Topology of Manifolds: Interactions between High and Low Dimensions

该奖项支持美国参与者参加将于2019年1月7日至18日在澳大利亚克雷斯维克墨尔本大学举行的“流形拓扑：高维和低维之间的相互作用”项目。这次会议将汇集来自世界各地的学生、博士后和研究人员，以激发对流形理论基本问题的研究。它将促进高维和低维拓扑研究人员之间的互动。会议安排如下：第一周由世界专家讲授小型课程，第二周举行会议。这两周都侧重于开放性问题和协作性工作。这种结构将极大地有利于早期职业研究人员。会议的另一个特点是节目的主题：促进高维度和低维度的互动将减轻技术会谈和问题的倾向。这次会议有两个主要的研究目的。首先，从高维和低维之间的相互作用中确定协同作用的设置，并在这些设置中的问题上取得进展。二是提出高质量的问题清单，以指导未来流形拓扑的研究。预计会议期间合作产生的精心设计和公开的问题清单将对数学界产生长期的好处。n流形是一个局部类似于n维欧几里德空间的空间。用几何方法研究了小于或等于3维的流形。维数大于或等于5的流形是通过外科理论来研究的，这涉及到代数和微分拓扑、代数和分析的混合。维度4介于两者之间；高维惠特尼技巧和低维几何技巧都只是部分成功。该计划的需求在于，这些领域在过去几十年中已经出现了分歧，以至于低/中/高维拓扑的研究人员往往并不总是意识到其他维度的当前研究/技术。该项目预计将产生协同效应，使专家和新一代早期职业研究人员都受益。该活动的网站可在https://www.matrix-inst.org.au/events/interactions-between-topology-in-high-and-low-dimensions/This上找到奖项反映了美国国家科学基金会的法定使命，并通过基金会的智力价值和更广泛的影响审查标准进行评估，认为值得支持。

This award supports funding for the participation of US-based participants in the program "Topology of Manifolds: interactions between high and low dimensions" to be held January 7-18, 2019 at the University of Melbourne, Creswick, Australia. This meeting will bring together students, postdocs, and researchers from all over the world to stimulate research on fundamental questions in manifold theory. It will promote the interaction between researchers in high and low dimensional topology. The meeting is structured as follows: mini-courses in the first week by world experts and a conference in the second week. Both weeks focus on open problems and collaborative work. This structure will greatly benefit early career researchers. Another feature of the meeting that will make it accessible is the theme of the program: promoting interactions high and low dimensions will mitigate the tendency of technical talks and problems. There are two main research aims for this meeting. The first is to identify settings for synergy from the interaction between high and low dimensions and to make progress on problems in these settings. The second is to produce a high-quality problem list to guide future research in manifold topology. It is expected that a well-crafted and publicized problem list arising from the collaboration during the meeting will be of long-term benefit to the mathematical community.An n-manifold is a space which locally resembles n-dimensional Euclidean space. Manifolds of dimension less or equal than 3 are studied using geometric techniques. Manifolds of dimension greater or equal than 5 are studied via surgery theory, which involves a mix of algebraic and differential topology, algebra, and analysis. Dimension 4 is in between; both the high dimensional Whitney Trick and the low-dimensional geometric techniques are only partially successful. The need for the program is that these areas have diverged in the last several decades, to the extent that, often researchers in low/middle/high dimensional topology are not always aware of the current research/techniques in other dimensions. This program is expected to lead to a synergy, benefiting both the experts and the new generation of early career researchers. The website for the event can be found at https://www.matrix-inst.org.au/events/interactions-between-topology-in-high-and-low-dimensions/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.