Low-Dimensional Topology and Fundamental Groups

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The PI will study problems in low-dimensional topology centered around the impact of the fundamental group on the geometric properties of a low dimensional manifold. Three general directions of investigation will be taken: an asymptotic geography problem for symplectic and smooth 4-manifolds with prescribed fundamental groups, surgery properties of the SU(3) Casson invariant, and the homotopy properties of the Atiyah-Patodi-Singer invariant associated to a unitary representation of the fundamental group of a manifold.The subject of topology is the study of properties of space that are unchanged under deformation or stretching. The fundamental group is a simple to define but very powerful mathematical object that measured how circles can be deformed in different spaces. Themost challenging spaces to study are the "low dimensional manifolds", i.e. those which have dimension 3 or 4, like the universe we live in. The PI proposes to study these using the fundamental group and its relation to the geometry and analysis of of a space.