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GO TO SCHEDULE/TITLE TO SEE LECTURES- Click on Title Japan-U.S. Mathematics Institute (JAMI) and Johns Hopkins University, Department of Mathematics
Conference on Asymptotic and
Effective Results in Complex Geometry
March 15-21, 2004 Organizers: J.P. Demailly, Y.T. Siu, and S. Zelditch
Alex Brudnyi - The classical Center-Focus problem posed by H. Poincare asks about the characterization of planar polynomial vector fields such that all their integral trajectories are closed curves whose interiors contain a fixed point, a center. In this talk I describe a new general approach to the Center Problem. Rob Lazarsfeld - will give an introduction to a circle of ideas involving asymptotic invariants of linear series, with a focus on the volume of a line bundle. The theme is that arbitrary big line bundles display a surprising number of properties analogous to those of ample divisors. The talk will focus on examples and open questions.
Andrew Sommese - We show how to use numerical continuation to compute the intersection $C=A\cap B$ of two complex algebraic sets $A$ and $B$. Enroute to this result, we first show how to find the irreducible decomposition of a system of polynomials restricted to an algebraic set. The intersection of components $A$ and $B$ then follows by considering the decomposition of the diagonal system of equations $u-v=0$ restricted to $\{u,v\}\in A \times B$. One offshoot of this new approach is that one can solve a large system of equations by finding the solution components of its subsystems and then intersecting these. It also allows one to find the intersection of two components of the two possibly identical polynomial systems, which is not possible with any previous numerical continuation approach. The above is joint work of Sommese with Jan Verschelde (University of Illinois at Chicago) and Charles W. Wampler (General Motors Research and Development). Jian Song - The global holomorphic \alpha-invariant introduced by Tian is closely related with the study in the existence of Kahler-Einstein metrics. We apply the result of Tian, Yau and Zelditch on polarized Kahler metrics to approximate plurisubharmonic functions and compute the \alpha-invariant of toric Fano manifolds.
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