Volume 2008, Issue 618

In this work we study the Positive Extension property and Hilbert's 17th problem for real analytic germs and sets. A real analytic germ X 0 of has the property if every positive semidefinite analytic function germ on X 0 has a positive semidefinite analytic extension to ; analogously one states the property for a global real analytic set X in an open set Ω of ℝ n . These properties are natural variations of Hilbert's 17th problem. Here, we prove that: (1) A real analytic germ has the property if and only if every positive semidefinite analytic function germ on X 0 is a sum of squares of analytic function germs on X 0 ; and (2) a global real analytic set X of dimension ≦ 2 and local embedding dimension ≦ 3 has the property if and only if it is coherent and all its germs have the property. If that is the case, every positive semidefinite analytic function on X is a sum of squares of analytic functions on X . Moreover, we classify the singularities with the property.

Let ( G , X ) be a Shimura pair of Hodge type such that G is the Mumford-Tate group of some elements of X . We assume that for each simple factor G 0 of G ad there exists a simple factor of G 0ℝ which is compact. Let N ≧ 3. We show that for many compact open subgroups , the Shimura variety Sh( G, X )/ K has a projective integral model over ℤ [1/ N ] which is a finite scheme over a certain Mumford moduli scheme . Equivalently, we show that if A is an abelian variety over a number field and if the Mumford-Tate group of A ℂ is G , then A has potentially good reduction everywhere. The last result represents significant progress towards the proof of a conjecture of Morita. If is smooth over ℤ [1/ N ], then it is a Néron model of its generic fibre. In this way one gets in arbitrary mixed characteristic, the very first examples of general nature of projective Néron models whose generic fibres are not finite schemes over abelian varieties.

We study a wide class of supersolutions of the porous medium equation. These supersolutions are defined as lower semicontinuous functions obeying the comparison principle. We show that they have a spatial Sobolev gradient and give sharp summability exponents. We also study pointwise behaviour.

Our aim is to prove that two formal power series of importance to quantum topology are Gevrey. These series are the Kashaev invariant of a knot (reformulated by Huynh and the second author) and the Gromov norm of the LMO of an integral homology 3-sphere. It follows that the power series associated to a simple Lie algebra and a homology sphere is Gevrey. Contrary to the case of analysis, our formal power series are not solutions to differential equations with polynomial coefficients. The first author has conjectured (and in some cases proved, in joint work with Costin) that our formal power series have resurgent Borel transform, with geometrically interesting set of singularities.

We study germs of singular varieties in a symplectic space. In [ V. I. Arnold , First step of local symplectic algebra, Differential topology, infinite-dimensional Lie algebras, and applications, D. B. Fuchs' 60th anniversary collection, American Mathematical Society Transl. Ser. 2, Am. Math. Soc. 194 (44) (1999), 1–8.], V. Arnold discovered so called “ghost” symplectic invariants which are induced purely by singularity. We introduce algebraic restrictions of differential forms to singular varieties and show that this ghost is exactly the invariants of the algebraic restriction of the symplectic form. This follows from our generalization of Darboux-Givental' theorem from non-singular submanifolds to arbitrary quasi-homogeneous varieties in a symplectic space. Using algebraic restrictions we introduce new symplectic invariants and explain their geometric meaning. We prove that a quasi-homogeneous variety N is contained in a non-singular Lagrangian submanifold if and only if the algebraic restriction of the symplectic form to N vanishes. The method of algebraic restriction is a powerful tool for various classification problems in a symplectic space. We illustrate this by complete solutions of symplectic classification problem for the classical A, D, E singularities of curves, the S 5 singularity, and for regular union singularities.