Mathematical Sciences: "Cohomological Topics in Finite Transformation Groups and Applications"

对称性在数学、物理和其他科学中扮演着重要的角色。在数学中，对称有许多不同但又相互关联的表现形式。空间X的有限对称群G诱导出X的代数不变量上的对称性，从而使变换群与表示论紧密地联系在一起。在这个项目中，研究人员将系统地研究对称几何(例如，不动点、稳定性子群等)之间的相互作用。以及这些表示的代数不变量。特别是，摩尔空间的对称性(S在1960年初提出的Steenrod问题)和代数几何和数论(除拓扑学外)中出现的代数曲线和曲面提供了下列一般理论将适用的具体问题。这个项目的一个重要特点是它的跨学科性质，这需要几个数学分支的工具和想法。这为这些领域的相互作用提供了一个自然的环境，包括代数几何在变换群和模表示理论中的新应用。其他应用包括(算法和归纳)构造重要的模表示类，代数函数域的数论性质，以及对代数曲面相对于4维流形的对称性的更深层次的理解。一个长期的目标是解决长期存在的Steenrod问题及其在流形对称性存在问题中的应用。

Symmetry plays an important role in mathematics, physics, and other sciences. In mathematics, symmetry has many different but related manifestations. A finite group G of symmetries of a space X induces symmetries on algebraic invariants of X. In this way, transformation groups and representation theory become intimately related. In this project, the investigator will look systematically at the interaction between the geometry of the symmetry (e.g., fixed points, stability subgroups, etc.) and the algebraic invariants of such representations. In particular, symmetries of Moore spaces (the Steenrod Problem posed in early 1960's) and algebraic curves and surfaces arising in algebraic geometry and number theory (besides topology) provide concrete problems to which the following general theory will apply. A significant feature of this project is its interdisciplinary nature, which requires tools and ideas from several branches of mathematics. This provides a natural setting for interaction of these areas, including new applications of algebraic geometry to transformation groups and modular representation theory. Other applications include (algorithmic and inductive) construction of important classes of modular representations, number theoretic properties of algebraic function fields, and a deeper understanding of symmetries of algebraic surfaces versus 4- dimensional manifolds. A long term objective is solution of the long-standing Steenrod Problem and its application to the problem of existence of symmetry for manifolds.