of surfaces and prove Gauss' Theorema Egregium (or "Remarkable Theorem"). In addition we may introduce higher dimensional topological and differentiable manifolds. Further topics will be announced later in the year.