Aberdeen Algebraic Geometry Group
Members
Academic staff
Postgraduate students
Research Interests
Derived categories of coherent sheaves (Bondal). A recent trend in algebraic geometry is to ask how geometric properties of algebraic varieties are reflected in homological properties of their derived categories of coherent sheaves. It was shown for example that the smoothness of an algebraic variety is related to the existence of a so-called strong generator in the corresponding derived category (Bondal and Van den Bergh). Bondal's research also explores how geometric properties of maps between algebraic varieties are encoded in the corresponding functors between derived categories. This has generated a new homological approach to the minimal model program in birational algebraic geometry. Future work may focus on extending the theory to include the derived categories associated with complex-analytic varieties. Homological properties of various classes of algebraic varities like Fano and Calabi-Yau varieties will be investigated using the insight given by mirror symmetry, a deep relation between complex-analytic and symplectic geometry.
Quantum field theories, string theory (Gorbounov, Shaikh). String theory is a quantum theory of gravity. A quantum field theory is something like an "explicit solution" of the equations or postulates of string theory. A programme outlined by Witten in the 1980's suggested that quantum field theories should be studied by means of certain reductions. One of these leads to topological quantum field theories, and this has been the subject of some spectacular research over the last 25 years. The other suggested reduction leads to "half twisted quantum field theories". In particular the so-called sigma-model associated with a manifold (an important type of quantum field theory) leads after half-twisted reduction to a family of vertex algebras (in their own right, algebraic approximations to quantum field theories), parametrised by the manifold. The relation of this family of vertex algebras to the geometry and topology of the manifold is very deep and is currently the subject of intensive research. Among the manifestations of this deep relationship is the "elliptic genus" which in quantum theory language is the partition function of the quantum field theory associated with the manifold. Such an interpretation uncovers unexpected properties of the elliptic genus. A major goal of this research is to use a quantum theoretic point of view to study other classical geometric invariants of manifolds.