Abstract. Let R be a regular local ring containing an infinite field. Let G be a reductive group scheme over R. We prove that a principal G-bundle over R is trivial if it is trivial over the fraction field of R. In other words, if K is the fraction field of R, then the map of non-abelian cohomology pointed setsinduced by the inclusion of R into K has a trivial kernel.

2'-5'-Oligoadenylate synthetases (OASs) produce the second messenger 2'-5'-oligoadenylate, which activates RNase L to induce an intrinsic antiviral state. We report on the crystal structures of catalytic intermediates of OAS1 including the OAS1·dsRNA complex without substrates, with a donor substrate, and with both donor and acceptor substrates. Combined with kinetic studies of point mutants and the previously published structure of the apo form of OAS1, the new data suggest a sequential mechanism of OAS activation and show the individual roles of each component. They reveal a dsRNA-mediated push-pull effect responsible for large conformational changes in OAS1, the catalytic role of the active site Mg(2+), and the structural basis for the 2'-specificity of product formation. Our data reveal similarities and differences in the activation mechanisms of members of the OAS/cyclic GMP-AMP synthase family of innate immune sensors. In particular, they show how helix 3103-α5 blocks the synthesis of cyclic dinucleotides by OAS1.

Abstract. Let G be a simple simply-connected group scheme over a regular local scheme U . Let E be a principal G-bundle over A 1 U trivial away from a subscheme finite over U . We show that E is not necessarily trivial and give some criteria of triviality. To this end, we define affine Grassmannians for group schemes and study their Bruhat decompositions for semi-simple group schemes. We also give examples of principal G-bundles over A 1 U with split G such that the bundles are not isomorphic to pullbacks from U . IntroductionIn 1976 Daniel Quillen and Andrei Suslin independently proved a conjecture of Serre that an algebraic vector bundle over an affine space is algebraically trivial (see [Qui, Sus]). A few years earlier, Hyman Bass (see [Bas, Problem IX]) asked a more general question (see also [Qui]): Let R be a regular ring, is every vector bundle over A 1 R := Spec R[t] isomorphic to the pullback of a vector bundle over Spec R? This is now known as Bass-Quillen problem. Note that it is enough to consider the case when R is a regular local ring (see [Qui, Thm. 1]). The problem was solved by Hartmut Lindel in [Lin] in the geometric case, that is, when R is a localization of a k-algebra of finite type, where k is a field.We can ask a more general question: consider a regular local k-algebra R and let G be a simple simply-connected group scheme over R. Question 1. Let E be a principal G-bundle over the affine line A 1 R . Is E isomorphic to the pullback of a principal G-bundle over Spec R?Using the standard relation between principal SL(n, R)-bundles and rank n vector bundles, we see that the above question reduces to the Bass-Quillen problem (=Lindel's Theorem if R is of essentially finite type), when G is the special linear group.Note that the answer to Question 1 is positive if R is a perfect field by a theorem of Raghunathan and Ramanathan (see [RR] and [Gil1]). We will see that the answer is in general negative even if we assume that G is a split group, and k is the field of complex numbers (see Theorem 3 and Example 2.4, where G = Spin(7, C)). To the best of my knowledge such examples were not known before (while examples with algebraically non-closed k, e.g. k being the field of real numbers, were known before, see Remark 2.6(ii)).The principal bundles over A 1 R we construct have the following property: they are isomorphic to the pullback of principal bundles over Spec R on the complement

Abstract. Consider the holomorphic bundle with connection on P 1 −{0, 1, ∞} corresponding to the regular hypergeometric differential operatorIf the numbers α i and β j are real and for all i and j the number α i − β j is not integer, then the bundle with connection is known to underlie a complex polarizable variation of Hodge structures. We calculate some Hodge invariants for this variation, in particular, the Hodge numbers. From this we derive a conjecture of Alessio Corti and Vasily Golyshev. We also use non-abelian Hodge theory to interpret our theorem as a statement about parabolic Higgs bundles.

Let X be a smooth projective curve over a field of characteristic zero. We calculate the motivic class of the moduli stack of semistable Higgs bundles on X. We also calculate the motivic class of the moduli stack of vector bundles with connections by showing that it is equal to the class of the stack of semistable Higgs bundles of the same rank and degree zero.We follow the strategy of Mozgovoy and Schiffmann for counting Higgs bundles over finite fields. The main new ingredient is a motivic version of a theorem of Harder about Eisenstein series claiming that all vector bundles have approximately the same motivic class of Borel reductions as the degree of Borel reduction tends to −∞.