Schedule.html

May 18	11:00-12:00 W. Fulton Ehrhart polynomials and toric varieties	LUNCH	14:00-15:00 M. Haiman Macdonald Polynomials.	Coffee Break	15:30-16:30 L. Gatto Schubert Calculus on Grassmann Algebras.
May 19	10:00-11:00 K. Ranestad An abelian fibration on the Hilbert scheme of degree 3 subschemes in a K3 surface of genus 9.	Coffee Break	11:30-12:30 T. Johnsen Schubert unions in flag varieties.	LUNCH	15:00-16:00 R. Piene TBA	DINNER
May 20	9:30-10:30 S.A. Stromme TBA	Coffee Break	11:00-12:00 T. Gustavsen The deformation relation on the set of Cohen-Macaulay modules on a quotient surface singularity.	LUNCH	13:30-14:30 E. Sernesi The rational connectedness of M_{15}.	P. Sarnak 15:30 E 1

W. Fulton. Ehrhart polynomials and toric varieties.

Ehrhart polynomials count the number of lattice points in polytopes and their integral multiples. We will discuss some old and new results and conjectures about them, especially as they relate to the geometry of toric varieties.

M Haiman. Macdonald Polynomials.

The classical Weyl character formula has 'q-analogs' which provide links between representation theory, geometry and the combinatorics of symmetric functions. In 1988, Macdonald discovered 'q,t-analogs' involving an extra parameter. Macdonald's functions appear in geometric and representation-theoretic contexts not visible from the one-parameter q-theory. The talk will be an introduction to some of these developments, with hints about tantalizing combinatorial aspects of Macdonald theory that we are just beginning to understand.

L. Gatto. Schubert Calculus on Grassmann Algebras.

The Grassmann algebra of a finitely generated free module of rank $n$ is a module over a natural commutative subring of endomorphisms. It turns out that any degree of the exterior algebra inherits a module structure which is isomorphic to the homology ring of some grassmannian, seen as a module over its cohomology. The talk will be mainly devoted to show how such a formalism works.

K. Ranestad. An abelian fibration on the Hilbert scheme of degree 3 subschemes in a K3 surface of genus 9.

Mukai gave a simple example of an abelian fibration on Hilb_2 of a general complete intersection of three quadrics in P^5. In work together with Atanas Iliev we describe a similar example for K3 surfaces of genus 9.

T. Johnsen. Schubert unions in flag varieties.

We study subsets of Grassmann varieties, such that these subsets are unions of Schubert varities, with respect to a fixed flag. We study the linear spans of, and in case of positive characteristic, the number of $F_q$-rational points on such unions. Moreover we study a geometric duality of such unions, and give a combinatorial interpretation of this duality. We discuss generalizations to (partial) flag varieties.

R. Piene.

S.A. Stromme.

T. Gustavsen. The deformation relation on the set of Cohen-Macaulay modules on a quotient surface singularity.

I will report on joint work with Runar Ile, and consider the set of isomorphism classes of rank $r$, maximal Cohen-Macaulay modules on a quotient surface singularity $X$. This set will be considered as a graph $\mathbf{G}^{\operatorname{def}}(X,r)$ with edges coming from the deformation relation. We conjecture that the number of connected components in the graph $\mathbf{G}^{\operatorname{def}}(X,r)$ is the absolute value of the determinant of the intersection matrix corresponding to the minimal resolution of $X.$ When $X$ is a rational double point, I will interpret a result of A. Ishii as an enrichment of the McKay correspondence and explain how this implies our conjecture. Using our work on the deformation theory of reflexive modules on a rational cone, I will prove the conjecture in this case.

E. Sernesi. The rational connectedness of M_{15}.

The talk will be devoted to an outline of the recent proof by A. Bruno and A. Verra of the rational connectedness of the moduli space ofcurves of genus 15.