MATH 427(F) Differential Topology

When mathematicians started to explore the world of geometric forms in higher dimensions, they found that neither the intuition nor the vocabulary of ordinary geometry was sufficient to describe and classify the new objects they were discovering. So the notion of a manifold was born. Manifolds are surfaces and shapes, that appear to be euclidean when a small region is examined. On a large scale, however, these objects fail to follow the rules for a euclidean geometry. Spheres, surfaces of donuts and pretzels, the earth's surface are examples of manifolds. Manifolds play very important roles in many different fields. They occur in several branches of mathematics, physics, chemistry, biology, geology, economics, etc. This course will be an introduction to the beautiful and inspiring theory of manifolds. The emphasis will be on doing calculus on manifolds. Topics include manifolds and differentiable maps; submanifolds and embeddings; tangent spaces, critical points and Sard's theorem; vector bundles and tubular neighborhoods. Morse theory and surgery on manifolds will also be discussed. In addition to learning the theory, we will look at open problems in the field.
Evaluation will be based primarily on homework and exams. Prerequisites: One of Mathematics 301, 305, 321 and one of Mathematics 312, 315; or Mathematics 324; or permission of instructor.

Hour:  CHKHENKELI