Workshop on Hypergeometric Motives and Calabi-Yau Differential Equations

Louisiana State University, Baton Rouge LA

Investigators

Abstract

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