Witt vectors were introduced by E. Witt back in 1930ies. In modern language, Witt's construction gives a funcntorial way to lift a commutative ring A in char p to a commutative ring W(A) of characteristic 0. Since the construction is functorial, it can be applied to the structure sheaf of an algebraic variety. However, this really became useful only in the 1970ies, when Deligne and Illusie showed how to extend it to differential forms. The result is a canonical "de Rham-Witt complex" WO* on any smooth algebraic variety X over a field of char p, and the cohomology of X with coefficients in WO* is canonically identified with Grothendieck's cristalline cohomology of X. I am going to report on some recent discoveries which can be roughly summarized as follows: even for a non-commutative ring A, one can define a functorial "Hochschild-Witt complex" with homology WHH_*(A); if A is commutative, then WHH_i(A)=WO^i(A) (this is analogous to the isomorphism HH_i(A)=O^i(A) discovered by Hochschild, Kostant and Rosenberg). Moreover, the construction of the Hochschild-Witt complex is actually simpler than the Deligne-Illusie construction, and it allows to clarify the structure of the de Rham-Witt complex. I will start very slowly, by introducing Witt vectors in the simplest possible case; at least in the first lecture, there will be definitely nothing non-commutative. I will assume very little prior knowledge. In particular, I will *not* assume knowledge of either Witt vectors or cristalline cohomology, and I will explain all the necessary material along the way. 

Andre Chatzistamatiou & Kay Rülling (Duisburg-Essen): Rational and Witt-rational singularities  

Let f be a birational proper morphism between smooth varieties. As an application of resolution of singularities Hironaka proved that the higher direct images of the structure sheaf along f vanish. The first part of the lectures will be devoted to giving a proof of this vanishing result in positive characteristic. The problem of bypassing resolution of singularities is solved by letting algebraic correspondences act on the higher direct images.
In the second part we will show how the same techniques can be applied to the sheaf of Witt vectors. Moreover, since the Witt vector cohomology is not a torsion group anymore, we can pass to rational coefficients. This allows us to replace resolution of singularities by alterations and yields an alternative definition of Witt rational singularities. In order to make this work, it is necessary to use the de Rham-Witt complex and results due to Ekedahl.

The purpose of these talks is to explain my construction of Higgs-bundles associated to representations of the geometric fundamental group of a curve over a p-adic field K.

A basic theorem in Hodge theory is the isomorphism between de Rham and Betti cohomology for complex manifolds; this follows directly from the Poincare lemma. The p-adic analogue of this comparison lies deeper, and was the subject of a series of extremely influential conjectures made by Fontaine in the early 80s (which have since been established by various mathematicians). In my talk, I will first discuss the geometric motivation behind Fontaine's conjectures, and then explain a simple new proof based on general principles in derived algebraic geometry --- specifically, derived de Rham cohomology --- and some classical geometry with curve fibrations. This work builds on ideas of Beilinson who proved the de Rham comparison conjecture this way.

We consider vector bundles V over a relative projective curve over an integral base scheme of fixed positive characteristic and discuss how the property of strong semistability varies in such a family. We present several examples where the bundle on the generic bundle is strongly semistable and on some special fiber semistable, but not strongly semistable. For a V-torsor given by a cohomology class in V we discuss the related question how the property of being an affine scheme may vary in the family.

Whenever a smooth projective variety lifts to characteristic zero, many characteristic-p-"pathologies" cannot happen. Now, lifting results are difficult to establish (and in general, starting from dimension 2, lifting does not hold in general), but sometimes it is easier, and even more natural, to lift a birational model, maybe even a slightly singular one, of a given variety. Thus, it is natural to study to what extent lifting is a birational invariant for varieties. We will see that it is not a birational invariant even for smooth projective and rational varieties if the dimension is large. Trivially, it holds for normal curves, and it is also not difficult to see that it holds for smooth surfaces. For surfaces with canonical singularities, we will see that this question is surprisingly subtle. We address this question from the point of view of classical resolutions of singularities, as well as "stacky" resolutions of singularities. This is joint work with Matthew Satriano.

I will discuss two modifications of the stable base locus of a big line bundle, giving numerical characterizations of these loci. The corresponding results in characteristic zero are proved via vanishing theorems. I will focus on how the use use of the Frobenius allows to recover these descriptions also in positive characteristic.

I will define a new variant of the Seshadri constant for ample line bundles in positive characteristic. We will then explore how lower bounds for this constant imply the global generation and/or very ampleness of the corresponding adjoint line bundle. As a consequence, we will deduce that the criterion for global generation and very ampleness of adjoint line bundles in terms of usual Seshadri constants holds also in positive characteristic (even though we may lack the usual vanishing theorems). This is joint work with Mircea Mustata.

Test ideals are a positive characteristic measure of singularities; they are (in a sense that can be made precise) the moral equivalent of the multiplier ideals often used in complex birational algebraic geometry. The aim of this talk is to present a global division theorem for test ideals which mirrors the analogous result for multiplier ideals (joint with Karl Schwede). However, due to the failure of various cohomology vanishing statements in positive characteristic, one is essentially forced to restrict the statement to certain naturally defined linear subsystems. The main new ingredient is a slight variation on well known characterizations of test ideals.

 I will compute the center of the ring of PD differential operators on a smooth variety over Z/p^nZ.  More generally, given an associative algebra A_0 over a finite field F_p and its flat deformation A_n over Z/p^{n+1} Z   I will prove that under a certain non-degeneracy condition the center of A_n is isomorphic to the ring of length n+1 Witt vectors over the center of A_0. If, in addition, A_0 is an Azumaya algebra over its center the higher Hochschild cohomology of A_n are identified with the de Rham-Witt differential forms on the center of A_0. This is a joint work with Allen Stewart.