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We give a complete solution for the reduced Gromov-Witten theory of resolved surface singularities of type A_n, for any genus, with arbitrary descendent insertions. We also present a partial evaluation of the T-equivariant relative…
In this paper, we study relations among known universal equations for Gromov-Witten invariants at genus 1 and 2.
We prove the famous Faber intersection number conjecture and other more general results by using a recursion formula of $n$-point functions for intersection numbers on moduli spaces of curves. We also present some vanishing properties of…
We give a closed formula for the Conway function of a splice in terms of the Conway function of its splice components. As corollaries, we refine and generalize results of Seifert, Torres, and Sumners-Woods.
We give a closed formula of the Littlewood-Richardson coefficients.
In math.AG/0207233, Okounkov and Pandharipande gave an operator formalism for computing the equivariant Gromov-Witten theory of the projective line. This thesis extends their result to orbifold lines. In the effective case the theory is…
We establish the existence of a symmetry within the Gromov-Witten theory of $\mathbb{CP}^n$ and its blowup along points. The nature of this symmetry is encoded in the Cremona transform and its resolution, which lives on the toric variety of…
A bicyclic pair is a smooth surface equipped with a pair of smooth divisors intersecting in two reduced points. Resolutions of self-nodal curves constitute an important special case. We investigate the logarithmic Gromov-Witten theory of…
In this paper, we will give a new perspective to the Cameron-Martin-Maruyama-Girsanov formula by giving a totally algebraic proof to it. It is based on the exponentiation of the Malliavin-type differentiation and its adjointness.
Stirling's formula is a powerful asymptotic approximation of the factorial function. Many well-known proofs of this formula are grounded in integral calculus. In this paper, we present an alternative proof of Stirling's formula using only…
We study Gromov-Witten theory of hypertoric Deligne-Mumford stacks from two points of view. From the viewpoint of representation theory, we calculate the operator of small quantum product by a divisor, following \cite{BMO}, \cite{MO},…
We conjecture an equivalence between the Gromov-Witten theory of 3-folds and the holomorphic Chern-Simons theory of Donaldson-Thomas. For Calabi-Yau 3-folds, the equivalence is defined by the change of variables, exp(iu)=-q, where u is the…
We propose a conjectural explicit formula of generating series of a new type for Gromov--Witten invariants of $\mathbb{P}^1$ of all degrees in full genera.
We prove a general magnetic Bourgain-Brezis-Mironescu formula. In particular, after developing a theory of magnetic bounded variation functions, we prove the validity of the formula in this class.
In this paper, one considers the change of orbifold Gromov-Witten invariants under weighted blow-up at smooth points. Some blow-up formula for Gromov-Witten invariants of symplectic orbifolds is proved. These results extend the results of…
We give an explicit formula for (T-equivariant) 3-pointed genus zero Gromov-Witten invariants for G/B. We derive it by finding an explicit formula for the equivariant Pontryagin product on the homology of the based loop group \Omega K.
The study of open/closed string duality and large $N$ duality suggests a Gromov-Witten theory for conifolds that sits on the border of both a closed Gromov-Witten theory and an open Gromov-Witten theory. In this work we employ the result of…
We study the spin Gromov-Witten (GW) theory of $\mathbb{P}^1$. Using the standard torus action on $\mathbb{P}^1$, we prove that the associated equivariant potential can be expressed by means of operator formalism and satisfies the 2-BKP…
In this note we show that any proof of Wallis's formula or of the probability integral formula proves both assertions.
In this paper we consider the orbifold curve, which is a quotient of an elliptic curve $\mathcal{E}$ by a cyclic group of order 4. We develop a systematic way to obtain a Givental-type reconstruction of Gromov-Witten theory of the orbifold…