Topology of Manifolds: Interactions between High and Low Dimensions

该奖项支持美国参与者参与将于2019年1月7日至18日在澳大利亚克雷斯威克墨尔本大学举行的项目《流形的拓扑：高维度与低维度之间的互动》。这次会议将汇集来自世界各地的学生、博士后和研究人员，以促进对多种理论中基本问题的研究。它将促进研究人员在高维和低维拓扑中的互动。会议安排如下：第一周由世界专家举办小型课程，第二周举行会议。这两周的重点都是开放问题和协作工作。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该计划的主题：促进高维度和低维度的互动将缓解技术会谈和问题的倾向。这次会议的主要研究目的有两个。第一个是从高维度和低维度之间的相互作用中确定协同的设置，并在这些设置中的问题上取得进展。第二是产生高质量的问题列表，以指导未来对流形拓扑的研究。预计在会议期间合作产生的精心制作和公开的问题清单将对数学界产生长期好处。n-流形是局部类似于n维欧几里德空间的空间。用几何方法研究了维度小于或等于3的流形。通过外科理论研究维度大于或等于5的流形，该理论涉及代数和微分拓扑、代数和分析的混合。维度4介于两者之间；高维的惠特尼技巧和低维的几何技巧都只取得了部分成功。对该计划的需求是，这些领域在过去几十年中出现了分歧，以至于低维/中维/高维拓扑中的研究人员往往不总是意识到其他维度的当前研究/技术。这一计划预计将产生协同效应，使专家和新一代早期职业研究人员受益。该活动的网站可在https://www.matrix-inst.org.au/events/interactions-between-topology-in-high-and-low-dimensions/This网站上找到，该奖项反映了国家科学基金会的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.