Conference: Monodromy and Its Applications

这次会议将讨论单行文字的一般话题。当一个数学对象在一个空间中平滑地变化，在一个点和附近的每个点看起来都是相同的，但在空间中绕着一个环移动会导致对象扭曲时，单行就发生了。单行现象在整个数学中都存在，在莫比乌斯条中尤其明显：每一小块都是一张普通的纸，但绕着圆圈运动会使纸翻到相反的一面。这次会议将提供历史的视角，回顾当前的成果，并展望未来的发展。会议的主题将是单行在所有化身中的关键作用：经典和L-进，局部和全局，算术和几何，它在数论和代数几何中的应用，以及它与群论和表示论的联系。近年来，这一领域出现了一些令人振奋的发展。其中包括Katz，Rojas-Le‘on和Tiep对有限(几乎拟)单群的完全分类，Landesmann和Litt对Putman-Wieland猜想的许多情况的证明，许多数学家在新环境下对Tannakian单群的计算及其应用，Lawrence和Sawin对Shafarevich猜想的推广，Bakker和Tsimerman对Grothendieck关于一族变种的周期猜想的相对类似的证明，以及Calegari，Dimitrov和Tang对无界分母猜想的证明。会议将讨论与这些和数学中的单元性的其他表现形式相关的发展。会议网站https://www.math.princeton.edu/katz80This奖反映了美国国家科学基金会的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This conference will discuss the general topic of monodromy. Monodromy occurs when a mathematical object varies smoothly over a space, appearing identical at a point and each nearby point, but traveling around a loop in the space causes the object to twist around. Monodromy occurs throughout mathematics and specifically can be seen in the M¨obius strip: each small piece is an ordinary piece of paper, but traveling around the circle causes the paper to flip around to the opposite side. This conference will provide a historical perspective, recount current results, and look forwad to future developments in monodromy.The main theme of the conference will be the key role of monodromy in all its incarnations: classical and l-adic, local and global, arithmetic and geometric, applications of it in number theory and algebraic geometry, and its connections to group theory and representation theory. In recent years there have been a number of exciting developments in this area. These include the complete classification of the finite (almost quasi) simple groups that occur as monodromy groups of hypergeometric sheaves by Katz, Rojas-Le´on and Tiep, the proof of many cases of the Putman-Wieland conjecture by Landesmann and Litt, the calculation of Tannakian monodromy groups in new settings by many mathematicians and their applications to generalizations of Shafarevich’s conjecture by Lawrence and Sawin, the proof of a relative analogue of Grothendieck’s period conjecture for a family of varieties by Bakker and Tsimerman, and the proof of the unbounded denominators conjecture by Calegari, Dimitrov, and Tang. The conference will discuss developments related to these and other manifestations of monondromy in mathematics. Conference website https://www.math.princeton.edu/katz80This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.