Geometry and Physics

提案dms -0305505标题：“几何与物理”摘要本基金资助了三位独立研究人员。教授和其他合作者首先研究扭曲k理论和环群，其次是微分k理论和狄拉克算子的几何不变量，最后是m理论和弦理论中的异常。hausel教授研究了四元数几何中的新数学，这些数学与复杂几何中的已知数学相对应，如希格斯束的模空间、平方可积调和形式、圆紧流形上的积分和超平面排列的面向量。乌伦贝克教授有两个关于双曲方程的项目，一个是寻找协变导数波算子的基本解，另一个是研究二维空间中的波映射爆炸。她还研究了TCFT中的Virasoro动作。这三位受资助的研究人员是一个大型几何小组的一部分，他们的研究项目涵盖了许多起源于高能理论物理或工程物理的几何主题。在数学方面，该小组的工作不仅限于几何分析，还涉及代数几何、几何拓扑、代数拓扑、表示理论、谐波分析和数学物理。小组研究的关键是数学、物理和生物学背景不同的数学家之间的合作。

Proposal DMS-0305505Title: "Geometry and Physics"ABSTRACTThere are three individual researchers supported on this grant. ProfessorFreed and various coauthors study first twisted K-theory and loop groups,secondly differential K-theory and geometric invariants of Dirac operators,and finally anomalies in M-theory and string theory. Professor Hauselresearches new mathematics in quaternionic geometry which corresponds toknown mathematics in complex geometry, such as the moduli space of Higgsbundles, square integrable harmonic forms, integration on circle compactmanifolds and face vectors of hyperplane arrangements. Professor Uhlenbeckhas two projects on hyperbolic equations, one of finding fundamentalsolutions of covariant-derivative wave operators and one studying wave mapblow-up in two space dimensions. She also studies Virasoro actions in TCFT.The three supported researchers are part of a large geometry group whoseresearch projects span many topics in geometry which originate in high energytheoretical physics or engineering physics. In mathematics proper the workof the group is not limited to geometric analysis, but also involvesalgebraic geometry, geometric topology, algebraic topology, representationtheory, harmonic analysis, and mathematical physics. Essential to the groupresearch are collaborations among mathematicians with diverse backgrounds inmathematics, physics and biology.