Enumerative Geometry Seminar (Fall 2018)
The goal of this seminar is to understand the celebrated GromovWitten/DonaldsonThomas correspondence.
 Organizers: Henry Liu, Melissa ChiuChu Liu
 Time/date: Tuesdays 2:403:55pm and Wednesdays 1:102:25pm
 Location: Math 507 (Tues) and Math 622 (Weds)
Talks on the GromovWitten side will generally be on Tuesdays. Talks on the DonaldsonThomas side will generally be on Wednesdays.
I am liveTeXing notes for this seminar, available here.
Please email me at hliu at math dot columbia dot edu if you would like to be on the mailing list.
Plan/References
The plan is to go through the MNOP and MOOP papers, which state and prove (primary, descendant) GW/DT correspondences for (toric, CY) 3folds.
 [MNOP1] GromovWitten theory and DonaldsonThomas theory, I
 [MNOP2] GromovWitten theory and DonaldsonThomas theory, II
 [MOOP] GromovWitten/DonaldsonThomas correspondence for toric 3folds
For the GW side, we need the GW vertex, which involves Hodge integrals. The 1leg and 2leg cases are in the following papers.
 [LLZ1] A Proof of a Conjecture of MariñoVafa on Hodge integrals
 [OP1] Hodge integrals and invariants of the unknot
 [LLZ2] A Formula of TwoPartition Hodge Integrals
To do GW/DT for arbitrary toric 3folds, not just CY3s, we need the theory for local curves and \(A_n\) resolutions.
 [BP] The local GromovWitten theory of curves
 [OP2] The local DonaldsonThomas theory of curves
 [M] GromovWitten theory of \(A_n\)resolutions
 [MO] DonaldsonThomas theory of \(A_n \times \mathbb{P}^1\)
We may go through additional related topics/papers as time permits.
Schedule
Tues Sept 11 
Melissa Liu Hurwitz numbers and the ELSV formula The ELSV formula, first proved by Ekedahl, Lando, Shapiro, and Vainshtein, relates Hurwitz numbers to Hodge integrals. In this talk, we explain what the ELSV formula is and how to prove it by virtual localization on moduli of relative stable maps to the projective line relative a point, following GraberVakil. Reference: Lectures on the ELSV formula 
Weds Sept 12 
Clara Dolfen GW/DT for local CY toric surfaces In their paper "GromovWitten theory and Donaldson Thomas theory I" (MNOP1), Maulik et al. conjecture a correspondence between the generating functions of GromovWitten and DonaldsonThomas invariants for 3folds, and prove it in the case of local CalabiYau toric surfaces. In this talk, we will give a brief introduction to DT theory, and talk about the fixed points of the DT moduli space under the torus action. We will use this insight to compute the DT counts via virtual localization following MNOP1. Reference: [MNOP1] 
Tues Sept 18 
Melissa Liu ELSV formula via relative virtual localization This is a sequel of my talk on September 11. In my talk on September 11, we defined Hodge integrals and simple Hurwitz numbers, and stated the ELSV formula (first proved by EkedahlLandoShapiroVainshtein) expressing simple Hurwitz numbers in terms of Hodge integrals. We then interpreted each simple Hurwitz number as the degree of a morphism from certain moduli of relative stable maps to a projective space. In this talk, we prove the ELSV formula by computing this degree via relative virtual localization. Reference: Lectures on the ELSV formula 
Weds Sept 19 
Clara Dolfen DT for local CY toric surfaces Last week we set up the machinery needed to compute the DT invariants for toric CY 3folds via virtual localization. In this talk, we will briefly recall the weight decomposition of the virtual tangent space and use it to derive an explicit formula for the DT counts in the case of local CY toric surfaces. Reference: [MNOP1] 
Tues Sept 25 
Melissa Liu GW/DT correspondence for the resolved conifold The resolved conifold is the total space of \(\mathcal{O}(1)\oplus \mathcal{O}(1)\) over \(\mathbb{P}^1\). It is a toric CalabiYau threefold whose GW/DT invariants can be interpreted as local GW/DT invariants of a superrigid smoothly embedded rational curve in a CalabiYau threefold. We will explain GW/DT correspondence for the resolved conifold, and its relation with the GopakumarVafa conjecture. References:

Weds Sept 26 
Ivan Danilenko GW/DT in relative setting and with descendants This time we follow "GromovWitten theory and Donaldson Thomas theory II" (MNOP2), Maulik et al. The main goal of the talk is to introduce DT invariants with insertions and relative DT invariants. We will state (partly conjectural) relations with their GW counterparts. Time permitting, we'll show how to compute zero dimensional contributions in DT theory by localization. References: [MNOP2] 
Tues Oct 02 
Melissa Liu GW/DT correspondence for the resolved conifold II This is a sequel of my talk on September 25. References:

Weds Oct 03 
Ivan Danilenko GW/DT in relative setting and with descendants II We'll define the DT rubber theory and use it to compute the equivariant vertex in the relative setting. References: [MNOP2] 
Tues Oct 09 
Melissa Liu Introduction to Relative GromovWitten Theory References:

Weds Oct 10 
Ivan Danilenko GW/DT in relative setting and with descendants III We'll define the DT rubber theory and use it to compute the equivariant vertex in the relative setting. References: [MNOP2], [OP2] 
Tues Oct 16 
Henry Liu The GW local curves TQFT We'll define the 2d TQFT associated to the GW partition function for local curves. If time permits we'll prove its semisimplicity. References: [BP]

Weds Oct 17 
Anton Osinenko Local DT theory of curves The local DonaldsonThomas theory of curves is solved by localization and degeneration methods. The results complete a triangle of equivalences relating GromovWitten theory, DonaldsonThomas theory, and the quantum cohomology of the Hilbert scheme of points of the plane. The quantum differential equation of the Hilbert scheme of points of the plane has a natural interpretation in the local DonaldsonThomas theory of curves. The solution determines the 1legged equivariant vertex. References: [OP2] 
Tues Oct 23 
Henry Liu Local curve computations We'll prove the GW local curves TQFT is semisimple and compute its pieces: tube, caps, and pair of pants. References: [BP]

Weds Oct 24 
Shuai Wang Local DT theory of curves The local DonaldsonThomas theory of curves is solved by localization and degeneration methods. The results complete a triangle of equivalences relating GromovWitten theory, DonaldsonThomas theory, and the quantum cohomology of the Hilbert scheme of points of the plane. The quantum differential equation of the Hilbert scheme of points of the plane has a natural interpretation in the local DonaldsonThomas theory of curves. The solution determines the 1legged equivariant vertex. References: [OP2] 
Tues Oct 30 
Henry Liu Cap and pants We'll first finish computing the level (1,0) cap. Then we'll finish computing the whole local curves TQFT by computing the level (0,0) pair of pants explicitly in one (simple) case, and then in general via a reconstruction result. References: [BP]

Weds Oct 31 
Shuai Wang Local DT theory of curves We will compute the cap and pair of pants. References: [OP2] 
Tues Nov 06  No meeting (election day) 
Weds Nov 07  No meeting (we're taking a break!) 
Tues Nov 13 
Melissa Liu The 1leg GromovWitten vertex References: [LLZ1], [OP1] 
Weds Nov 14 
Yakov Kononov Relative DT theory of \(A_n \times \mathbb{P}^1\) The talk will be devoted to the relative DonaldsonThomas theory of \(A_n \times \mathbb{P}^1\). I will express the action of divisor operators in terms of the action of \(\widehat{\mathfrak{gl}}(n+1)\) on the Fock space. References: [MO] 
Tues Nov 20 
Melissa Liu The 2leg GromovWitten vertex References:

Weds Nov 21  No meeting (Thanksgiving) 
Tues Nov 27 
Melissa Liu A mathematical theory of the topological vertex References:

Weds Nov 28 
Andrei Okounkov Toric GW/DT correspondence I will describe the full GW/DT correspondence for toric 3folds using capped vertices and then answer all your questions. References: [MOOP] 
Tues Dec 04 
Henry Liu Capping and quantum differential equation We'll see in detail the derivation of the QDE for the capping operator, using rubber calculus. Then we'll match the operator \(M_D\) on the GW/DT sides and use it to match capped edges. References: [MOOP], [OP2] 
Weds Dec 05 
Henry Liu Capping and quantum differential equation Continuation of Tuesday. References: [MOOP], [OP2] 