Workshop on Hypergeometric Motives and Calabi-Yau Differential Equations
超几何动机和卡拉比-丘微分方程研讨会
基本信息
- 批准号:1642598
- 负责人:
- 金额:$ 2万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Standard Grant
- 财政年份:2016
- 资助国家:美国
- 起止时间:2016-09-15 至 2017-08-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
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
该奖项是为了支持来自美国的研究人员参加2017年1月8日至28日在澳大利亚墨尔本举行的会议中心MATRIX为期3周的项目《超几何动机和Calabi-Yau微分方程式》。这项活动将汇集两个重叠数学领域的资深和初级研究人员,即超几何动机和Calabi-Yau微分方程式,讨论这两个领域的最新发展和未来方向。本期节目的话题与镜面对称现象有关,同时属于数论和数学物理的前沿。Calabi-Yau变种家族自然而然地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,超几何动机是方便的对象,可以在其上测试重要的标准猜想,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,其中包括相当多来自美国的初级参与者。除了理论发展,研讨会还将促进软件MAGMA、SageMath和Pari/GP在研究中的作用。作为回报,该程序将加强超几何动机的实现。更具体地说,该程序的目标是:研究数论中的中心对象-一般L函数,将所谓的超几何动机的L函数作为一类重要但更接近的类;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积分和模性现象。根据有影响力的朗兰兹程序,动机L函数期望与源自自同构形式的L函数重合,从而满足函数方程,并且可以解析地继续;换句话说,它们的行为类似于经典的Riemann Zeta函数。多年来,人们已经花费了大量的精力来创建一个有用的L函数库和自同构形式库,最近的例子是LMFDB项目(由美国国家科学基金会赞助的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显式地研究和计算超几何动机的L函数,这些超几何动机有望涵盖广泛的已知L函数。他们开发了高效的计算技术,其中大多数已经在计算机代数软件中实现。通过研究有限超几何函数,将它们理解为有限域上的周期,并将它们与经典超几何函数进行类比,从而实现了理解这些L函数的局部因子的另一步。一个总体的期望是,有限域上的周期形成了理解镜像对称中出现的完整性现象的新方向。镜像对称性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中心研究主题之一。大量的例子表明,镜像映射在所谓的标准坐标下的表达式具有丰富的算术性质,如模块化。这个表达式涉及到奇点附近的一族Calabi-Yau流形上的Picard-Fuchs微分方程解。解释模块化是镜像对称性算术方面的最终目标。非常值得注意的是,Calabi-Yau方程在其他几种情况下也出现了,就像统计力学模型中对圆周率、马勒测度和随机游动的母函数的有理逼近一样。更多信息可以在会议website:http://www.matrixatmelbourne.org.au/events/hypergeometric-motives-and-calabi-yau-differential-equations上找到
项目成果
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Ling Long其他文献
On Hopf Algebras of Dimension 4p Table of Contents
关于 4p 维 Hopf 代数 目录
- DOI:
- 发表时间:
- 期刊:
- 影响因子:0
- 作者:
Yi;S. Ng;L. Hogben;Jonathan D. H. Smith;Sung Y. Song;Ling Long - 通讯作者:
Ling Long
A Comparison of the Performance of Two Kinds of Waterborne Coatings on Bamboo and Bamboo Scrimber
两种水性涂料在竹材及重组竹上的性能比较
- DOI:
10.3390/coatings9030161 - 发表时间:
2019-03 - 期刊:
- 影响因子:3.4
- 作者:
Jianfeng Xu;Ru Liu;Huagui Wu;Hongyun Qiu;Yanglun Yu;Ling Long;Yonghao Ni - 通讯作者:
Yonghao Ni
Research on fluid flow and heat transfer characteristics in a three-dimensional condenser
- DOI:
10.1016/j.anucene.2024.110967 - 发表时间:
2025-02-01 - 期刊:
- 影响因子:
- 作者:
Zhiqiang Duan;Yuan Tian;Siyuan Wang;Ling Long;Jianjun Deng - 通讯作者:
Jianjun Deng
On Atkin-swinnerton-dyer Congruence Relations
论阿特金-斯温纳顿-戴尔同余关系
- DOI:
10.1515/crelle-2016-0037 - 发表时间:
2009 - 期刊:
- 影响因子:0
- 作者:
LI WEN;Ling Long;Zifeng Yang - 通讯作者:
Zifeng Yang
Honeycomb-like 3D N,P-codoped porous carbon anchored with ultrasmall Fe2P nanocrystals for efficient Zn-air battery.
蜂窝状 3D N、P 共掺杂多孔碳锚定有超小 Fe2P 纳米晶体,用于高效锌空气电池。
- DOI:
10.1016/j.carbon.2019.11.073 - 发表时间:
2020 - 期刊:
- 影响因子:10.9
- 作者:
Lulu Chen;Yelong Zhang;Lile Dong;Xiangjian Liu;Ling Long;Siyu Wang;Changyu Liu;Shaojun Dong;Jianbo Jia - 通讯作者:
Jianbo Jia
Ling Long的其他文献
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{{ truncateString('Ling Long', 18)}}的其他基金
The Arithmetic of Hypergeometric Varieties and Noncongruence Modular Forms
超几何簇和非全等模形式的算术
- 批准号:
1602047 - 财政年份:2016
- 资助金额:
$ 2万 - 项目类别:
Standard Grant
Applications of Automorphic Forms in Number Theory and Combinatorics
自守形式在数论和组合学中的应用
- 批准号:
1363265 - 财政年份:2014
- 资助金额:
$ 2万 - 项目类别:
Standard Grant
Noncongruence Modular Farms and Supercongruences
非全等模块化农场和超全等
- 批准号:
1303292 - 财政年份:2013
- 资助金额:
$ 2万 - 项目类别:
Continuing Grant
Modular Forms for Noncongruence Subgroups
非同余子群的模形式
- 批准号:
1001332 - 财政年份:2010
- 资助金额:
$ 2万 - 项目类别:
Standard Grant
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