Special Meeting: Fields Program in Geometric Applications of Homotopy Theory - International US Participation

同伦理论的应用总是定义了这门学科最引人注目的方面。菲尔兹计划的目的是研究和发展同伦理论在代数几何、数学物理和相关学科中的新应用。单层和预层的同伦理论是基元同伦理论的基础，也是近十年来代数K-理论的主要计算成果之一。近年来，在椭圆上同调理论和拓扑模形式的定义中，出现了形式群律堆叠上的谱的预定义。单纯集及其相关的组合结构也是高级范畴理论的基础。所有这些科目在数学物理中都有当前和预期的应用。同伦理论在代数组合学和计算机科学中的新应用中也有类似的想法，特别是在超平面排列或图着色的分析中，以及在并行处理系统的并发行为的模型、计算几何和复杂性中。该计划将在菲尔兹研究所举行，该研究所将在2007年1-6月期间开展一项关于同伦理论几何应用的密集计划。NSF的资助将允许相当数量的年轻美国研究人员(博士后、初级教师和研究生)参与这一计划，并接受这一领域的培训。同伦理论是拓扑学数学学科中的一个核心问题，但如上所述，它影响着广泛的领域，包括代数几何、逻辑和计算机科学，以及数学物理(例如在拓扑量子场论中)。将同伦理论的研究人员与应用领域的研究人员聚集在一起，将刺激这两个领域的令人兴奋的进步。

The applications of homotopy theory have always defined the most compelling aspects of the subject. The purpose of the Fields program is to study and develop new applications of homotopy theory in algebraic geometry, mathematical physics and related disciplines. The homotopy theories of simplicial sheaves and presheaves are the foundation for motivic homotopy theory, and as such have contributed to the major calculational successes of the last decade in algebraic K-theory. In recent years, presheaves of spectra on the stack of formal group laws have emerged in the definitions of elliptic cohomology theories and topological modular forms. Simplicial sets and related combinatorial constructions are also the foundation for higher category theory. All of these subjects have both current and intended applications in mathematical physics. Similar ideas are present in new applications of homotopy theory in algebraic combinatorics and computer science, particularly in analysis of hyperplane arrangements or graph colouring, and in models for concurrent behaviour of parallel processing systems, computational geometry, and complexity.The program will take place at the Fields Institute, which will mount an intensive program on Geometric Applications of Homotopy Theory during the period January-June 2007. NSF funding will allow a significant number of young U.S. researchers (postdocs, junior faculty, and graduate students) to participate in this program, and receive training in this area. Homotopy theory is a core topic within the mathematical subject of topology, but as described above, it impacts on a broad range of fields, including algebraic geometry, logic and computer science, and mathematical physics (eg in topological quantum field theory). Bringing together researchers in homotopy theory with those from the areas of application will stimulate exciting progress in both fields.