Conference: Trisections Workshop: Connections with Symplectic Topology

该奖项为会议提供支持“三分研讨会：辛拓扑连接”，将于2023年6月26日至30日在加州大学戴维斯分校举行。会议将侧重于发展三分理论，在四流形拓扑研究的新兴领域，和辛四流形的拓扑和几何的研究之间的联系。它将通过汇集所有职业阶段的数学家（包括知名专家，早期职业研究人员和学生）来促进研究发展，并将具有许多旨在积极吸引参与者的功能。首先，将有研讨会前，虚拟，介绍性的小型课程，由该领域的专家提供，旨在预览和激励研讨会的中心议题。第二，将有全体会议和简短的闪电式会谈，旨在强调在上午会议期间广泛的研究人员的最新发展。第三，工作坊将以下午的工作小组会议为特色，与会者将合作研究与三分理论和辛拓扑以及四维流形几何有关的开放性问题。辛几何最初起源于哈密顿力学，但后来发展成为一个丰富的领域，应用于数学的几个领域，如复几何，代数几何，奇异光滑结构的四维流形，奇异性理论，动力学和镜像对称。尽管经过几十年的发展和许多开创性的进展，看似基本的问题-如辛合痕问题-仍然是开放的。三分理论提供了新的组合方法来研究辛流形，并提供了一个新的视角来阐明困难的开放问题。这为许多新的方向和应用提供了一个起点，并为通过这两个领域的协同作用取得进展提供了丰富的机会。自引入以来的十年中，三分理论已经发展到几乎触及四流形拓扑的每一个方面-从比较光滑结构和引入新的不变量到表示打结表面和描述外科手术。该研讨会将建立在最近的成功的理论突出之间的相互作用辛和复杂的几何形状的四流形和组合功能的三分。其中包括：一个表征辛表面在复杂的射影平面作为那些承认横向阴影图;一个证明存在的三分辛四流形是兼容的环境辛结构;和一个既定的对应关系，某些复杂的曲线在复杂的射影平面和六边形格的环面，产生一个组合的方法辛合痕问题。该会议的网站是：https://sites.google.com/view/tw2023/This奖反映了NSF的法定使命，并已被认为是值得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的支持。

This award provides support for the conference “Trisections Workshop: Connections with Symplectic Topology” that will take place at the University of California, Davis, during June 26-30, 2023. The conference will focus on developing connections between trisection theory, an emerging area of research in four-manifold topology, and the study of the topology and geometry of symplectic four-manifolds. It will catalyze research developments by bringing together mathematicians of all career stages – including established experts, early-career researchers, and students – and will have a number of features that are designed to actively engage the participants. First, there will be pre-workshop, virtual, introductory mini-courses given by experts in the field and designed to preview and motivate the central topics of the workshop. Second, there will be both plenary talks and shorter, lightning-style talks aimed at highlighting recent developments by a wide range of researchers during the morning sessions. Third, the workshop will feature afternoon working-group sessions in which participants will collaborate on open problems relating to the theory of trisections and symplectic topology and geometry of four-manifolds.Symplectic geometry originally arose from Hamiltonian mechanics but has since developed into a rich field of its own with applications to several areas of mathematics, such as complex geometry, algebraic geometry, exotic smooth structures on four-manifolds, singularity theory, dynamics, and mirror symmetry. Despite decades of development and many seminal advances, seemingly basic problems – such as the symplectic isotopy problem – remain wide open. Trisection theory provides new combinatorial methods to study symplectic manifolds and offers a new perspective to shed light on difficult open problems. This provides a starting point for many new directions and applications and a rich opportunity for making advances through the synergy of these two fields. In the ten years since its introduction, trisection theory has developed to touch nearly every facet of four-manifold topology – from comparing smooth structures and introducing new invariants to representing knotted surfaces and describing surgery operations. The workshop will build on recent successes of the theory highlighting the interplay between the symplectic and complex geometry of four-manifolds and the combinatorial features of trisections. These include: a characterization of symplectic surfaces in the complex projective plane as those admitting transverse shadow diagrams; a proof of the existence of trisections on symplectic four-manifolds that are compatible with the ambient symplectic structure; and an established correspondence between certain complex curves in the complex projective plane and hexagonal lattices on the torus that yields a combinatorial approach to the symplectic isotopy problem. The conference website is at: https://sites.google.com/view/tw2023/This 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.