Mathematical Sciences: Group Actions, Scalar Curvature and Surgery-Theoretic Problems

研究流形的对称性和流形上具有弱曲率性质的黎曼度量的存在性问题。对称性研究将包括关于低维流形的问题，以及外科理论和同伦理论在涉及高维流形的问题中的应用；后者中的大部分将涉及其奇异集具有相对较大维度的对称，并将此类行为的可能分类与奇异集维度相对较小的情况进行比较。所有这些问题都将用代数和几何拓扑学相结合的方法来研究，输入来自稳定和不稳定同伦理论、流形上的运算理论、拓扑和可微变换群以及包括规范理论在内的低维拓扑方法。这项工作的目的是更好地理解关于正曲线度量的存在以及某些对称的存在和分类的一些问题；这些问题通常是与几何拓扑学的最新进展有关的。流形在数学中是无处不在的，自然地作为高维空间中的方程组的解而产生。至于低维流形，我们生活在一维、三维或四维中，这取决于是否像考虑三个空间维度一样考虑时间维度。因此，关于流形的对称性、流形的曲率和其他性质的更多信息肯定是有用的。//

The investigator intends to study questions involving the symmetry properties of manifolds and the existence of riemannian metrics with weak curvature properties on manifolds. The symmetry investigations will include questions about low- dimensional manifolds and the applications of surgery theory and homotopy theory to problems involving higher-dimensional manifolds; much of the latter will involve symmetries whose singular sets have relatively large dimension and comparing the possible classifications of such actions with cases where the singular set dimension is relatively small. All these problems will be studied by a combination of methods from algebraic and geometric topology, with input from stable and unstable homotopy theory, the theory of surgery on manifolds, topological and differentiable transformation groups, and low-dimensional topological methods, including gauge theory. The objective of the work is an improved understanding of some questions about the existence of positively curved metrics and the existence and classification of certain symmetries; these questions have generally arisen in connection with recent advances in geometric topology. Manifolds are ubiquitons in mathematics, arising naturally as the solutions of systems of equations in higher dimensional spaces. As to low-dimensional manifolds, we live in one, three-dimensional or four-dimensional, depending on whether the time dimension is considered as well as the three space dimensions. More information about symmetries of manifolds, their curvature, and other properties is thus bound to be useful. //