The proposal is directed towards the geometry underlying the Langlands program. A major and expanding influence are the ideas from quantum field theory. The ``Critical Quantization'' program aims to develop a formalism, that would unify approaches to modular representation theories and to representations of affine Lie algebras at a critical level, and should have wide including the study of derived categories of coherent sheaves. The ``Quadratic Koszul Duality'' project attempts to give a geometric approach to a family of dualities that appear throughout representation theory, culminating with the Cherednik Fourier transform. Finally, we propose certain ``completely algebro-geometric'' approach to Class Field Theory and Langlands program.
The project aims towards relating and unifying developments in mathematics (representation theory, algebraic geometry, number theory) and theoretical particle physics (quantum field theory and in particular string theory). The main development in mathematics in the last decade was the influence of predictions derived from Quantum Field Theory. This has resulted in some inroads and I expect the direction to reverse with mathematics contributing essentially to the understanding of particle physics and the ``Grand Unification'' project of giving a consistent description of known physical forces. The main goal of the project is to take certain steps in this direction by explaining mathematically certain elements of String Theory and Quantum Field Theory in general.
