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个问题最引人注目的方面。菲尔兹计划的目的是研究和开发同伦理论在代数几何，数学物理和相关学科中的新应用。单纯层和预层的同伦理论是动机同伦理论的基础，因此在过去的十年里，它为代数K理论的主要计算成功做出了贡献。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栈上的谱的预层。单纯集和相关的组合构造也是高级范畴理论的基础。所有这些主题在数学物理中都有当前和预期的应用。类似的想法也出现在同伦理论在代数组合学和计算机科学中的新应用中，特别是在超平面排列或图形着色的分析中，以及在并行处理系统的并发行为模型、计算几何和复杂性中。该课程将在2007年1月至6月期间进行同伦理论的几何应用的强化课程。美国国家科学基金会的资助将使大量年轻的美国研究人员（博士后，初级教师和研究生）参加这一计划，并接受这一领域的培训。同伦理论是数学拓扑学中的一个核心主题，但如上所述，它对广泛的领域产生了影响，包括代数几何，逻辑和计算机科学，以及数学物理（例如拓扑量子场论）。把同伦理论的研究人员与应用领域的研究人员聚集在一起，将刺激这两个领域令人兴奋的进展。

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.