Abelianization of Connections in Two and Three Dimensions

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The PI studies problems of geometry using methods imported from particle physics. In joint work with his collaborators, he has recently developed a new geometric technique of "abelianization" -- so called because it reduces nonabelian problems (involving operations for which the order in which we do the operations matters) to simpler abelian ones (where the order does not matter). The PI, together with his collaborators and graduate students, will work on several new applications of abelianization. One application is a new approach to solving certain differential equations, including the Schrodinger equation which governs the physics of some quantum systems. A second application is a new way of measuring the topology of 3-dimensional spaces. The results of this work will be disseminated broadly both in the mathematics and high-energy physics communities, helping to bring these two areas closer together. The work will also contribute to the training of graduate students in both fields.The PI's recent joint work with collaborators introduced a new ingredient to the theory of flat connections: a way of "abelianizing" flat connections on a rank N complex vector bundle over a surface, replacing them by almost-flat connections on a line bundle over an N-fold branched covering surface. The full scope of this new theory is not yet known: it appears that there are many more uses of abelianization yet to be discovered. The PI aims to develop some of these. First, he will study a family of special connections on surfaces called "opers," which can be abelianized in a canonical way. On the one hand, this is a warmup for the abelianization of the twistor lines in the Hitchin system. On the other hand, it gives a new way of understanding the locus of opers and thus a new perspective on many related issues, from the classical theory of linear scalar differential operators to nonperturbative extensions of topological string theory. Second, he will consider abelianization on a 3-manifold instead of a surface. One immediate application is the development of new formulas for classical complex Chern-Simons invariants. Third, the PI aims to develop a new relation between abelianization and Floer theory on cotangent bundles.