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We describe the small quantum cohomology ring of complete flag varieties by algebro-geometric methods, as presented in our previous work Quantum cohomology of flag varieties (Internat. Math. Res. Notices, no. 6 (1995), 263–277). We also give a geometric proof of the quantum Monk formula.

0. Introduction. The main goal of this paper is to give a unified description for the structure of the small quantum cohomology rings for all projective homogeneous spaces SLn(C)/P , where P is a parabolic subgroup. The quantum cohomology ring of a smooth projective variety, or more generally of a symplectic manifoldX, has been introduced by string… (More)

- Ionut Ciocan-Fontanine, BUMSIG KIM, Davesh Maulik, Hyo Chul Myung
- 2011

We construct new compactifications with good properties of moduli spaces of maps from nonsingular marked curves to a large class of GIT quotients. This generalizes from a unified perspective many particular examples considered earlier in the literature.

- Ionut Ciocan-Fontanine, BUMSIG KIM
- 2009

We construct new “virtually smooth” modular compactifications of spaces of maps from nonsingular curves to smooth projective toric varieties. They generalize Givental’s compactifications, when the complex structure of the curve is allowed to vary and markings are included, and are the toric counterpart of the moduli spaces of stable quotients introduced by… (More)

Following the work of Okounkov-Pandharipande [OP1, OP2], Diaconescu [D], and the recent work [CDKM] studying the equivariant quantum cohomology QH∗ (C∗)2(Hilbn) of the Hilbert scheme and the relative Donaldson-Thomas theory of P ×C, we establish a connection between the J-function of the Hilbert scheme and a certain combinatorial identity in two variables.… (More)

- Ionut Ciocan-Fontanine, Matjaz Konvalinka, Igor Pak
- J. Comb. Theory, Ser. A
- 2011

Abstract. Based on the ideas in [CKP], we introduce the weighted analogue of the branching rule for the classical hook length formula, and give two proofs of this result. The first proof is completely bijective, and in a special case gives a new short combinatorial proof of the hook length formula. Our second proof is probabilistic, generalizing the (usual)… (More)

- Ionut Ciocan-Fontanine, BUMSIG KIM
- 2013

For each positive rational number ε, the theory of ε-stable quasimaps to certain GIT quotients W/G developed in [CKM] gives rise to a Cohomological Field Theory. Furthermore, there is an asymptotic theory corresponding to ε → 0. For ε > 1 one obtains the usual Gromov-Witten theory of W//G, while the other theories are new. However, they are all expected to… (More)

The famous hook-length formula is a simple consequence of the branching rule for the hook lengths. While the Greene-Nijenhuis-Wilf probabilistic proof is the most famous proof of the rule, it is not completely combinatorial, and a simple bijection was an open problem for a long time. In this extended abstract, we show an elegant bijective argument that… (More)

- Ionut Ciocan-Fontanine, Bumsig Kim
- 2014

We provide a short introduction to the theory of ε-stable quasimaps and its applications via wall-crossing to Gromov-Witten theory of GIT targets. Mathematics Subject Classification (2010). Primary 14D20, 14D23, 14N35.

- Ionut Ciocan-Fontanine, BUMSIG KIM
- 2013

In previous work we have conjectured wall-crossing formulas for genus zero quasimap invariants of GIT quotients and proved them via localization in many cases. We extend these formulas to higher genus when the target is semi-positive, and prove them for semi-positive toric varieties, in particular for toric local Calabi-Yau targets. The proof also applies… (More)