Representations of algebraic groups over 2-dimensional fields, G-bundles on surfaces and mathematical physics

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The PI plans to study several problems related to algebraic groups over 2-dimensional local and global fields and their representations. More specifically the PI would like to introduce and study the notion of Hecke eigen-forms for 2-dimensional semi-global fields. In the second part of his project the PI suggests to attack several (mathematically well-posed) problems related to 4-dimensional gauge theory. Both subjects can be reformulated in terms of various questions about G-bundles on algebraic surfaces. The PI believes that above questions may also be connected to the (not yet formulated) 2-dimensional geometric Langlands duality.The research lies on the border of such fields as number theory, representation theory and mathematical physics; sucesful implementation of the project might shed some new light on the connection between these fields. For example, number theory is perhaps one of the oldest mathematical subjects and one of its most important parts is called the Langlands program. Recently it has been realized that geometric aspects of the Langlands program have many connections with modern mathamtical physics (such as 4-dimensional quantum field theory). The research project should confirm the existence of such links as well broaden and generalize them.