#### Jan Denef: Geometric proof of Ax-Kochen's theorem on p-adic forms

We will sketch a new proof of the Theorem of Ax and Kochen that any projective hypersurface over the p-adic numbers has a p-adic rational point, if it is given by a homogeneous polynomial with more variables than the square of its degree d, assuming that p is large enough with respect to the degree d. Our proof is purely geometric and (unlike all previous ones) does not use methods from mathematical logic. It is based on a theorem of Abramovich and Karu about weak toroidalization of morphisms. The method also yields a proof of a conjecture of Colliot-Thélène and generalizations. An earlier more complicated version of our proof was based on Cutkosky's Theorem on Local Monomialization of Morphisms, but weak toroidalization yields stronger results.

#### Paul Hacking: Vector bundles associated to degenerations of surfaces

Surface singularities occurring in codimension one in the moduli space are of two main types: there is the ordinary double point (x

^{2}+y

^{2}+z

^{2}=0) and a class of cyclic quotient singularities first studied by J. Wahl in 1980. For the ordinary double point there is a vanishing cycle: a 2-sphere on the nearby smooth surface Y that shrinks to a point on the singular surface X, and generates the kernel of the specialization map H

_{2}(Y,Z) --> H

_{2}(X,Z). For Wahl singularities there are no vanishing cycles: the rational homology of the fiber is constant. We describe a construction of a vector bundle on the smooth surface that plays a role analogous to the vanishing cycle for Wahl singularities.

In the case of del Pezzo surfaces the relevant vector bundles were classified by the Russian school in the 1990s. We will explain the geometric interpretation of their results.

#### Stefan Kebekus: Extension properties of differential forms on singular spaces, and applications to moduli

The talk is concerned with differential forms on log canonical varieties. It is shown that any p-form defined on the smooth locus of a variety with canonical or klt singularities extends regularly to any resolution of singularities. Many of the fundamental theorems and techniques known for sheaves of logarithmic differentials on smooth varieties also hold in the setting of minimal model theory.

These results apply to hyperbolicity properties of moduli stacks, and allow to generalise Shafarevich hyperbolicity to higher dimensions. The lectures concentrate on methods and results that relate moduli theory with recent progress in higher dimensional birational geometry.

#### Allen Knutson: Frobenius splittings and degeneration

Let f be a polynomial of degree n in n variables (same n), over a field. From the hypersurface f=0, we can construct a whole stratification by varieties: decompose into components, intersect them (set-theoretically), and repeat.

Theorem: (1) => (2) => (3) in the following:

(1) The leading term init f (with respect to some term order on monomials) is the product of the variables.

(2) The number of F_p points on the hypersurface is not a multiple of p.

(3) Every intersection Y encountered in the process above is already reduced.

Indeed, (1) => every init Y is reduced.

I'll use reducedness to motivate the definition of Frobenius splitting, with which the above results are proved.

#### János Kollár: A local version of the Kawamata-Viehweg vanishing theorem

The, by now classical, Kawamata--Viehweg vanishing theorem says that

*global* cohomologies vanish for divisorial sheaves which are linearly equivalent to a divisor of the form (nef and big)+Δ. We show that

*local* cohomologies vanish for divisorial sheaves which are linearly equivalent to a divisor of the form Δ. If

*X* is a cone over a Fano variety, one can set up a perfect correspondence between the global and local versions. More generally, we study the depth of various sheaves associated to a log canonical pair (

*X*, Δ).

#### Rob Lazarsfeld: Positivity of cycles on abelian varieties

The cones of divisors and curves defined by various positivity conditions on a smooth projective variety have been the subject of a great deal of work in algebraic geometry, and by now they are quite well understood. However the analogous cones for cycles of higher codimension have started to come into focus only recently. I will discuss a couple of computations on abelian varieties where one can work out the picture fairly completely -- already here one sees some non-classical phenomena. I will also discuss some of the many open problems that present themselves around this circle of ideas. (This is joint work with Olivier Debarre, Lawrence Ein and Claire Voisin.)

#### Davesh Maulik: Relative Behrend functions

Given a smooth projective threefold, Donaldson-Thomas invariants encode virtual enumerative information about its moduli space of sheaves. In the case of Calabi-Yau threefolds, one lucrative approach to studying these invariants (and various conjectures) is to express them in terms of certain constructible functions, canonically associated to the moduli space. In this talk, we'll try to recap some of these ideas and explain an extension to other geometries.

#### Yu-jong Tzeng: Universal Formulas for Counting Nodal Curves on Surfaces

The problem of counting nodal curves on algebraic surfaces has been studied since the nineteenth century. On the projective plane, it asks how many curves defined by homogeneous degree d polynomials have only nodes as singularities and pass through points in general position. On K3 surfaces, the number of rational nodal curves was predicted by the Yau-Zaslow formula. Goettsche conjectured that for sufficiently ample line bundles L on algebraic surfaces, the numbers of nodal curves in |L| are given by universal polynomials in four topological numbers. Furthermore, based on the Yau-Zaslow formula he gave a conjectural generating function in terms of quasi-modular forms and two unknown series. In this talk, I will discuss how degeneration methods can be applied to count nodal curves and sketch my proof of Goettsche's conjecture.