CIRGET Low dimensional topology learning seminar

For autumn 2015 semester, we will meet Thursdays 1515 - 1630 in PK 5115, which is the main seminar room next to the lifts. Our goal is to introduce Heegaard Floer homology for 3-manifolds, knots and links, and to understand some new invariants that relate to knot and link concordance such as the Upsilon and nu invariants, as well as some older creatures such as tau and d.

heegaardmoves hdtrefoil

Talks

Here is what happened.

Thursday 24th September: Mark Powell, introduction to knot concordance, the Seifert form, the Alexander polynomial and Levine-Tristram signatures.

Thursday 1st October: Nima Hoda, Heegaard splittings and Heegaard diagrams of 3-manifolds.

Thursday 8th October: Nima continued, Heegaard diagrams of knot complements, followed by Matthias Nagel, the symmetric product.

Thursday 15th October: Matthias Nagel, lagrangian tori, Whitney discs and the definition of Floer chain complexes.

Thursday 22nd and Thursday 29th October: Clovis Kari, Spin-c structures on 3-manifolds and their relation to Heegaard Floer groups.

Thursday November 5th: Ying Hu, the Maslov index, understanding discs in the symmetric product in terms of domains on the Heegaard diagram, computing some basic examples.

Thursday November 12th: Matthias Nagel, knot Floer homology, Alexander grading, filtered chain complexes with U action, examples, definition of tau.

Thursday November 19th: Liam Watson, definition of Upsilon, some properties.

Thursday November 26th: Mark Powell, Absolute grading of Gripp and Huang via homotopy classes of vector fields.

Thursday December 3rd: Matthias Nagel, Surgery exact triangle and maps induced by cobordisms.

FRIDAY December 4th, Guest lecture CIRGET seminar by Adam Levine. Satellite operators and piecewise linear concordance.

Thursday December 10th: Ying Hu, Absolute rational grading and d-invariants.

References

Clay Lectures on Heegaard Floer homology: Part 1 and Part 2, Peter Ozsvath, Zoltan Szabo.

Lectures from Park City, page 209 of this.

Introduction to the basics of Heegaard Floer homology, Bijan Sahamie.

Survey article, a leisurely introduction, apparently: Heegaard diagrams and holomorphic disks, Peter Ozsvath, Zoltan Szabo.

Background on Floer theory and how this led to Heegaard Floer: Floer theory and low dimensional topology, Dusa McDuff.

ICM proceedings paper: Heegaard diagrams and Floer homology, Peter Ozsvath, Zoltan Szabo.

Original Annals reference 1: Holomorphic disks and topological invariants for closed three-manifolds, Peter Ozsvath and Zoltan Szabo.

Original Annals reference 2: Holomorphic disks and three-manifold invariants: properties and applications, Peter Ozsvath, Zoltan Szabo.

Not by Ozsvath and Szabo: A survey of Heegaard Floer homology, Andras Juhasz.

Understanding domains: A cylindrical reformulation of Heegaard Floer homology, Robert Lipshitz.

Understanding domains: An algorithm for computing some Heegaard Floer homologies, Sucharit Sarkar, Jiajun Wang.

Undertanding gradings: An absolute grading on Heegaard Floer homology by homotopy classes of oriented 2-plane vector fields , Vinicius Gripp and Yang Huang.

Knot Floer homology: Holomorphic disks and knot invariants, Peter Ozsvath, Zoltan Szabo.

d invariants and 4-manifolds: Absolutely Graded Floer homologies and intersection forms for four-manifolds with boundary , Peter Ozsvath, Zoltan Szabo.

All the formal properties in one place: The geography and botany of Heegaard Floer homology, Matthew Hedden, Liam Watson.

The tau invariant: Knot Floer homology and the four-ball genus, Peter Ozsvath, Zoltan Szabo.

First paper on Upsilon: Concordance homomorphisms from knot Floer homology, Peter Ozsvath, Andras Stipsicz, Zoltan Szabo

A readable account: Notes on the concordance invariant Upsilon, Charles Livingston.

Upsilon used to compute four ball genera:The four-genus of connected sums of torus knots, Charles Livingston, Cornelia Van Cott.

Two papers on Upsilon by Shida Wang: The genus filtration in the smooth concordance group and On the First Singularity for the Upsilon Invariant of Algebraic Knots.

An application of Upsilon to unoriented surfaces: Unoriented knot Floer homology and the unoriented four-ball genus, Peter Ozsvath, Andras Stipsicz, Zoltan Szabo.

The epsilon invariant: The knot Floer complex and the smooth concordance group, Jennifer Hom.

Another new concordance invariant: Four-ball genus bounds and a refinement of the Ozsvath-Szabo tau-invariant, Jennifer Hom, Zhongtao Wu.

An interesting recent paper: Concordance maps in knot Floer homology, Andras Juhasz, Marco Marengon.