CalabiYau 3folds are a special kind of 6dimensional curved space, with a lot of geometrical structure. They are of great interest to mathematicians working in Algebraic and Differential Geometry, and to physicists working in String Theory. The greatest problem in fundamental physics is to find a single theory which successfully combines Einstein's General Relativity  the physics of very large things, such as galaxies  and Quantum Theory  the physics of very small things, such as atoms. String Theory is the leading candidate for doing this. It predicts that the dimension of spacetime is not 4 (3 space plus one time), but 10. The extra 6 dimensions are rolled up in a CalabiYau 3fold, with very small radius. So according to String Theory, CalabiYau 3folds describe the vacuum of space itself. Using physical reasoning, String Theorists made extraordinary mathematical predictions about CalabiYau 3folds, known as Mirror Symmetry , which have been verified in many cases, and cause much excitement among mathematicians. Mirror Symmetry says that two quite different CalabiYau 3folds X, X* can have identical Quantum Theories (so far, this is not a welldefined idea), and when this happens, we can set up a correspondence between aspects of the geometry of X and X* which affect their Quantum Theories. Often these correspondences relate objects which seem quite different  a nonmathematical analogue would be to conjecture a onetoone correspondence between giraffes in Kenya and bananas in Zambia. One chapter of the Mirror Symmetry story which is still work in progress relates two kinds of invariants on X and X*: the DonaldsonThomas invariants of X, which are numbers counting algebraic objects called coherent sheaves on X, should be equal to other invariants counting special Lagrangian 3folds on X*. Special Lagrangian 3folds are nonalgebraic objects, superficially as different from coherent sheaves as giraffes are from bananas. When mathematicians talk about invariants they mean a number, such as 42, computed by counting some kind of geometric object, which has the important property that you can make big changes to the underlying geometry, but for mysterious reasons, the number remains the same. This invariance property makes mathematicians very excited (perhaps we should get out more?) as it suggests there is some underlying mathematical reality which is independent of these big changes, which we don't yet understand, and we want to know what it is. DonaldsonThomas invariants have this kind of invariance property. Funded by another EPSRC grant, the Principal Investigator has recently proved that if we deform a different part of the geometry of the CalabiYau 3fold, DonaldsonThomas invariants are not fixed, but change by a rigid wallcrossing formula . That is, when we cross a wall in the space of CalabiYau 3folds, the DonaldsonThomas invariants on one side of the wall can be written as sums of products of DonaldsonThomas invariants on the other side. The goal of this project is to prove some conjectures which will first help to explain this wallcrossing formula, and secondly allow us to generalize DonaldsonThomas invariants to a larger class of new invariants containing much more information, which will also satisfy a wallcrossing formula of a similar shape. It turns out that a very nice way of understanding multiplicative properties of invariants, such as DonaldsonThomas invariants, is to encode them in an algebra morphism from a very large universal algebra , which is far too big to understand or compute, to a much smaller, explicit algebra, where the invariants take their values. Previous work by the Principal Investigator constructed a Lie algebra morphism from a subspace of the universal algebra. We want to extend this to an algebra morphism on the full universal algebra, and generalize it to morphisms to some larger explicit algebras.
