Workshop on Hypergeometric Motives and Calabi-Yau Differential Equations

该奖项将支持来自美国的研究人员参加为期三周的“超几何动机和Calabi- Yau微分方程”项目，该项目将于2017年1月8日至28日在澳大利亚墨尔本的MATRIX会议中心举行。该活动将汇集两个重叠数学领域的高级和初级研究人员，即超几何动机和Calabi-Yau微分方程，讨论这两个领域的最新发展和未来方向。本课题与镜像对称现象有关，同时属于数论和数学物理的前沿。卡拉比-丘品种的家族自然出现在镜像对称的背景下，通常看起来是超几何的。另一方面，超几何动机是一种方便的对象，可以用来检验重要的标准猜想，甚至可能发现一些新的特征。该项目的预期参与者来自不同的地理位置和学术水平，包括相当多来自美国的初级参与者。除了理论发展之外，研讨会还将促进Magma、SageMath和Pari/GP软件在研究中的作用。作为回报，该程序将增强超几何动机的实现。更具体地说，该程序的目标是：研究一般的l函数，它是数论的中心对象，使用所谓的超几何动机的l函数作为一个重要但更容易接近的类；并研究Calabi- Yau流形族的算术和几何，特别是在镜像对称中出现的完整性和模块化现象。通过有影响力的Langlands规划，期望动机l -函数与由自同构形式产生的l -函数重合，从而满足泛函方程并可解析连续；换句话说，它们的行为类似于经典的黎曼函数。在创建有用的l函数和自同态形式库方面已经花费了许多年的努力，最近的例子是由NSF赞助的LMFDB项目（http://www.lmfdb.org/）。自2009年以来，由Fernando Rodriguez Villegas （http://users.ictp.it/~villegas/hgm/index.html）领导的一组研究人员一直在研究和计算超几何动机的l函数，这有望涵盖广泛的已知l函数。他们开发了高效的计算技术，其中大部分已经在计算机代数软件中实现。通过研究有限的超几何函数，将它们理解为有限域上的周期，并将它们与经典的超几何函数进行类比，进一步了解了这些l函数的局部因子。总的期望是，有限域上的周期形成了理解镜像对称中出现的整体性现象的新方向。镜像对称最初是由物理学家在20世纪80年代中期发现的，至今仍是弦理论和代数几何的核心研究主题之一。大量的实例表明，所谓规范坐标下的镜像映射表达式具有丰富的算术性质，如模性。这个表达式涉及到奇异点附近Calabi—Yau流形族的Picard—Fuchs微分方程的特解。在镜像对称的算术方面，解释模块化是一个终极目标。值得注意的是，Calabi- Yau微分方程出现在其他几个不同的背景下，如π的有理近似、马勒测度和统计力学模型中随机游走的生成函数。更多信息可访问会议网站：http://www.matrixatmelbourne.org.au/events/hypergeometric-motives-and-calabi-yau-differential-equations

The award is to support the attendance of researchers from the United States in a 3-weeks-long program `Hypergeometric motives and Calabi--Yau differential equations' at the conference center MATRIX at Melbourne, Australia, January 8-28 2017. This activity will bring together both senior and junior researchers in two overlapping fields of mathematics, namely hypergeometric motives and Calabi-Yau differential equations, to discuss recent developments and future directions in both areas. The topic of this program is related to the phenomenon of mirror symmetry and belongs to the cutting edge of number theory and mathematical physics at the same time. Families of Calabi--Yau varieties naturally arise in the context of mirror symmetry and often appear to be hypergeometric. On the other hand, hypergeometric motives are handy objects on which important standard conjectures could be tested and perhaps even some new features could be discovered. Expected participants of the program represent a wide variety of geographical locations and academic levels, including quite a few junior participants from the United States. In addition to theoretic developments, the workshop will foster the role of software Magma, SageMath, and Pari/GP in research. In return, the program will enhance the implementation of hypergeometric motives.More particularly, the goals of the program are: to study general L-functions, which are central objects in number theory, using the L-functions of so-called hypergeometric motives as an important but more approachable class; and to investigate the arithmetic and geometry of families of Calabi--Yau manifolds, in particular the integrality and modularity phenomena arising in mirror symmetry. By the influential Langlands program, motivic L-functions are expected to coincide with L-functions arising from automorphic forms and consequently satisfy functional equations and can be continued analytically; in other words, they behave similarly to the classical Riemann zeta function. Already for many years much effort was spent on creation of a useful library of L-functions and automorphic forms, the most recent example being the LMFDB project (http://www.lmfdb.org/) sponsored by NSF. Since 2009 a group of reserachers led by Fernando Rodriguez Villegas (http://users.ictp.it/~villegas/hgm/index.html) was investigating and computing explicitly the L-functions of hypergeometric motives, which are expected to cover a wide range of known L-functions. They developed efficient computational techniques, most of which are already implemented in computer algebra software. Another step towards understanding the local factors of these L-functions was brought to the scene by investigating finite hypergeometric functions, understanding them as periods over finite fields and drawing analogies between them and classical hypergeometric functions. An overall expectation is that periods over finite fields form a new direction in understanding the integrality phenomenon arising in mirror symmetry. Originally discovered by physicists the mid-1980s, mirror symmetry remains one of the central research themes binding string theory and algebraic geometry. Numerous examples show that the expression of the mirror map in so-called canonical coordinates possesses rich arithmetic properties, such as modularity. This expression involves particular solutions to Picard--Fuchs differential equations attached to families of Calabi--Yau manifolds near a singular point. Explaining modularity is an ultimate goal on the arithmetic side of mirror symmetry. Quite remarkably, Calabi--Yau differential equations show up in several other contexts as diverse as rational approximations to pi, Mahler measures and generating functions of random walks in models of statistical mechanics.More information can be found on the conference website:http://www.matrixatmelbourne.org.au/events/hypergeometric-motives-and-calabi-yau-differential-equations