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

国际研究奖学金计划使美国科学家和工程师能够在国外进行9到24个月的研究。 该计划的奖项提供了联合研究的机会，以及使用独特或互补的设施，专业知识和国外的实验条件。刘易斯与奥地利维也纳大学的卢德米尔·卡察尔科夫博士合作。久贺-佐竹霍奇猜想假设两个非常不同的几何对象之间存在代数对应关系？一个K3曲面和一个阿贝尔曲面这个猜想？霍奇猜想的一个特例，克莱数学研究所的一个？著名的千年难题？被广泛认为是正确的，但已知的例子非常少。 这个项目使用镜像对称的K3曲面，以揭示新的光的分类Fano品种和Kuga-Satake霍奇猜想。镜像对称，一个令人惊讶的数学对偶性首次预测的弦理论家，往往涉及复杂的代数几何的一个空间，一个更简单的建设涉及凯勒几何的？镜子？空间 PI与合作者最近进行的工作表明，大多数已知的久贺-佐竹霍奇猜想的案例都承认镜像对称的重新解释。 作为该项目的一部分，除了重新解释已知的猜想案例外，还正在开发新的例子，最终目标是为问题提供一个通用框架。这项工作需要深入了解同调镜像对称，Katzarkov博士和他的研究小组成员是这方面的专家。它还需要与复曲面品种，镜像映射和超几何函数，这是在PI的研究对象的设施。 该项目还涉及其他问题有关镜像对称的K3曲面，包括使用某些家庭的K3曲面的分类法诺三倍皮卡德数一。霍奇猜想是一个伟大的开放问题，在代数几何，实际上在所有的数学。虽然Kuga-Satake Hodge猜想是完整Hodge猜想的严格子问题，但它仍然非常有趣。同样地，镜像对称在代数几何中的重要性也在发展;它需要进一步的研究，但已经带来了令人着迷的发现。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.