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#### MAT 620:Topics in Topology: Morse Homology

Course Instructor: Mark McLean

Tuesday, Thursday 10:00am-11:20am, Mathematics 5127

#### Introduction to the Course

Suppose we have some complicated (but smooth) outdoor sculpture or statue and suppose that it is raining. When a rain drop hits the sculpture it sticks to the sculpture and then runs down the side creating a flow line. One can ask: is there a systematic way of finding the shape of such a statue using these flow lines and conversely can one obtain information about these flow lines from knowledge of the shape of the sculpture? There are higher dimensional (and even infinite dimensional) versions of this question, where the 'sculpture' is a manifold and the 'rain' is represented by a smooth function called a Morse function. We will show how you can calculate homology groups using this data and conversely show how one can obtain information such as the number of critical points of a Morse function given the Betti numbers of a manifold, and will prove these facts rigorously. At the end of the course there will be a brief sketch of how one can generalize this to infinite dimensions enabling us to find periodic orbits of Hamiltonian systems. Some of the proofs in this course may not be so efficient as they have infinite dimensions in mind although we will not pursue the infinite dimensional case in any detail in this course.

#### Prerequisites

You should be very familiar with topics in Differential geometry such as Sards Theorem, Implicit Function theorem, connections and curvature etc. You must also have some familiarity with analysis such as Banach spaces.

Office Hours: Tuesday 2pm-3pm (Math 4-101B).

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