Summer School in Gromov-Witten Theory 2014

Pingree Park, Colorado
June 23 - July 4, 2014





Beer Talks

Getting There

Confirmed Participants



Huai-Liang Chang

Landau Ginzburg type theories from algebraic geometry


The Landau Ginzburg model unifies differerent moduli spaces and their counting in A side of mirror symmetry.  Algebro geometric approach to define it via p field and cosection localization unifies these different theories, as like Gromov Witten theory, FJRW theory, and also others.  We will brief on their constructions, comparisons, and also relations which is work under progress.



Bohan Fang

Eynard-Orantin recursion and all genus mirror symmetry of a projective line

The mirror of a toric variety is a Landau-Ginzburg model. In case this mirror could be reduced to an affine curve, one could run the Eynard-Orantin recursion and obtain higher genus invariants, which should predict GW invariants of the original toric variety. When the toric variety is a CY 3-fold, this is the BKMP conjecture. I will illustrate this phenomenon through a more basic example -- the equivariant projective line.


Jeremy Guere

Genus-zero quantum invariants of chain polynomials : how to overcome non-concavity and matrix factorizations


Yunfeng Jiang

On the crepant transformation conjecture for toric birational transformations.


Atsushi Kanazawa

Trilinear forms and Chern classes of Calabi--Yau threefolds


I will talk about the interplay of the intersection trilinear forms and Chern classes of a Calabi-Yau threefold. A natural question is, what kind of trilinear forms, 0-th approximation of GW invariants, occur on a Calabi-Yau threefold? This question is also related to topology of Calabi-Yau threefolds due to Wall's structure theorem on real 6-folds. I will provide new formulae that hold for an arbitrary compact Calabi-Yau threefold. This is a joint work with P.H.M. Wilson. 


Byenongho Lee

Frobenius manifolds and symmetries.


Cristina Manolache

Comparing Gromov-Witten and stable quasimap invariants


Howard Nuer

Calabi-Yau 3-folds containing Enriques surfaces and degenerations to a strange family

We give several constructions of Calabi-Yau 3-folds containing Enriques surfaces.  For each family we describe several birational models, and for one of these families we can identify all of its minimal models, exhibiting interesting behavior from the point of view of MMP.  Using an alternative description of this family, we find two degenerations which admit crepant resolutions.  One of these degenerations is a 1-parameter family of Calabi-Yau 3-folds that has very strange properties from the point of view of mirror symmetry.  We discuss these strange features, in particular predictions of vanishing odd-degree GW invariants for the mirror and strange chern invariants, as well as some possible explanations for their occurrence.”


Andrea Petracci

Quantum periods of del Pezzo surfaces with 1/3(1,1) singularities


Recent work of Coates-Corti-Galkin-Kasprzyk uses quantum cohomology to reproduce the Iskovskikh-Mori-Mukai classification of smooth Fano 3-folds. The central idea is that the quantum period of a smooth Fano 3-fold corresponds to the classical period of certain Laurent polynomials supported on 3-dimensional reflexive polytopes. It is conjectured that a similar correspondence holds between del Pezzo surfaces with isolated quotient singularities and a certain class of Laurent polynomials supported on Fano polygons. In this talk I will show some examples of such a correspondence. This is joint work with Alessandro Oneto.


Yefeng Shen

Mirror symmetry for exceptional unimodular singularities


I would like to talk about the mirror theorem between the Saito- Givental theory of exceptional unimodular singularities on Landau-Ginzburg B-side and the Fan-Jarvis-Ruan-Witten theory of their BHK-mirror partners on Landau-Ginzburg A-side. On the B-side, we develop a perturbative method to compute the genus-zero correlation functions associated to the primitive forms. This is applied to the exceptional unimodular singularities, and we show that the numerical invariants match the orbifold-Grothendieck-Riemann- Roch and WDVV calculations in FJRW theory on the A-side. The coincidence of the full data at all genera is established by reconstruction techniques. This is joint work with Changzheng Li, Si Li and Kyoji Saito.


Amit Solomon
Topology of gradient flow near a critical point

Morse theory illustrates the intimate relationship between the critical

points of a smooth function on a manifold and the topology of the manifold:

Given a generic function with non-degenerate critical points, one can construct a

chain complex, known as the Morse complex, whose homology equals the singular

homology of the manifold.

Unfortunately, when the critical points of the function are degenerate, the Morse

complex construction fails miserably.  In recent work with J. Solomon, we make

the first step towards extending the Morse complex to a class of degenerate functions.

Namely, we endow the stable set of a degenerate critical point with a natural 

stratification generalizing the concept of the stable manifold.


Michel Van Garrel

Integrality of relative BPS state counts of toric Del Pezzo surfaces


Zhengyu Zong

The orbifold topological vertex

For smooth toric Calabi-Yau 3-folds, the Gromov-Witten theory is obtained by gluing the GW vertex, a generating function of cubic Hodge integrals, and the Donaldson-Thomas theory is obtained by gluing the DT vertex, a generating function of 3d partitions. The mathematical theory of GW vertex was first built by Li-Liu-Liu-Zhou and the GW/DT correspondence for topological vertex was proved for one-leg and two-leg cases. Later the GW/DT correspondence for smooth toric 3-fold was prove in MNOP.

For toric Calabi-Yau 3-orbifolds, the orbifold GW theory is obtained by gluing the GW orbifold vertex, a generating function of cubic abelian Hurwitz-Hodge integrals, and the orbifold DT theory is obtained by gluing the DT orbifold vertex, a generating function of colored 3d partitions. In this talk, I will talk about the computation of one-leg and two-leg orbifold GW vertex. In particular, this computation yields a proof of GW/DT correspondence for orbifold topological vertex in one-leg and two-leg cases. This work is joint with Dustin Ross.