Microseminar 1:
Topology of Milnor Fibrations
Description: In this student seminar we're going to understand the symplectic topology of isolated singularities. We'll first go through the first ten chapters of [Mil68] to learn about the differential topological interpretation of isolated singularties, introduce the notion of Milnor fibration, and give a complete description of the topology of the Milnor fiber(i.e. a combinatorial formula for the homotopy type of the Milnor fiber). Then if time permits, we would move into the second part of this seminar: the topology of isolated singularties. The reference of the second part would be Looijenga's book [Loo84], which applies Milnor's theory to further study properties of algebraic singularties. Detailed schedule goes as follows:
Schedule
| Time | Contents | Speakers | References |
| May 24th | Introduction and some Algebraic Geometry | Siyang Liu | [Mil68], Chapter 1 and 2 |
| May 31st | The curve selection lemma and the fibration theorem | Shuhao Li | [Mil68], chapter 3 and 4 |
| June 7th | Topology of Milnor fibration and the special case of isolated singularties | Dashen Yan | [Mil68], chapter 5 and 6 |
| June 14th | More about topology of Milnor fibrations and Brieskorn varieties | Siyang Liu | [Mil68], chapter 7-9 |
| June 19th - June 23rd | Kylerrec 2022 |
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| June 28th | isolated singularities and Milnor fibrations revisited | Haosen Wu | [Loo84], chapter 1 and 2 |
| July 5th | Picard-Lefschetz Theory and Critical Spaces | Jiawei Hu | [Loo84], chapter 3-4 |
| July 11th - July 22nd | Séminaire de Mathématiques Supérieures 2022: Floer Homotopy Theory |
| July 26th | Monodromy and Deformation theory | Haosen Wu | [Loo84], chapter 5-6 |
| Aug. 2nd | Gauss-Mannin Connection, Ⅰ | Siyang Liu | [Loo84], chapter 7-9 |
| Aug. 8th - Aug. 12th | SYNC Early Career Workshop |
| Aug. 16th | Gauss-Mannin Connection, Ⅱ | Shengzhen Ning | [Loo84], chapter 7-9 |
References
- [Mil68] Milnor, J. (1968). Singular Points of Complex Hypersurfaces. Princeton University Press.
- [Loo84] Looijenga, E. J. N. (1984). Isolated Singular Points on Complete Intersections. Cambridge University Press. doi: 10.1017/CBO9780511662720.
- [Ust99] Ustilovsky, I. (1999). Infinitely many contact structures on S4m + 1. Int. Math. Res. Not., 1999(14), 781–791. doi: 10.1155/S1073792899000392
- [Huy05] Huybrechts, D. (2005). Complex Geometry. Springer. doi: 10.1007/b137952
- [Oht05] Ohta, H., & Ono, K. (2005). Simple Singularities and Symplectic Fillings. J. Differential Geom., 69(1), 001–042. doi: 10.4310/jdg/1121540338
- [Kea15] Keating, A. (2015). Homological mirror symmetry for hypersurface cusp singularities. arXiv, 1510.08911. Retrieved from https://arxiv.org/abs/1510.08911v2
- [KwK16] Kwon, M., & van Koert, O. (2016). Brieskorn manifolds in contact topology. Bull. London Math. Soc., 48(2), 173–241. doi: 10.1112/blms/bdv088