The symposium is supported by the NSF, Rice University and the University of Texas Austin. Everyone is welcome. There is no registration fee. We offer housing, but no other travel support. If you would like a room, for friday or saturday night, please send email to Sean Keel, before Feburary 15, 2007.
Abstract: I shall explain some of Joyce's recent preprint hep-th/0507039 about holomorphic generating functions formed from invariants counting semistable objects in an abelian category relative to a given stability condition.
Lawrence Ein: Valuations via arcs.
Abstract: We show that we can associate every divisorial valuation over an algebraic variety with an irreducible closed subset in the arc space of the variety. We show that each valuation can be defined by a finite number of equations, though the graded ring associated with the valuation may not be finitely generated.
Yi Hu: Compactification of the space of algebraic maps from P^1 to P^n.
Abstract: We procide a nonsingular projective compactification of the space of algebraic maps from P^1 to P^n by adding a divisor with simple normal crossings. We then calclate its Hodge polynomial in both recursive and closed forms. The compactification provides an alternative to Kontsevich's stable map compactification, and in fact, we expect our compactification is also a coarse moduli space of some moduli functor.
Daniel Huybrechts: Derived equivalencs of K3 surfaces and orientation.
Abstract: I shall report on work in progress with Macri and Stellari showing that any autoequivalence of the bounded derived category of a K3 surface induces an orientation preserving Hodge isometry.
János Kollár: Quotients of Calabi-Yau Varieties.
Abstract: (Joint with Michael Larsen) Let X be a complex Calabi-Yau variety, that is, a complex projective variety with canonical singularities whose canonical class is numerically trivial. Let G be a finite group acting on X and consider the quotient variety X/G. The aim is to determine the place of X/G in the birational classification of varieties. That is, we determine the Kodaira dimension of X/G and decide when it is uniruled or rationally connected. We also give a rough classification of possible stabilizer groups which cause X/G to have Kodaira dimension -\infty. These stabilizers are closely related to unitary reflection groups.
Rob Lazarsfeld: Local syzygies of multiplier ideals.
Abstract: Given a polynomial f and a weighting coefficient c > 0, one can construct the multiplier ideal J(F^c) of f with coefficient c. Thanks to the vanishing theorems they come with, these multiplier ideals hav found many applications in local and global algebraic geometry. Multiplier ideals are always integrally closed, but up to now they have not been known to satisfy any other local properties. I will discuss some joint work with K. Lee showing that there are some surprising restrictions on the higher syzygies of any multiplier ideal. It follows in particular that in dimensions at least three, many integrally closed ideals do not arise as multiplier ideals.