Quantum Topology of Fano Varieties

我们今年在媒体上有很大的推广，也有新的令人兴奋的结果：我们对所有105个光滑Fano三重族构造了镜像，对其中的104个族给出了环面完全交或其“非阿贝尔化”的良好表示，从而计算了它们的亏格为零的Gromov-Witten不变量，因此我们证明了镜像对称的假设（在微分方程方面）对于这104个家庭，我们引入了一个新的但非常基本的极值局部系统的概念-这些系统是非循环的并且它等价于一些泛函的最小化，通过对104个光滑Fano三折叠族的研究，证明了我们的猜想：所有通过镜像对称与奇维Fano簇相联系的局部系统都是极值系统。我们的方法不依赖于维度，现在帝国理工学院的机器正在计算5亿个四维候选者（计算基本不变量大约需要2-3个月）。

We had great outreach in media this year and also new exciting results : we constructed mirrors for all 105 families of smooth Fano threefolds, for 104 of these families we give a nice representation as toric complete intersection or its "unabelianization", thus we compute their genus zero Gromov-Witten invariants, and so we prove hypothesis of mirror symmetry (in what concerns differential equations) for these 104 families, we introduce a new but very fundamental notion of extremal local systems - those are acyclic and it is equivalent to minimization of some functional, we support our conjecture that all local systems associated with odd-dimensional Fano varieties via mirror symmetry are extremal by case of 104 families of smooth Fano threefolds. Our method doesn't depend on dimension and now machines in Imperial College are working out half billion four-dimensional candidates (computing the basic invariants will take approximately 2-3 more months).