Topological Quantum Field Theory studies maps from Riemann surfaces to targets - algebraic stacks. Homological Mirror Symmetry allows us to look deeper into geometry of these stacks and algebraic varieties. In particular this project studies two main questions:
1. Rationality and unirationality of algebraic varieties. Program maximum in this direction is understanding non-rationality of generic four dimensional cubic and non-unirationality of generic three dimensional quartic.
2. Theory of algebraic cycles - computations of Chow and Griffiths groups, integral and Hodge cycles. Program minimum in this direction is understanding Bloch's conjecture for Chow groups of surfaces of general type and Griffiths groups of so called Fano - Calabi Yau manifolds.
By now Topological Quantum Field Theory and Homological Mirror Symmetry have been established as closed and beautiful mathematical parts of String theory. Regardless of the success of the experiments in the high energy collider in CERN and confirmation of String Theory as a theory of everything, one can still use Topological Quantum Field Theory and Homological Mirror Symmetry in order to get nice mathematical consequences. In particular one can use them to get answers of some long standing questions in Algebraic geometry.
