18.937 Topics in Geometric Topology

In Spring 2019, Prof. Mrowka was teaching Instanton Floer Homology at MIT.

In this webpage, you can find the lecture notes that I transcribed. You can also read them in a single document.

I would like to thank Zhenkun Li and Gregory Parker for their lecture notes, their help with some exercises and for proofreading. I would also like to thank Chris Gerig for correcting some mistakes and his help on editing some documents.

Feel free to send me e-mails if you find any mistakes or typos.

Feb. 6. Lecture 1: An Overview of Morse Homology.

Feb. 8. Lecture 2: The Stable Manifold Theory.

Feb. 11. Cancelled.

Feb. 13. Cancelled.

Feb. 15. Lecture 3: Morse Homology and Its Variants.

Feb. 18. Lecture 4: Floer Homology: An Overview.

Feb. 20. Lecture 5: Connections and Curvatures.

Feb. 22. Lecture 6: The Chern-Simons Functional.

Feb. 25. Lecture 7: Representations.

Feb. 27. Lecture 8: The Energy Identity and Instantons on S^4.

March 1. Lecture 9: Instantons on S^4 continued.

March 4. Lecture 10: The ADHM Construction.

March 6. Lecture 11: Gauge Transformations

March 8. Lecture 12: The Configuration Space is Hausdorff.

March 11. Lecture 13: The Construction of Slices.

March 13. Lecture 14: The Big-Slice Theorem.

March 15. Lecture 15: Uhlenbeck's Fundamental Lemma.

March 18. Lecture 16: Uhlenbeck's Compactness Theorem I.

March 20. Lecture 17: Uhlenbeck's Compactness Theorem II.

March 22. Lecture 18: The Structure of the Moduli Space.

March 25. Spring Break.

March 27. Spring Break.

March 29. Spring Break.

April 1. Lecture 19: Donaldson's Diagonalization Theorem.

April 3. Lecture 20: The Topology of Configuration Spaces I.

April 5. Lecture 21: The Topology of Configuration Spaces II.

April 8. Lecture 22: Orienting the Moduli Space.

April 10. Lecture 23: The Convention of Orientations.

April 12. Lecture 24: Evaluation on the Fundamental Class.

April 15. Patriots' Day.

April 17. Cancelled.

April 19. Lecture 25: Donaldson's Polynomial Invariants.

April 22. Lecture 26: The ASD Equation on Cylinders.

April 24. Lecture 27: The Spectrum Flow and the Index.

April 26. Lecture 28: 4-Manifolds with Cylindrical Ends

April 29. Lecture 29: Instanton Floer Homology

May 1. Lecture 30: Functoriality I

May 3. Lecture 31: Functoriality II

May 6. Lecture 32: Flat Connections on Surfaces.

May 8. Lecture 33: Symplectic Reductions.

May 10. Lecture 34: The Yang-Mills Equation over Riemann Surfaces.

May 13. Lecture 35: Orbifolds.

May 15. Lecture 36: Classification of SO(3) Bundles Over Orbifolds.