West Coast Algebraic Topology Summer School

该奖项将资助西海岸代数拓扑暑期学校，这是2013-2016年夏季的一系列项目，针对在代数拓扑和相关领域工作的早期职业数学家。这些项目将积极吸引参与者，并在规模和规模之间交替进行。在规模更大的为期一周的项目中，目标是50到100名参与者，一个大型组委会将计划讲座，由一些参与者准备并发表演讲。每周大约只有15场讲座，参与者有时间解决组委会制定的问题，向彼此（特别是演讲者）和组织者提问，阅读辅助材料并复习自己的笔记，彼此非正式地谈论与项目相关的工作等。为期三天的小型课程将主要由来自提供实物和额外财政支持的机构的研究生组织，这些机构包括斯坦福大学、俄勒冈大学、华盛顿大学和不列颠哥伦比亚大学。2013年9月6日至8日，俄勒冈大学的课程将是关于无限范畴的，这个主题最近已经成为代数拓扑及其在代数几何和几何拓扑中的应用中一些最重要的突破的关键技术部分。在伯克利数学科学研究所即将到来的为期一年的项目中，学生组织者选择学习近年来在该学科中开发的最重要的新机器。2014年7月，在英属哥伦比亚大学为期一周的项目中（太平洋数学科学研究所提供了大量额外的资金和实物支持），该项目将研究拓扑场论。这些理论的发展需要数学和物理社区之间的相互作用，以及数学内部代数拓扑学家、几何拓扑学家、微分几何学家和表示理论家之间的相互作用。一个旨在解决理论的不同方面而又连贯的程序将帮助一代研究人员与一个非常富有成果的子领域相结合。2015年和2016年的主题将继续根据其重要性和时效性以及与其他领域的联系来选择。2016年的项目将集中于色同伦理论，而确定2015年项目的主题将涉及参与者。在所有这些项目中，我们将有额外的专业学习机会，以满足早期职业数学家作为教师发展和职业发展的需求。例如，在研究早期宇宙的结构，设计具有最少数量控制器的机械臂，以及在新型数据分析中，拓扑知识是必不可少的。作为形状的基础研究，代数拓扑学在数学的其他领域有着重要的分支，从基本的计数问题（组合学）到方程的解（代数），再到微积分的应用和扩展（分析）。西海岸代数拓扑暑期学校选择的主题与这些联系在一起。早期职业数学家非常需要研究水平的专业学习，这可以开辟新的研究途径，也非常需要扩大他们的专业网络，这可以开辟新的合作。如果没有这样的早期职业机会，研究项目可能会缩小或完全停滞不前。对于美国西半部的早期职业数学家来说，这些需求可能更为突出，因为那里在地理上比其他地区更孤立，对于来自代表性不足的群体的数学家来说，他们被积极招募为参与者。这些暑期学校建立在一系列成功项目的基础上，这些项目在学习的深度和培养的专业关系方面都超出了预期。许多参与者报告说，这些经历是变革性的，我们希望这些经历在2013-2016年的系列中继续存在。这些经历的主动学习方面不仅为参与者的研究计划提供了信息，也为他们的教学方法提供了信息。因此，这个项目的影响超越了数学和科学研究，延伸到了高等教育甚至K-12教育。

This award will fund the West Coast Algebraic Topology Summer School, a series of programs in the summers from 2013-2016 aimed at early-career mathematicians who work in algebraic topology and related areas. The programs will actively engage the participants, and alternate between larger and smaller scale. In the larger week-long programs, aiming for between 50 and 100 participants, a large organizing committee will plan lectures which some participants will prepare and then present. There will only be about 15 lectures for the week, leaving time for participants to work on problems formulated by the organizing committee, to ask questions of each other (especially the presenters) and the organizers, to read supporting materials and go over their own notes, to talk informally with each other about their work related to the program, etc. The smaller three-day programs will be primarily organized by graduate students from institutions providing both in-kind and additional financial support, namely Stanford University, the University of Oregon, the University of Washington, and the University of British Columbia. In September 6-8, 2013 at the University of Oregon the program will be on Infinity Categories, a topic which has recently become a key technical piece of some of the most significant recent breakthroughs in algebraic topology and its applications to algebraic geometry and geometric topology. With the upcoming yearlong program at the Mathematical Sciences Research Institute in Berkeley, the student organizers have chosen to learn about arguably the most important new machinery developed in the subject in recent years. In July 2014, the week-long program at the University of British Columbia (with significant additional financial and in-kind support from the Pacific Institute for the Mathematical Sciences), the program will be on Topological Field Theories. Development of these theories requires interplay both between the mathematics and physics communities, and within mathematics between algebraic topologists, geometric topologists, differential geometers, and representation theorists among others. A program designed to address the different aspects of the theory while being coherent will aid a generation of researchers in interfacing with a remarkably fruitful subfield. The topics in 2015 and 2016 will continue to be chosen based on their importance and timeliness and connections with other fields. The 2016 program will likely focus on Chromatic Homotopy Theory, while determining the topic for the 2015 program will involve the participants. At all of these programs, we will have additional opportunities for professional learning addressing needs for early-career mathematicians to develop as teachers and to position themselves for career growth.Knowledge of topology is essential for example in studying the structure of the early universe, in designing a robot arm with a minimal number of controllers, and in new kinds of data analysis. In its role as a basic study of shape, algebraic topology has significant ramifications in other areas of mathematics, from basic counting problems (combinatorics) to solutions of equations (algebra) to applications and extensions of the calculus (analysis). The West Coast Algebraic Topology Summer School chooses topics with such connections in mind. Early career mathematicians have a great need for research-level professional learning, which can open up new research pathways, and for enlarging their professional networks, which can open up new collaborations. Without such early-career opportunities, research programs can narrow or stall out altogether. These needs can be more prominent for early career mathematicians in the western half of the United States, which is more geographically isolated than other regions, and for mathematicians from under-represented groups, who are actively recruited to be participants. These summer schools build on a series of successful programs which have exceeded expectations in both the depth of learning and the strength of professional relationships fostered. Many participants have reported them to be transformative experiences, and we expect those kinds of experiences to continue in our series from 2013-2016. The active learning aspects of these experiences inform not only the research programs of participants but their approach to pedagogy as well. Thus the program's impact goes beyond mathematical and scientific research to postsecondary and even K-12 education.