Conference: Trisections Workshop: Connections with Symplectic Topology

该奖项为将于2023年6月26日至30日在加州大学戴维斯分校举行的“三节研讨会：与辛拓扑的连接”会议提供支持。会议将聚焦于发展四流形拓扑研究的新兴领域三分理论与辛四流形拓扑和几何研究之间的联系。它将汇集所有职业阶段的数学家——包括成熟的专家、职业生涯早期的研究人员和学生——从而促进研究的发展，并将具有许多旨在积极吸引参与者的功能。首先，将有由该领域专家提供的研讨会前、虚拟、介绍性迷你课程，旨在预览和激励研讨会的中心主题。其次，在上午的会议上，将有全体会议和简短的闪电式会谈，旨在强调广泛的研究人员的最新进展。第三，研讨会将以下午的工作组会议为特色，参与者将在与三分理论和四流形的辛拓扑和几何有关的开放问题上进行合作。辛几何最初起源于哈密顿力学，但后来发展成为一个自己的丰富领域，应用于数学的几个领域，如复杂几何、代数几何、四流形上的奇异光滑结构、奇点理论、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- -例如辛同位素问题- -仍然悬而未决。三分理论为辛流形的研究提供了新的组合方法，并为揭示复杂的开放问题提供了新的视角。这为许多新的方向和应用提供了起点，并为通过这两个领域的协同发展提供了丰富的机会。自引入以来的十年里，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手术操作。研讨会将以最近的成功理论为基础，强调四流形的辛几何和复杂几何与三分线的组合特征之间的相互作用。这些包括：在复射影平面上的辛曲面的表征为允许横向阴影图的辛曲面；辛四流形上与环境辛结构相容的三分存在性的证明并建立了复射影平面上的某些复曲线与环面上的六边形晶格之间的对应关系，从而给出了求解辛同位素问题的组合方法。该奖项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知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.