Topological Applications of Floer Theory

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Gauge theory is the study of a particular kind of non-linear differential equations, that originally appeared in quantum physics. In mathematics, the solutions to these equations can be used to understand the topology (the shape) of the underlying space. When studying three-dimensional shapes, the information from gauge theory is encoded into an algebraic structure called Floer homology. The project aims to study different versions of Floer homology and their applications to topology. In particular, although Floer homology is associated to three-dimensional spaces, by an indirect route it can give insights into the triangulations of spaces of dimension five or higher; classifying these triangulations is a major open problem. A triangulation is a decomposition of the space into polyhedra, and gives a simple combinatorial description of the space. The project also aims at constructing new variants of Floer homology, with inspiration drawn from recent advances in quantum physics. In a different direction, through collaboration with computer scientists, it is proposed to apply topology to the classification of distributed computing models.The project concerns Floer theory and its applications to the study of both low dimensional and high dimensional manifolds. In particular, the PI will investigate the Seiberg-Witten Floer stable homotopy types of three-manifolds and the associated Pin(2)- equivariant Seiberg-Witten Floer homology. These theories can be used to get information about the homology cobordism group in three dimensions. In turn, homology cobordism gives insight into the classification of triangulations for manifolds of dimension at least five. Similar methods will be used to study the intersection forms of spin four-manifolds. The PI will also work on constructing Pin(2) versions of Heegaard Floer homology and knot Floer homology, and on the development of new computational techniques for Heegaard Floer homology. Furthermore, in an ongoing collaboration with computer scientists, the PI is exploring applications of topology to distributed computing; there, the solvability of a given task by a system of several computers can be rephrased in terms of a question in homotopy theory.