Workshop on Homotopy theory and Derived Algebraic Geometry

2007年5月，菲尔兹研究所举办了一个关于几何和拓扑堆栈的研讨会；通过这次研讨会，我们看到了派生代数几何这个新兴领域的一个快照。在一系列引人注目的演讲中，许多是由具有相对较新博士学位的数学家进行的，我们看到了派生方案、派生堆栈、更高类别以及伴随而来的同伦理论在广泛的几何和拓扑学科中的实现和应用。这次新的研讨会是2007年会议的后续活动：主要目的是在三年后重新审视这个领域，评估发生了什么，看看我们要去哪里。特别是，自2007年以来，衍生代数几何领域及其与高范畴理论领域的相互作用迅速发展，现在是代数拓扑几个发展领域的核心。现在是探索这种相互作用的理想时机。同意参与的研究人员包括马克·贝伦斯（麻省理工学院），华盛顿特区。西辛斯基（巴黎13号）、拉尔夫·科恩（斯坦福大学）、安德烈·亨利克斯（乌得勒支）、格德·劳伦斯（波鸿）、泰勒·劳森（明尼苏达州）、迈克·曼德尔（印第安纳州）、尼科·瑙曼（雷根斯堡）和查尔斯·雷兹克（UIUC）。代数几何是数学的一个经典领域，起源于对多变量多项式方程组解的研究。对多项式的关注使得所研究的几何对象非常刚性，而拓扑学研究的是在任何连续变形下保持不变的现象。衍生代数几何试图将代数拓扑技术引入到代数几何中，以捕获和计算一些明显被固有刚性隐藏的更精细的结构。最近取得了显著的成功。这笔拨款将用于资助美国研究数学家在其职业生涯初期的出席，以这种方式促进这些思想在更广泛的研究社区中的传播。

In May 2007 there was a workshop at the Fields Institute on stacks in geometry and topology; with this workshop, we saw a snapshot of the emerging field of derived algebraic geometry. In a remarkable series of talks, many by mathematicians with relatively recent PhDs, we saw the implementation and application of derived schemes, derived stacks, higher categories, and the attendant homotopy theory across a broad spectrum of geometric and topological subjects. This new workshop is a follow-up to the 2007 conference: the main point is to revisit the field three years later, to assess what has happened and to to see where we are going. In particular, the field of derived algebraic geometry and its interplay with higher category theory field has grown rapidly since 2007 and is now central to several developing areas of algebraic topology. It is an ideal moment to explore this interplay. Researchers who have agreed to participate include Mark Behrens (MIT), D-C. Cisinski (Paris 13), Ralph Cohen (Stanford), Andre Henriques (Utrecht), Gerd Laures, (Bochum), Tyler Lawson (Minnesota), Mike Mandell (Indiana), Niko Naumann (Regensburg), and Charles Rezk (UIUC).Algebraic geometry is a classical field of mathematics, arising from the study of solutions of systems of polynomial equations in many variables. The focus on polynomials make the geometric objects studied very rigid, in contrast to topology, which is the study of phenomena which remain unchanged under any continuous deformation. Derived algebraic geometry seeks to import techniques from algebraic topology into algebraic geometry in order to capture and calculate some of the finer structure apparently hidden by the inherent rigidity. There have been remarkable recent successes. This grant will be used to fund the attendance of US research mathematicians near the beginning of their careers, in this way promoting the spread of these ideas among the broader research community.