Refined floor diagrams from higher genera and lambda classes

We show that, after the change of variables $q=e^{iu}$, refined floor diagrams for $\mathbb{P}^2$ and Hirzebruch surfaces compute generating series of higher genus relative Gromov-Witten invariants with insertion of a lambda class. The proof uses an inductive application of the degeneration formula in relative Gromov-Witten theory and an explicit result in relative Gromov-Witten theory of $\mathbb{P}^1$. Combining this result with the similar looking refined tropical correspondence theorem for log Gromov-Witten invariants, we obtain some non-trivial relation between relative and log Gromov-Witten invariants for $\mathbb{P}^2$ and Hirzebruch surfaces. We also prove that the Block-G\"ottsche invariants of $\mathbb{F}_0$ and $\mathbb{F}_2$ are related by the Abramovich-Bertram formula.Comment: 44 pages, 8 figures, revised version, exposition greatly improved, main results unchanged, published in Selecta Mathematic

The quantum tropical vertex

Gross-Pandharipande-Siebert have shown that the 2-dimensional Kontsevich-Soibelman scattering diagrams compute certain genus zero log Gromov-Witten invariants of log Calabi-Yau surfaces. We show that the $q$-refined 2-dimensional Kontsevich-Soibelman scattering diagrams compute, after the change of variables $q=e^{i \hbar}$, generating series of certain higher genus log Gromov-Witten invariants of log Calabi-Yau surfaces. This result provides a mathematically rigorous realization of the physical derivation of the refined wall-crossing formula from topological string theory proposed by Cecotti-Vafa, and in particular can be seen as a non-trivial mathematical check of the connection suggested by Witten between higher genus open A-model and Chern-Simons theory. We also prove some new BPS integrality results and propose some other BPS integrality conjectures.Comment: v2: 68 pages, revised version (minor mistake in Section 5 corrected), published in Geometry and Topolog

On an example of quiver DT/relative GW correspondence

We explain and generalize a recent result of Reineke-Weist by showing how to reduce it to the Gromov-Witten/Kronecker correspondence by a degeneration and blow-up. We also refine the result by working with all genera on the Gromov-Witten side and with refined Donaldson-Thomas invariants on the quiver side.Comment: 37 pages, comments welcom

Quivers and curves in higher dimension

We prove a correspondence between Donaldson-Thomas invariants of quivers with potential having trivial attractor invariants and genus zero punctured Gromov-Witten invariants of holomorphic symplectic cluster varieties. The proof relies on the comparison of the stability scattering diagram, describing the wall-crossing behavior of Donaldson-Thomas invariants, with a scattering diagram capturing punctured Gromov-Witten invariants via tropical geometry.Comment: 38 pages, 3 figures. Comments welcome

Fock-Goncharov dual cluster varieties and Gross-Siebert mirrors

Cluster varieties come in pairs: for any $\mathcal{X}$ cluster variety there is an associated Fock-Goncharov dual $\mathcal{A}$ cluster variety. On the other hand, in the context of mirror symmetry, associated with any log Calabi-Yau variety is its mirror dual, which can be constructed using the enumerative geometry of rational curves in the framework of the Gross-Siebert program. In this paper we bridge the theory of cluster varieties with the algebro-geometric framework of Gross-Siebert mirror symmetry. Particularly, we show that the mirror to the $\mathcal{X}$ cluster variety is a degeneration of the Fock-Goncharov dual $\mathcal{A}$ cluster variety and vice versa. To do this, we investigate how the cluster scattering diagram of Gross-Hacking-Keel-Kontsevich compares with the canonical scattering diagram defined by Gross-Siebert to construct mirror duals in arbitrary dimensions. Consequently, we derive an enumerative interpretation of the cluster scattering diagram. Along the way, we prove the Frobenius structure conjecture for a class of log Calabi-Yau varieties obtained as blow-ups of toric varieties.Comment: 51 pages, revised version published in Journal f\"ur die reine und angewandte Mathematik (Crelles Journal

Contributions à la résolution de problèmes à une machine avec fonctions temporelles de type exponentiel

Notre travail s’inscrit dans une classe particulière des problèmes d’ordonnancement : les problèmes à une machine dans lesquels les durées d’exécution des tâches ne sont plus des constantes mais dépendent du temps. La complexité de ces problèmes dépend de la nature de la fonction temporelle modélisant la durée des tâches. Nous étudions le cas particulier des fonctions exponentielles. Pour un type particulier de fonctions exponentielles, nous démontrons que des problèmes deviennent polynomiaux sous certaines conditions.