International Research Fellowship Program: K3 Surfaces, Normal Forms and the Kuga-Satake Hodge Conjecture

[09:65 . 183]刘易斯国际研究奖学金计划使美国科学家和工程师能够到国外进行9至24个月的研究。该计划的奖励为联合研究提供了机会，并利用独特或互补的设施、专业知识和国外的实验条件。该奖项将支持雅各布·m·刘易斯博士与奥地利维也纳大学的拉德米尔·卡扎科夫博士开展为期24个月的研究。Kuga-Satake Hodge猜想假设两个非常不同的几何物体之间存在代数对应关系？一个K3曲面和一个阿贝尔变体。这个猜想?霍奇猜想的一个特例，克莱数学研究所的一个？著名的“千年问题”是什么？被广泛认为是正确的，但众所周知的例子非常少。这个项目使用镜面对称的K3表面来揭示Fano品种的分类和Kuga-Satake Hodge猜想。镜像对称是一种令人惊讶的数学对偶，最早由弦理论家预测，它经常将一个空间的复杂代数几何与涉及Kähler镜像几何的更简单结构联系起来。空间。PI与合作者最近进行的工作表明，大多数已知的Kuga-Satake Hodge猜想都可以用镜像对称的方式重新解释。作为该项目的一部分，除了重新解释该猜想的已知案例外，还正在开发新的例子，其最终目标是为该问题提供一个总体框架。这项工作需要对同源镜像对称有深入的了解，卡扎科夫博士和他的研究小组成员是这方面的专家。它还需要具备环面变化、镜像映射和超几何函数的能力，这些都是PI研究的对象。该项目还涉及到与K3曲面的镜像对称相关的其他问题，包括使用某些K3曲面族在Fano三倍与皮卡德数1的分类中。霍奇猜想是代数几何中，乃至整个数学领域中最重要的开放问题之一。虽然Kuga-Satake Hodge猜想是完整Hodge猜想的严格子问题，但它仍然是非常有趣的。同样，镜像对称在代数几何中也越来越重要；它需要进一步的研究，但已经有了令人着迷的发现。在霍奇猜想和分类问题中寻找镜像对称的作用为这些令人印象深刻的研究领域提供了一个新的视角。

0965183LewisThe International Research Fellowship Program enables U.S. scientists and engineers to conduct nine to twenty-four months of research abroad. The program's awards provide opportunities for joint research, and the use of unique or complementary facilities, expertise and experimental conditions abroad.This award will support a twenty-four-month research fellowship by Dr. Jacob M. Lewis to work with Dr. Ludmil Katzarkov at the University of Vienna in Austria.The Kuga-Satake Hodge conjecture posits an algebraic correspondence between two very different geometric objects?a K3 surface and an abelian variety. This conjecture?a special case of the Hodge conjecture, one of the Clay Mathematical Institute?s famous Millenium Problems?is widely believed to be true, but remarkably few examples are known. This project uses Mirror Symmetry for K3 surfaces to shed new light on the classification of Fano varieties and on the Kuga-Satake Hodge conjecture. Mirror symmetry, a surprising mathematical duality first predicted by string theorists, often relates complicated algebraic geometry of one space to a much simpler construction involving the Kähler geometry of the ?mirror? space. Recent work in progress by the PI with collaborators suggests that most of the known cases of the Kuga-Satake Hodge conjecture admit a re-interpretation in terms of mirror symmetry. As part of this project, in addition to reinterpreting the known cases of the conjecture, new examples are being developed, with the ultimate goal of providing a general framework for the problem. This work requires deep knowledge of homological mirror symmetry, in which Dr. Katzarkov and members of his research group are experts. It also requires facility with toric varieties, mirror maps, and hypergeometric functions, which are objects of study in the PI's research. The project also deals with other questions related to mirror symmetry for K3 surfaces, including the use of certain families of K3 surfaces in the classification of Fano threefolds with Picard number one.The Hodge conjecture is one of the great open problems in algebraic geometry, and indeed in all of mathematics. While the Kuga-Satake Hodge conjecture is a strict sub-problem of the full Hodge conjecture, it is nevertheless of great interest. Similarly, mirror symmetry is developing importance in algebraic geometry; it needs further study but is already leading to fascinating discoveries. Finding a role for mirror symmetry within the Hodge conjecture and classification problems provides a fresh perspective to these imposing areas of research.