Workshop on Homotopy theory and Derived Algebraic Geometry

2007年5月，在菲尔兹研究所举办了一个关于几何和拓扑堆栈的研讨会；通过这个研讨会，我们看到了派生代数几何这一新兴领域的快照。在一系列引人注目的演讲中，许多是由拥有相对较新的博士学位的数学家所做的，我们看到了派生方案、派生堆栈、更高范畴以及随之而来的同伦理论在广泛的几何和拓扑主题中的实施和应用。这一新的研讨会是2007年会议的后续行动：主要观点是三年后再次访问该领域，评估已经发生的情况，并看看我们将走向何方。特别是，派生代数几何及其与更高范畴理论领域的相互作用自2007年以来迅速发展，现在是代数拓扑学几个发展领域的核心。现在是探索这种相互作用的理想时刻。同意参与的研究人员包括麻省理工学院的Mark Behrens，D-C。Cisinski(巴黎13)，Ralph Cohen(Stanford)，Andre Henrique(Utrecht)，Gerd Laures(Bochum)，Tyler Lawson(明尼苏达)，Mike Mandell(印第安纳州)，Niko Naumann(Regensburg)和Charles Rezk(UIUC)。代数几何是一个经典的数学领域，起源于对多元多项式方程组解的研究。与拓扑学相比，对多项式的关注使所研究的几何对象非常严格，拓扑学是研究在任何连续变形下保持不变的现象。派生的代数几何试图将代数拓扑中的技术引入到代数几何中，以便捕捉和计算一些明显隐藏在固有刚性中的更精细的结构。最近取得了令人瞩目的成功。这笔赠款将用于资助美国研究数学家在职业生涯即将开始时参加会议，以这种方式促进这些想法在更广泛的研究社区中传播。

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.