Conference: Monodromy and Its Applications

这次会议将讨论单一性的一般主题。单一性是指一个数学对象在空间中平滑地变化，在一个点和附近的每个点上看起来是相同的，但在空间中绕一个圈运动会导致对象扭曲。单一性在数学中随处可见，特别是在莫比乌斯条中：每一小块都是一张普通的纸，但绕着圆圈旋转会使纸翻转到相反的一边。这次会议将提供一个历史的视角，叙述当前的结果，并展望未来的发展。会议的主题将是单一性在其所有化身中的关键作用：经典和l-adic，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理论的联系。近年来，这一领域取得了许多令人兴奋的进展。这些成果包括Katz， roja - le´on和Tiep对作为超几何轴的单群出现的有限（几乎是拟）单群的完全分类，Landesmann和Litt对Putman-Wieland猜想的许多例子的证明，许多数学家在新环境下对Tannakian单群的计算及其在Lawrence和Sawin对Shafarevich猜想的推广中的应用，Bakker和Tsimerman对Grothendieck的周期猜想的一个相对类似的证明，以及Calegari、Dimitrov和Tang对无界分母猜想的证明。会议将讨论与数学中这些和其他单一性表现有关的发展。会议网站https://www.math.princeton.edu/katz80This奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，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.