I will be speaking at the Young Researchers in Mathematics conference, at Warwick, Saturday 16th April, 10:15-11:15. A few weeks ago I provided a short, workmanlike abstract:
We discuss the role of topology in algebraic geometry. There are simple but surprising arguments which show that many properties of a complex submanifold of projective space are controlled by its topology, in fact by the second homology group. More subtle properties are determined by a convex cone, the “cone of curves”. For algebraic surfaces, the cone of curves can be studied using hyperbolic geometry.
This description now looks a little dull to me. So why, if you’re a young researcher in mathematics, should you come to my talk? Here’s an abstract that describes the point of the talk rather than just summarizing its technical content:
In algebraic geometry, questions — e.g. can I deform this curve? — seem to have stark yes-or-no answers. This binary landscape is far from the real picture. Topology — in particular, obstruction theory — gives us a language and framework for more calibrated answers. Using topology, when the answer is no, we can describe how far from yes it is. When the answer is yes, we can understand why and by how much. Obviously, this is useful if you’re an algebraic geometer. If you’re not an algebraic geometer, the example of topology’s value in algebraic geometry may be suggestive of the benefits for you of knowing some topology.