Workshops: Homotopy Harnessing Higher Structures

该奖项支持来自美国的初级研究型数学家前往英国剑桥艾萨克·牛顿数学科学研究所参加四个研讨会。这些研讨会都是同伦利用高级结构的学期课程的一部分，分别是“同伦理论中的高级结构”（2018年7月2-6日）、“等变与动同伦理论”（2018年8月13-17日）、“派生代数几何与色同伦理论”（2018年9月24-28日）和“流形”（2018年12月3-7日）。最近十五年见证了代数拓扑的复兴。旧问题用新方法解决了，新方法引出了新思想，新思想引出了新问题，新问题引出了新定理。在每一种情况下，在引入和研究更高的结构之后，都取得了巨大的进步。牛顿研究所的更大项目将评估已经完成的工作，突出哪些工作做得好，发展该领域，举办教学和研究研讨会，并为该领域各个分支的各级研究人员提供会议场所。将有数学家常驻，但更大的社区将通过四个讲习班合并。该奖项将仅用于支持美国研究生，博士后研究员和初级教师参加研讨会，没有其他重要支持。NSF资助的直接影响将是培训和职业发展多达30名初级研究人员，他们将有机会参加牛顿研究所的重大项目，牛顿研究所是数学领域的主要国际联系。第二个影响是进一步发展在美国和欧洲的代数拓扑和派生代数几何新兴研究小组之间的合作。四个工作坊各有不同的侧重点和风格。第一部分是关于高层结构的，重点是基础和基础的应用，并明确针对该领域的新研究人员。第二个会议是关于等变和动力同伦理论的，它将成为该领域一个至关重要和极其活跃的领域的重大研究会议。第三，关于衍生代数几何和色同伦理论，是一个新兴课题和经典领域的融合；年轻的研究人员在这一新的混合领域发挥了重要作用。最后，流形的代数拓扑已经成为同伦理论、衍生几何和数学物理的一个智力交叉点；本次研讨会旨在作为跨学科的对话，庆祝丰富的发展。所有这些研讨会的明确目标是吸纳新的声音，不仅作为听众参与者，而且作为演讲者。研讨会的网站可在https://www.newton.ac.uk/event/hhh/workshopsThis上找到奖项反映了美国国家科学基金会的法定使命，并通过基金会的知识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 travel for junior research mathematicians from the United States to participate in four workshops to be held at the Isaac Newton Institute for Mathematical Science in Cambridge, England. These workshops, all part of the semester program on Homotopy Harnessing Higher Structures are "Higher Structures in Homotopy Theory" (July 2-6, 2018), "Equivariant and Motivic Homotopy Theory" (August 13-17, 2018), "Derived Algebraic Geometry and Chromatic Homotopy Theory" (September 24-28, 2018), and "Manifolds" (December 3-7, 2018). The last fifteen years have seen a renaissance for algebraic topology. Old problems have been solved using new methods, new methods led to new ideas, new ideas to new problems, and new problems to new theorems. In each case, there was enormous progress after the introduction and the study of higher structures. The larger program at the Newton Institute will assess what has been done, highlight what is working well, develop the field, give instructional and research workshops, and provide a meeting place for researchers at all levels of the various branches of the field. There will be mathematicians in residence, but the larger community will be incorporated through the four workshops.This award will be used solely to support the workshop participation of US graduate students, postdoctoral fellows, and junior faculty without other significant support. The direct impact of NSF funding will be the raining and career development of up to 30 junior researchers, who will gain the opportunity to participate in a major program at the Newton Institute, which is a major international nexus in mathematics. A secondary impact is to further develop collaboration between emerging research groups in algebraic topology and derived algebraic geometry in the US and Europe. Each of the four workshops has a different emphasis and flavor. The first, on higher structures, is focused on the foundations and applications of foundations, and is expressly aimed at researchers new to the field. The second, on equivariant and motivic homotopy theory, is set to be a major research conference on a vital and extremely active area in the field. The third, on derived algebraic geometry and chromatic homotopy theory, is a blend of an emerging topic and a classical area; younger researchers have been important in this new mixture of fields. Finally, the algebraic topology of manifolds has become an intellectual crossroads for homotopy theory, derived geometry, and mathematical physics; this workshop is intended as cross-disciplinary dialog celebrating a wealth of developments. It is the expressed aim of all of these workshops to incorporate new voices not simply as audience participants, but as speakers as well. The website of the workshops can be found at https://www.newton.ac.uk/event/hhh/workshopsThis 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.