Henry Liu

Enumerative Geometry Seminar (Fall 2018)

The goal of this seminar is to understand the celebrated Gromov-Witten/Donaldson-Thomas correspondence.

Talks on the Gromov-Witten side will generally be on Tuesdays. Talks on the Donaldson-Thomas side will generally be on Wednesdays.

I am live-TeXing 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) 3-folds.

For the GW side, we need the GW vertex, which involves Hodge integrals. The 1-leg and 2-leg cases are in the following papers.

To do GW/DT for arbitrary toric 3-folds, not just CY3s, we need the theory for local curves and \(A_n\) resolutions.

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 Graber-Vakil.

Reference: Lectures on the ELSV formula
Weds Sept 12 Clara Dolfen
GW/DT for local CY toric surfaces

In their paper "Gromov-Witten theory and Donaldson Thomas theory I" (MNOP1), Maulik et al. conjecture a correspondence between the generating functions of Gromov-Witten and Donaldson-Thomas invariants for 3-folds, and prove it in the case of local Calabi-Yau 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 Ekedahl-Lando-Shapiro-Vainshtein) 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 3-folds 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 Calabi-Yau threefold whose GW/DT invariants can be interpreted as local GW/DT invariants of a super-rigid smoothly embedded rational curve in a Calabi-Yau threefold. We will explain GW/DT correspondence for the resolved conifold, and its relation with the Gopakumar-Vafa conjecture.

References:
Weds Sept 26 Ivan Danilenko
GW/DT in relative setting and with descendants

This time we follow "Gromov-Witten 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 Gromov-Witten 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 Donaldson-Thomas theory of curves is solved by localization and degeneration methods. The results complete a triangle of equivalences relating Gromov-Witten theory, Donaldson-Thomas 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 Donaldson-Thomas theory of curves. The solution determines the 1-legged equivariant vertex.

References: [OP2]
Tues Oct 23 Henry Liu
Title: TBA

Abstract: TBA

References: [BP]
Weds Oct 24 Shuai Wang
Title: TBA

Abstract: TBA

References: [OP2]
Tues Oct 30 Speaker: TBA
Title: TBA

Abstract: TBA

References: TBA
Weds Oct 31 Shuai Wang
Title: TBA

Abstract: TBA

References: TBA