Prof. K.R. Nagarajan's 80th Birthday Meet

CMI, Monday, December 9, 2013

Programme

• 12.45–14.00: Lunch

• 14.00–15.00: Hilbert coefficients of local rings by Krishna Hanumanthu, CMI

  Abstract:

  Hilbert functions of graded commutative rings are interesting and important objects of study. They are related to classical geometric invariants, such as the dimension and degree of a projective variety. Their study naturally extends to local rings, where we look at the Hilbert functions of the associated graded algebras.

  We consider a Cohen-Macaulay local ring and study the Hilbert functions and coefficients associated to them. There are several interesting questions on these coefficients, including bounds on them. We will present some results obtained in a joint work with Craig Huneke.

• 15.00–15.15: Tea Break

• 15.15–16.15: Higgs bundles on Elliptic surfaces by Rohith Varma, CMI

  Abstract:

  Consider triples (X,C,f) where X is a smooth projective surface (complex), C is a smooth curve and f: X → C is a surjective map, whose general fibers are elliptic curves. It is known that in the case of Euler characteristic of X positive, we have an isomorphism of the fundamental group of X and orbifold fundamental group of C (the orbifold stucture is over the points where f has multiple fibers). Hence the representation spaces (into GL_n) of the two groups are naturally isomorphic (as complex analytic spaces). Both these spaces following the work of Hitchin, Simpson et al can be identified (topologically) as certain moduli spaces of Higgs bundles on X and moduli spaces of parabolic bundles on C. The aim of the talk is to demonstrate an algebraic geometric correspondence between these moduli spaces under certain assumptions on X.

• 16.15–17.00: Felicitations

• 17.00–17.30: High Tea