| Date | Topic | Reading assignment and HW | More reading materials | |
|---|---|---|---|---|
| Jan 19-22 | Overview, survery on students' academic background and goal. Metric spaces, open ball approach to topology and continuity. |
Homework 1, | ||
| Jan 25-29, | axiomatic treatment of the concept of a topological space.
Taylor expansion of a curve, Fundamental Theorem of the local theory of curves |
Lecture notes for week 2 | ||
| Feb 2-9, | Continuity, homeomorphisms, quotient spaces, and product spaces. | Lecture notes for week 3 Homework 2, | ||
| Feb 9-15, | Bases and subbases. A revisit of continuity and open sets. |
A categorical review of topology. Lecture notes for week 4 | ||
| Feb 16-22 | Compactness; Separation axioms (Hausdorff property). Connectedness, path-connectedness, and their relation. |
Lecture notes for week 5 Homework 3, | ||
| Feb 23-28 | Topological surfaces, definition and basic properties | Lecture notes for week 6 Homework 4, | ||
| Mar 2-10 | Fundamental polygon, cut and paste surgery, classification theorem. | Lecture notes for week 7. | Hitchin's notes introduction to surfaces. | |
| Mar 11-15 | Homotopy of countinous maps. | Lecture notes for week 8 |
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| Mar 16-22 | Fundamental group, first definition, and homotopy invariance. | |||
| Mar 23-29 | More fundamental group, computations: the circle, figure 8, and surfaces. | Homework 5, |
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| Apr 1-5 | A quick introduction to homology theory | |||
| Apr 6-14 | Selected topics: Morse theory, reeb graph, and persistent homology. | |||
| Apr 15-21 | Final presentation week. |