Abstract: An algebraic zip datum is a tuple Z = (G;P;Q;') con- sisting of a reductive group G together with parabolic subgroups P and Q and an isogeny ' : P=RuP ! Q=RuQ. We study the action of the group EZ := � (p;q) 2 PQ �'(�P(p)) = �Q(q) � on G given by ((p;q);g) 㜡 pgq −1 . We define certain smooth EZ-invariant sub- varieties of G, show that they define a stratification of G. We deter- mine their dimensions and their closures and give a description of the stabilizers of the EZ-action on G. We also generalize all results to non-connected groups. We show that for special choices of Z the algebraic quotient stack (EZnG) is isomorphic to (GnZ) or to (GnZ 0 ), where Z is a G-variety studied by Lusztig and He in the theory of character sheaves on spher- ical compactifications of G and where Z 0 has been defined by Moonen and the second author in their classification of F-zips. In these cases the EZ-invariant subvarieties correspond to the so-called "G-stable pieces" of Z defined by Lusztig (resp. the G-orbits of Z 0 ).

Abstract: We show the existence of stationary solutions for some reaction-diffusion type equations in the appropriate H2 spaces using the fixed point technique when the elliptic problem contains second order differential operators with and without Fredholm property.

Abstract: In [Nek], Nekovař gave formulations of analogues of Tate and Poitou-Tate duality for finitely generated modules over a complete commutative local Noetherian ring R with finite residue field of characteristic a fixed prime p. In the usual formulation of these dualities, one takes the Pontryagin dual, which does not in general preserve the property of finite generation. Nekovař takes the dual with respect to a dualizing complex of Grothendieck so as to have a duality between bounded complexes of R-modules with finitely generated cohomology groups. This paper is devoted to a generalization of this result to the setting of nonabelian p-adic Lie extensions. Recall that a dualizing complex ωR is a bounded complex of R-modules with cohomology finitely generated over R that has the property that for every complex M of finitely generated R-modules, the Grothendieck dual RHomR(M,ωR) in the derived category of R-modules D(ModR) has finitely generated cohomology, and moreover, the canonical morphism M −→ RHomR(RHomR(M,ωR), ωR) is an isomorphism in D(ModR). Such a complex exists and is unique up to quasiisomorphism and translation (see [Har1]). One can choose ωR to be a bounded complex of injectives, in which case the derived homomorphism complexes are represented by the complexes of homomorphisms themselves. If R is regular, then R itself, as a complex concentrated in degree 0, is a dualizing complex, but R is not in general R-injective. If R = Zp, for instance, then the complex

Abstract: We show that there exists ane moduli space for torsion- free sheaves on a projective surface which have agood framing" on a big and nef divisor. This moduli space is a quasi-projective scheme. This is accomplished by showing that such framed sheaves may be considered as stable pairs in the sense of Huybrechts and Lehn. We characterize the obstruction to the smoothness of the moduli space and discuss some examples on rational surfaces. 2010 Mathematics Subject Classication: 14D20; 14D21;14J 60

Abstract: In a previous article, we introduced notions of finiteness obstruction, Euler characteristic, and L 2 -Euler characteristic for wide classes of categories. In this sequel, we prove the compatibility of those notions with homotopy colimits of I-indexed categories where I is any small category admitting a finite I-CW-model for its I- classifying space. Special cases of our Homotopy Colimit Formula include formulas for products, homotopy pushouts, homotopy orbits, and transport groupoids. We also apply our formulas to Haefliger complexes of groups, which extend Bass-Serre graphs of groups to higher dimensions. In particular, we obtain necessary conditions for developability of a finite complex of groups from an action of a finite group on a finite category without loops.

Abstract: This work discusses combinatorial and arithmetic aspects of cohomology vanishing for divisorial sheaves on toric varieties. We obtain a refined variant of the Kawamata-Viehweg theorem which is slightly stronger. Moreover, we prove a new vanishing theorem related to divisors whose inverse is nef and has small Iitaka dimension. Fi- nally, we give a new criterion for divisorial sheaves for being maximal Cohen-Macaulay.

Abstract: In this article we introduce a very simple an widely applicable criterion for extending natural transformations to higher K-theory. More precisely, we prove that every natural transformation defined on the Grothendieck group and with values in an additive the- ory admits a unique extension to higher K-theory. As an application, the higher trace maps and the higher Chern characters originally con- structed by Dennis and Karoubi, respectively, can be obtained in an elegant, unified, and conceptual way from our general results.

Abstract: Let X be a minuscule homogeneous space, an odd- dimensional quadric, or an adjoint homogenous space of type different from A and G2. Le C be an elliptic curve. In this paper, we prove that for d large enough, the scheme of degree d morphisms from C to X is irreducible, giving an explicit lower bound for d which is optimal in many cases.

Abstract: We consider a W � -dynamical system (M�;�), which modelsnitely many particles coupled to an innitely exten ded heat bath. The energy of the particles can be described by an unbounded operator, which has innitely many energy levels. We show ex istence of the dynamicsand existence of a (�;�) -KMS state under very explicit conditions on the strength of the interaction and on the inverse temperature �.

Abstract: In a first result, we describe all finitely generated factorial algebras over an algebraically closed field of characteristic zero that come with an effective multigrading of complexity one by means of generators and relations. This enables us to construct systematically varieties with free divisor class group and a complexity one torus action via their Cox rings. For the Fano varieties of this type that have a free divisor class group of rank one, we provide explicit bounds for the number of possible deformation types depending on the dimension and the index of the Picard group in the divisor class group. As a consequence, one can produce classification lists for fixed dimension and Picard index. We carry this out expemplarily in the following cases. There are 15 non-toric surfaces with Picard index at most six. Moreover, there are 116 non-toric threefolds with Picard index at most two; nine of them are locally factorial, i.e. of Picard index one, and among these one is smooth, six have canonical singularities and two have non-canonical singularities. Finally, there are 67 non- toric locally factorial fourfolds and two one-dimensional families of non-toric locally factorial fourfolds. In all cases, we list the Cox rings explicitly.