**Title:** Gromov-Witten/Donaldson-Thomas correspondence for toric threefolds

**Speaker:** Alexei Oblomkov

**Speaker Info:** Princeton

**Brief Description:**

**Special Note**:

**Abstract:**

There are two ways to obtain a curve $C$ in some ambient space $Y$. We can give the ideal $I$ of defining equation or we can choose an "etalon" curve $C$ and define a map $f: C\to Y$. In the case when $dim Y=3$ the spaces of $f$'s and $I$'s (with fixed topological invariants) admit natural compatification.For a collection of curves $K_1,\dots, K_m\subset Y$ and fixed topological invariants of $C$ we can ask the question: how many there are curves $C\subset Y$ such that $C$ meets all of $K_i$. The answer, depends on our choice of the compactification of the space of curves. If we use the first methods to describe the curves $C$ then the answer is called Donaldson-Thomas invariant, if we use the second method the answer is Gromov-Witten invariant.

Eventhough the DT and GW invariants are very far from being equal, Nekrasov, Maulik, Okounkov and Panharipande (MNOP) conjectured the correspondence between them. This correspondence is a particular case gauge/string theory duality. The conjecture is proved for the large class of threefolds (for toric threefolds) by Maulik, Okounkov, O. and Pandharipande.

The aim of the talk is to discuss the definition of DT, GW invariant and to sketch the proof in the toric case.

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