Hodge theory and the Hodge conjecture

Abstract: The Hodge conjecture is probably the most important open question in the field of algebraic cycles -- and it comes with a fairly big prize. However, the Lefschetz 1-1 theorem is still the main piece of evidence for the conjecture. The 1-1 theorem is the Hodge conjecture in dimension 2. While there is a very easy proof of it using elementary sheaf theory, Lefschetz's original idea was to use the theory of normal functions. I will discuss this idea, how hard it is to make it work, why it does not work in higher dimensions, and some new ideas due to M. Green and P. Griffiths for relating normal functions to the Hodge conjecture. I will also discuss recent work by H. Fang, Z. Nie, G. Pearlstein and myself fleshing out these new ideas.

Refreshments will be served at 2:45 p.m. (Math Lounge, MATX 1115).