Summer School in Gromov-Witten Theory 2014
Pingree Park, Colorado
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.
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.
Genus-zero quantum invariants of chain polynomials : how to overcome non-concavity and matrix factorizations
On the crepant transformation conjecture for toric birational transformations.
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.
Frobenius manifolds and symmetries.
Comparing Gromov-Witten and stable quasimap invariants
Calabi-Yau 3-folds containing Enriques
surfaces and degenerations to a strange family
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.
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.
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.
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
The orbifold topological vertex