Travel support grant for the program on "Interactions between topological recursion, modularity, quantum invariants and low-dimensional topology"

该奖项为2016年澳大利亚项目“拓扑递归、模块化、量子不变量和低维拓扑之间的相互作用”的美国参与者提供旅行支持，该项目将于2016年11月28日至2016年12月23日在墨尔本新成立的澳大利亚数学研究所MATRIX@Melbourne举行。该项目开始于为期一周的（澳大利亚）暑期学校，目标是研究生和博士后学者，随后是为期两周的国际会议，由来自世界各地的专家就标题中列出的主题进行演讲，并具有不同的背景。该计划的重点是找到一个系统的数学理解“量化”的物理思想应用于两个变量的函数。量子化是20世纪初由普朗克、爱因斯坦、海森堡、薛丁格、狄拉克等巨人发现的，是将经典牛顿力学转变为当时新发现的量子力学的一种方法。一百年后，同样的量子化发挥了发现新键或不变量的作用，以识别我们周围空间的更精细结构。经典不变量被称为空间的同伦和同伦，它们告诉我们粗糙的结构。量子不变量告诉我们关于给定空间的更精确的信息。美国研究人员和澳大利亚数学家之间精心设计的互动有望为这一令人兴奋的新研究领域的研究提供养分。同时，该计划旨在激励早期职业研究人员，特别是女科学家进入这一领域。该项目将聚集来自数学和物理学的顶尖专家，讨论最新进展，并探索拓扑、数论和拓扑递归之间的新联系。在低维拓扑中的量子不变量被发现，而拓扑递归和量子曲线正在迅速发展的背景下，这样的会议显得尤为及时。这些领域之间的联系现在正成为严密的数学理论。在拓扑学方面，在过去的四年里，我们见证了数论与理想三角剖分（主要是双曲型）3-流形之间联系的令人兴奋的活动。在拓扑递推方面，最显著的成就是最近求解了三维环形Calabi-Yau轨道的重构猜想。在相互作用的量子化方面，解决了Gaiotto关于希钦分量希格斯束量子化的猜想。该计划旨在研究突出的猜想，包括(1)双曲体积猜想；(2) AJ-Conjecture;(3)量子模块化猜想。来自拓扑递归和量子曲线的新思想有望在研究这些猜想方面发挥关键作用。为期四周的活动预计将在该领域取得重大进展。该程序的URL如下。http://www.matrix-inst.org.au/events/interactions-between-topological-recursion-modularity-quantum-invariants-and-low-dimensional-topology/

This award provides travel support for U.S. based participants to the 2016 Australian program titled "Interactions between Topological Recursion, Modularity, Quantum Invariants and Low-Dimensional Topology," to be held at the newly inaugurated Australian Mathematical Research Institute in Melbourne, MATRIX@Melbourne, from November 28, 2016, to December 23, 2016. The Program starts with a week of (Australian) summer school aimed at graduate students and postdoctoral scholars, followed by two week-long international conferences featuring talks by experts from all over the world on the subjects listed in the title, and with diverse background. The focus of the Program is to find a systematic mathematical understanding of the physical idea of `quantization' applied to functions in two variables. Quantization was discovered by Planck, Einstein, Heisenberg, Schoedinger, Dirac, and other giants in the early 20th century, as a procedure to change classical Newtonian mechanics to then newly discovered quantum mechanics. One hundred years later, the same quantization plays a role of discovering new keys, or invariants, to identify finer structures of spaces surrounding us. The classical invariants are called homology and homotopy of the space, and these tell us the coarse structure. The quantum invariants tell us far more refined information of the given space. The designed interaction between U.S. researchers and Australian mathematicians is expected to fertilize the study of this exciting new area of research. At the same time, the Program aims at inspiring early career researchers, in particular, women scientists, into this field.The Program will gather leading experts from both mathematics and physics to address recent advances and explore new connections between topology, number theory, and topological recursion. Such a meeting is particularly timely as quantum invariants in low-dimensional topology are discovered, while topological recursion and quantum curves are making rapid advances. Connections between these fields are now becoming rigorous mathematical theory. On the topology side, in the past four years we have witnessed exciting activity on the connection between number theory and ideal triangulations of (mostly hyperbolic) 3-manifolds. On topological recursion side, the most notable achievement is the recent solution of the Remodeling Conjecture of toric Calabi-Yau orbifolds of dimension three. And on the quantization aspect of the interplay, Gaiotto's conjecture on quantizing Higgs bundles of Hitchin component is solved. The Program aims at studying outstanding conjectures, including (1) Hyperbolic Volume Conjecture; (2) AJ-Conjecture; and (3) Quantum Modularity Conjecture. New ideas from topological recursion and quantum curves are expected to play a key role on studying these conjectures. A significant advance on the field is expected as a result of the 4-week activity. The URL of the Program is the following. http://www.matrix-inst.org.au/events/interactions-between-topological-recursion-modularity-quantum-invariants-and-low-dimensional-topology/