Richard E. Phillips Lecture Series 2008

Abstract: The zeroes of a polynomial, and the locus of simultaneous zeros of several polynomials, has been studied for over 500 years. These loci are simplest when one uses complex numbers, compactifies, and perturbs the equations to remove singularieies. The loci are then smooth projective varieties. These are the central objects in algebraic geometry. They are studied by a combination of algebraic and geometric methods, and a central problem is their classification.

These talks will describe the progress toward classification.The first lecture will focus on the classical case of complex curves (Riemann surfaces) and some general features of the problem.The second lecture will describe the classification of two-dimensional varieties (complex surfaces) done at the beginning of the 20-th century and introduce some of the ideas of S. Mori's program for extending the classification to higher dimensions.The last talk will discuss the Mori program in more detail, ending with a recent theorem provided by C. Birkar, P. Cascini, J. M^cKernan and myself, that (in most cases) completes this program in all dimensions.

Christopher Hacon recieved his PhD from U.C.L.A. in 1998. His outstanding work in algebraic geometry has been recognized by a Sloan Fellowship (2003) and AMS Centennial Fellowship (2006). Hacon and his collaboarator M^c Kernan were recently awarded the very prestigious 2007 Clay Research Award for their work on "advancing our understanding of the birational geometry of algebraic varieties". Their theorem stating that the canonical ring of any smooth algebraic variety is finitely-generated is a major advance in mathematics.

Classifying Algebraic Varieties

Christopher Hacon

Department of Mathematics, University of Utah


Classifying Algebraic Varieties Christopher Hacon Department of Mathematics, University of Utah Abstract: The zeroes of a polynomial, and the locus of simultaneous zeros of several polynomials, has been studied for over 500 years. These loci are simplest when one uses complex numbers, compactifies, and perturbs the equations to remove singularieies. The loci are then smooth projective varieties. These are the central objects in algebraic geometry. They are studied by a combination of algebraic and geometric methods, and a central problem is their classification. These talks will describe the progress toward classification.The first lecture will focus on the classical case of complex curves (Riemann surfaces) and some general features of the problem.The second lecture will describe the classification of two-dimensional varieties (complex surfaces) done at the beginning of the 20-th century and introduce some of the ideas of S. Mori's program for extending the classification to higher dimensions.The last talk will discuss the Mori program in more detail, ending with a recent theorem provided by C. Birkar, P. Cascini, J. M^cKernan and myself, that (in most cases) completes this program in all dimensions. The dates of the lectures are: Lecture 1: Monday, March 17 4:10-5 p.m B 104 WH Lecture 2: Tuesday, March 18 4:10-5 pm A 304 WH Lecture 3: Thursday, March 20 4:10-5 pm A 304 WH Christopher Hacon recieved his PhD from U.C.L.A. in 1998. His outstanding work in algebraic geometry has been recognized by a Sloan Fellowship (2003) and AMS Centennial Fellowship (2006). Hacon and his collaboarator M^c Kernan were recently awarded the very prestigious 2007 Clay Research Award for their work on "advancing our understanding of the birational geometry of algebraic varieties". Their theorem stating that the canonical ring of any smooth algebraic variety is finitely-generated is a major advance in mathematics.
For additional information: Department of Mathematics Michigan State University East Lansing, MI 48824-1027 (517)355-9680 ginther@math.msu.edu www.math.msu.edu/Lecture_Series