Related papers: The GW/PT conjectures for toric pairs

We provide a direct proof of a conjecture of Brini relating the Gromov-Witten theory of the resolved conifold to the Ablowitz-Ladik integrable hierarchy at the level of primaries. In doing so, we use a functional representation of the…

We prove the existence and give a classification of toric non-commutative crepant resolutions (NCCRs) of Gorenstein toric singularities with divisor class group of rank one. We prove that they correspond bijectively to non-trivial upper…

We establish a genus zero correspondence between the equivariant Gromov-Witten theory of the Deligne-Mumford stack $[\mathbb{C}^N/G]$ and its blowup at the origin. The relationship generalizes the crepant transformation conjecture of…

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…

We prove that a smooth rationally connected projective threefold of Picard number two is toric if and only if it admits an int-amplified endomorphism. As a corollary, we show that a totally invariant smooth curve of a non-isomorphic…

We prove the transformation formula of Donaldson-Thomas (DT) invariants counting two dimensional torsion sheaves on Calabi-Yau 3-folds under flops. The error term is described by the Dedekind eta function and the Jacobi theta function, and…

The logarithmic Kazhdan-Lusztig correspondence is a conjectural equivalence between braided tensor categories of representations of small quantum groups and representations of certain vertex operator algebras. In this article we prove such…

We verify the Morrison--Kawamata conjecture for a certain class of rational threefolds, namely blowups of P^3 in the base locus of a net of quadrics with no reducible members. This seems to be the first verified case of the conjecture for…

It was pointed out by Eliashberg in his ICM 2006 plenary talk that the integrable systems of rational Gromov-Witten theory very naturally appear in the rich algebraic formalism of symplectic field theory (SFT). Carefully generalizing the…

We conjecture the existence of almost integer invariants governing the all-genus equivariant Gromov-Witten theory of Calabi-Yau fivefolds with a torus action. We prove the conjecture for skeletal, locally anti-diagonal torus actions by…

We consider four approaches to relative Gromov-Witten theory and Gromov-Witten theory of degenerations: Jun Li's original approach, Bumsig Kim's logarithmic expansions, Abramovich-Fantechi's orbifold expansions, and a logarithmic theory…

Consider a log Calabi-Yau pair $(X,D)$ consisting of a smooth del Pezzo surface $X$ of degree $\geq 3$ and a smooth anticanonical divisor $D$. We prove a correspondence between genus zero logarithmic Gromov-Witten invariants of $X$…

The probability that a zero of a random real polynomial of increasing degree is real tends to zero. However, passing from polynomials to Laurent polynomials yields a surprising result: the probability that a root is real tends not to zero,…

We prove a conjecture of Goncharov, which says that any multiple polylogarithm can be expressed via polylogarithms of depth at most half of the weight. We give an explicit formula for this presentation, involving a summation over trees that…

The generalized Donaldson-Thomas invariants counting one dimensional semistable sheaves on Calabi-Yau 3-folds are conjectured to satisfy a certain multiple cover formula. This conjecture is equivalent to Pandharipande-Thomas's strong…

In this exposition we give a simple and complete treatment of A. Knutson and T. Tao's recent proof (http://front.math.ucdavis.edu/math.RT/9807160) of the saturation conjecture, which asserts that the Littlewood-Richardson semigroup is…

A well-known conjecture of Orlov asks whether the existence of a full exceptional collection implies rationality of the underlying variety. We prove this conjecture for arithmetic toric varieties over general fields. We also investigate a…

We develop some ideas of Morrison and Plesser and formulate a precise mathematical conjecture which has close relations to toric mirror symmetry. Our conjecture, we call it Toric Residue Mirror Conjecture, claims that the generating…

We develop a contour integral formalism for computing the K-theoretic equivariant 3-vertex. Within the Jeffrey--Kirwan (JK) residue framework, we show that, by an appropriate choice of the reference vector, both the equivariant…

We solve several open problems concerning integer points of polytopes arising in symplectic and algebraic geometry. In this direction we give the first proof of a broad case of Ewald's Conjecture (1988) concerning symmetric integral points…