• Publications
  • Influence
Canonical bases for cluster algebras
In [GHK11], Conjecture 0.6, the first three authors conjectured that the ring of regular functions on a natural class of affine log Calabi-Yau varieties (those with maximal boundary) has a canonicalExpand
Large Complex Structure Limits of K3 Surfaces
Motivated by the picture of mirror symmetry suggested by Strominger, Yau and Zaslow, we made a conjecture concerning the Gromov-Hausdorff limits of Calabi-Yau n-folds (with Ricci-flat K\"ahlerExpand
From real affine geometry to complex geometry
We construct from a real ane manifold with singularities (a tropical manifold) a degeneration of Calabi-Yau manifolds. This solves a fundamental problem in mirror symmetry. Furthermore, a strikingExpand
The tropical vertex
Elements of the tropical vertex group are formal families of symplectomorphisms of the 2-dimensional algebraic torus. We prove ordered product factorizations in the tropical vertex group areExpand
Logarithmic Gromov-Witten invariants
ing, now let π : Y → W be a proper morphism of schemes and let αi : Mi → OY , i = 1, 2, be two fine saturated log structures on Y . Consider the functor (2.1) LMorY/W (M1, M2) : (Sch/W ) −→ (Sets)Expand
Topological mirror symmetry
This paper focuses on a topological version on the Strominger-Yau-Zaslow mirror symmetry conjecture. Roughly put, the SYZ conjecture suggests that mirror pairs of Calabi-Yau manifolds are related byExpand
Mirror symmetry for log Calabi-Yau surfaces I
We give a canonical synthetic construction of the mirror family to pairs (Y,D) where Y is a smooth projective surface and D is an anti-canonical cycle of rational curves. This mirror family isExpand
Examples of Special Lagrangian Fibrations
  • M. Gross
  • Mathematics, Physics
  • 1 December 2000
We explore a number of examples of special Lagrangian fibrations on non-compact Calabi-Yau manifolds invariant under torus actions. These include fibrations on crepant resolutions of canonical toricExpand
Mirror Symmetry via Logarithmic Degeneration Data I
Introduction. 1 1. Derivations and differentials 6 2. Log Calabi-Yau spaces: local structure and deformation theory 16 2.1. Local structure 16 2.2. Deformation theory 25 3. Cohomology of logExpand
Dirichlet Branes and Mirror Symmetry
Research in string theory over the last several decades has yielded a rich interaction with algebraic geometry. In 1985, the introduction of Calabi-Yau manifolds into physics as a way to compactifyExpand