The notion of a parameter or moduli space is one of the key mathematical ideas of the 20th century. In algebraic geometry, Hilbert Schemes are the basic example of such a parameter space, and serve as a starting place for many constructions of other parameter spaces, including the moduli space of curves.
The point of the talk is to go over what we mean by a parameter or moduli space, what the Hilbert scheme parameterizes, and how to construct it.
Working Group: Brill-Noether theory
Abstract:
Brill-Noether theory is the study of special linear series on curves.
A line bundle on a curve is called special if its degree puts it in the range where its behaviour is not completely determined by Riemann-Roch, and a special linear series is essentially the global sections of a special line bundle.
The special linear series on a curve control the geometry of the curve under different maps, and in some ways the special linear series on a specific curve distinguish it from all other curves of genus g.
There is a rich theory of these special series, both for a fixed curve and as the curve varies in moduli, and the idea of the working group is to explore this theory.
I think that it is a good topic because it brings in many fundamental ideas in algebraic geometry:
The end goals are the descriptions of the constructions of the various types of varieties parameterizing special linear series, as well as proving the basic theorems about their dimensions, connectedness, and behaviour on a general curve. The source material is the book by Arbarello, Cornalba, Griffiths, and Harris, as well as papers by Fulton, Lazarsfeld, and others.