Quantum Topology of Fano Varieties

今年我们在媒体上有了很大的推广，也有了新的令人兴奋的结果：我们为所有105个光滑Fano三重家族构造了镜像，对于其中104个家族，我们给出了一个很好的表示为环完全交或其非交换化，从而我们计算了它们的亏零Gromov-Witten不变量，因此我们证明了这104个家族的镜像对称性假设(在有关微分方程的情况下)，我们引入了一个新的但非常基本的极值局部系统的概念-那些是非循环的，它等价于某些泛函的最小化，我们通过104个光滑Fano三重簇的情形支持我们的猜想，即所有通过镜像对称与奇维Fano簇相关的局域系统都是极值的。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(计算基本不变量将需要大约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).