Stability conditions on derived categories

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Calabi-Yau 3-folds are a special kind of 6-dimensional 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 space-time is not 4 (3 space plus one time), but 10. The extra 6 dimensions are rolled up in a Calabi-Yau 3-fold, with very small radius. So according to String Theory, Calabi-Yau 3-folds describe the vacuum of space itself. Using physical reasoning, String Theorists made extraordinary mathematical predictions about Calabi-Yau 3-folds, known as Mirror Symmetry , which have been verified in many cases, and cause much excitement among mathematicians. Mirror Symmetry says that two quite different Calabi-Yau 3-folds X, X* can have identical Quantum Theories (so far, this is not a well-defined 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 non-mathematical analogue would be to conjecture a one-to-one correspondence between giraffes in Kenya and bananas in Zambia. The Homological Mirror Symmetry Conjecture is a mathematically precise version of part of Mirror Symmetry. It says that two mathematical structures T and T* associated to X and X*, called triangulated categories , should be the same.This research project concerns the existence of stability conditions (Z,P) on a triangulated category T, an enhancement of the structure of T. Our major goal is to construct examples of stability conditions (Z,P) on one side of the mirror symmetry picture. We can then use stability conditions to define interesting invariants -- numbers -- of semistable objects in T. Without using a stability condition to reduce the number of objects, there is no way to define sensible numbers of objects in T, as the number would be infinite. Knowing about such stability conditions would enable us to state a more powerful version of the Homological Mirror Symmetry Conjecture: the triangulated categories T, T* of X, X* should have stability conditions (Z,P) and (Z*,P*), and the mirror map from T to T* should identify (Z,P) and (Z*,P*). It follows that the invariants -- numbers of semistable objects -- we can compute on each side should be identified under the mirror map. That is, the new conjecture predicts that some numbers of geometric objects on X, which we can hopefully compute, should be the same as some numbers of quite different geometric objects on X*, which again we can hopefully compute. This could be checked in examples.