Mathematical Sciences: 4-Manifolds, Groups Actions, and Scalar Curvature

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The main purpose of this research project is to study symmetries of manifolds, with particular attention paid to symmetries of 4-dimensional manifolds. The problems to be studied vary from general questions concerning the geometric structure of exotic space forms and 4-dimensional manifolds to many specific problems connected with the existence of symmetric and asymmetric manifolds. The broad spectrum of methods which will be employed in this research include the techniques of surgery and homotopy theory, transformation groups, differential geometry and gauge theory. Since manifolds are very natural geometric objects, including spheres and tori, for example, and since symmetry is a very natural geometric property, techniques for treating symmetry of manifolds are important to cultivate.