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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.

- IONUŢ CIOCAN-FONTANINE, BUMSIG KIM
- 2009

We construct new " virtually smooth " modular com-pactifications of spaces of maps from nonsingular curves to smooth projective toric varieties. They generalize Givental's compactifica-tions, 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… (More)

- IONUŢ CIOCAN-FONTANINE, BUMSIG KIM, Shigeru Mukai
- 2014

We introduce a new big I-function for certain GIT quotients W//G using the quasimap graph space from infinitesimally pointed P to the stack quotient [W/G]. This big I-function is expressible by the small I-function introduced in [6, 10]. The I-function conjecturally generates the Lagrangian cone of GromovWitten theory for W//G defined by Givental. We prove… (More)

- IONUŢ 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.

Following the work of Okounkov-Pandharipande [OP1, OP2], Dia-conescu [D], and the recent work [CDKM] studying the equivariant quantum coho-mology QH * (C *) 2 (Hilb n) of the Hilbert scheme and the relative Donaldson-Thomas theory of P 1 × C 2 , we establish a connection between the J-function of the Hilbert scheme and a certain combinatorial identity in… (More)

- IONUŢ 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)

IONU¸T CIOCAN-FONTANINE 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 SL n (C)/P , where P is a parabolic subgroup. The quantum cohomology ring of a smooth projective variety, or more generally of a symplectic manifold X, has been… (More)

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

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) hook walk… (More)

- IONUŢ 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)

- Ionuţ 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.