The Graduate Student Topology Seminar meets in Math 528, at 2:30 pm on Tuesdays unless noted otherwise (in red).

Plan The plan is for the talks to cover a wider range of topics, including symplectic topology, Heegaard Floer theory and other facets of low-dimensional topology. Ideally, the talks should be accessible to all topology students, including first- and second-years.

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Schedule The following schedule is tentative and may be modified as the semester progresses.Date | Speaker | Title | References |
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Sep 5 (Fri) 1:30 pm |
Organizational Meeting |
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Sep 9 | Mike Wong | Morse theory I | Hutchings: Lecture notes on Morse homology Schwarz: Morse Homology |

Sep 15 (Mon) 2:30 pm |
Kristen Hendricks (UCLA) Math 622 |
A spectral sequence for fixed point Floer cohomology | Hendricks: A spectral sequence for the Floer cohomology of symplectomorphisms of trivial polarization class |

Sep 23 | Mike Wong | Morse theory II | Hutchings: Lecture notes on Morse homology Schwarz: Morse Homology |

Sep 30 | Robert Castellano | Other structures of Morse theory | Fukaya: Morse homotopy, A-infinity category, and Floer homologies Cieliebak, Frauenfelder: A Floer homology for exact contact embeddings Schwarz: On the action spectrum for closed symplectically aspherical manifolds |

Oct 7 | Sara Venkatesh | Introduction to Floer homology | Audin, Damian: Morse Theory and Floer Homology Salamon: Lectures on Floer homology |

Oct 14 | James Cornish | Introduction to characteristic classes | Milnor, Stasheff: Characteristic Classes |

Oct 21 | Jingyu Zhao | Kirby calculus I | Gompf, Stipsicz: 4-Manifolds and Kirby Calculus |

Oct 28 | Zhechi Cheng | Kirby calculus II | Gompf, Stipsicz: 4-Manifolds and Kirby Calculus Scorpion: The Wild World of 4-Manifolds |

Nov 4 | Election Day – University Holiday |
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Nov 12 (Wed) 3:00 pm |
Sara Venkatesh | The Maslov index | Audin, Damian: Morse Theory and Floer Homology McDuff, Salamon: Introduction to Symplectic Topology |

Nov 18 | Zhechi Cheng | Introduction to Heegaard Floer homology | Ozsváth, Szabó: Holomorphic disks and topological invariants for closed three-manifolds Lipshitz: A cylindrical reformulation of Heegaard Floer homology Auroux: A beginner's introduction to Fukaya category |

Nov 25 | Mike Wong | Knot Floer homology | Manolescu, Ozsváth, Sarkar: A combinatorial description of knot Floer homology Manolescu, Ozsváth, Szabó, Thurston: On combinatorial link Floer homology |

Dec 2 | Paul Siegel | Exotic 4-manifolds via spectral theory | Atiyah, Patodi, Singer: Spectral asymmetry and Riemannian geometry Stolz: Exotic structures on 4-manifolds detected by spectral invariants |

## Abstracts

Date Sep 9

Speaker Mike Wong

Title Morse theory I

Abstract We will review the basic concepts in Morse theory in the modern approach.

Date Sep 15

Speaker Kristen Hendricks

Title A spectral sequence for fixed point Floer cohomology

Abstract If M is an exact symplectic manifold with stably trivial tangent bundle, then a symplectomorphism of M induces a map from M to the infinite symplectic group via the induced map on the tangent bundle. We show that if this map is nullhomotopic (and other common technical requirements are satisfied), Seidel and Smith's localization theory for Floer cohomology implies the existence of a spectral sequence from the Floer cohomology of the square of the symplectomorphism to the Floer cohomology of the symplectomorphism itself, and a corresponding rank inequality.

Date Sep 23

Speaker Mike Wong

Title Morse theory II

Abstract We will review the technical difficulties encountered in defining Morse homology, and briefly mention their counterparts in Floer homology theories.

Date Sep 30

Speaker Robert Castellano

Title Other structures of Morse theory

Abstract You have now seen the definition of Morse homology, but that is just the tip of the iceberg! I will show how familiar (and perhaps not familiar) structures in singular homology can be recovered Morse theoretically. I will also describe invariants that are new in Morse homology. Finally, I will introduce variants of Morse homology that are useful in some circumstances. These are all, of course, discussed with an eye towards Floer theory, but no knowledge of Floer theory is necessary. Potential topics include: Morse-Bott homology, spectral invariants, A-infinity structures, spectral sequences.

Date Oct 7

Speaker Sara Venkatesh

Title Introduction to Floer homology

Abstract Morse homology serves as a prototype for Floer homology. With Morse homology our inspiration, we construct the Floer homology of a Hamiltonian function on a symplectic manifold.

Date Oct 14

Speaker James Cornish

Title Introduction to characteristic classes

Abstract An introduction to the Stiefel–Whitney, Euler, Chern and Pontryagin classes, as well as some uses for these characteristic classes.

Date Oct 21

Speaker Jingyu Zhao

Title Kirby calculus I

Abstract I will talk about handle decompositions and their applications in 3- and 4-dimensional topology, such as Heegaard splittings and Kirby diagrams.

Date Oct 28

Speaker Zhechi Cheng

Title Kirby calculus II

Abstract Last time, we saw that any 4-manifold can be represented by a Kirby diagram. This week, I will continue the subject and talk more about the framing of a 2-handle and Kirby moves, and show how to manipulate them. We will study applications to both 3- and 4-manifolds.

Date Nov 12

Speaker Sara Venkatesh

Title The Maslov index

Abstract We define the Maslov index and discuss its application to Hamiltonian Floer homology.

Date Nov 18

Speaker Zhechi Cheng

Title Introduction to Heegaard Floer homology

Abstract I will give a very gentle introduction to Lagrangian intersection Floer homology and use it as a prototype to build Heegaard Floer homology of closed 3-manifolds. I will show how this construction is analogous to Morse homology. I will say something about its applications and calculations if time permits.

Date Nov 25

Speaker Mike Wong

Title Knot Floer homology

Abstract We will define the grid homology of a link, and relate it to the knot Floer homology, defined using Heegaard diagrams. If time permits, we will discuss the many flavors of knot Floer homology.

Date Dec 2

Speaker Paul Siegel

Title Exotic 4-manifolds via spectral theory

Abstract The famous Atiyah–Singer index theorem computes geometric and topological invariants using analysis of differential operators on manifolds. There are unexpected subtleties involved in generalizing this theorem to manifolds with boundary which are captured by Atiyah–Patodi–Singer's η-invariant, an invariant built out of the spectrum of a differential operator. In this talk I will review the theory of η-invariants and discuss a theorem due to Stolz which uses eta invariants to detect exotic structures on ℝP^4.