Topological Quantum Field Theory in dimensions two and three

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The proposed research is on Topological Quantum Field Theory (TQFT) in dimensions two and three. TQFTs produce topological invariants of geometric objects such as knots and 3-dimensional manifolds. We shall study applications of TQFTs to classical topological problems such as finding whether a given Serre fibration over a surface (or more, generally, over a 2-dimensional orbifold) has a section. This will require a careful study of Homotopy QFTs introduced by the author in 1999. In another direction, we will give a general algebraic framework for 3-dimensional state-sum TQFTs and connect the latter to 3-dimensional surgery TQFTs as follows: the state sum TQFT derived from a finite semisimple monoidal category C is isomorphic to the surgery TQFT derived from the Drinfeld double of C. The proof will use the technique of bottom tangles due to Bruguieres-Virelizier-Habiro.Quantum physics has invariably served as a source of new mathematical insights and techniques. One such insight introduced about 20 years ago by the Fields medalists E. Witten and M. Atiyah is called Topological Quantum Field Theory. It allows us to approach real-world geometric objects such as surfaces of solid bodies or knotted circles in the space from a new, physical perspective. We will use this viewpoint to address certain geometric problems concerning the surfaces. In particular, we will answer several classical questions raised well before the introduction of Quantum Field Theory. We will also study a connection between mathematical techniques arising from Quantum Field Theory and from Quantum Gravity. This study will involve deep algebraic techniques introduced by another Fields medalist V. Drinfeld. This project will exhibit new facets of the fundamental connection between physics, geometry, and algebra.