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. 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. 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.