Band 2009, Heft 630

We relate a classic algebro-geometric degeneration technique, dating at least to Hodge 1941 (J. London Math. Soc. 16: 245–255), to the notion of vertex decompositions of simplicial complexes. The good case is when the degeneration is reduced, and we call this a geometric vertex decomposition . Our main example in this paper is the family of vexillary matrix Schubert varieties , whose ideals are also known as (one-sided) ladder determinantal ideals. Using a diagonal term order to specify the (Gröbner) degeneration, we show that these have geometric vertex decompositions into simpler varieties of the same type. From this, together with the combinatorics of the pipe dreams of Fomin-Kirillov 1996 (Discr. Math. 153: 123–143), we derive a new formula for the numerators of their multigraded Hilbert series, the double Grothendieck polynomials, in terms of flagged set-valued tableaux . This unifies work of Wachs 1985 (J. Combin. Th. (A) 40: 276–289) on flagged tableaux, and Buch 2002 (Acta. Math. 189: 37–78) on set-valued tableaux, giving geometric meaning to both. This work focuses on diagonal term orders, giving results complementary to those of Knutson-Miller 2005 (Ann. Math. 161: 1245–1318), where it was shown that the generating minors form a Gröbner basis for any anti diagonal term order and any matrix Schubert variety. We show here that under a diagonal term order, the only matrix Schubert varieties for which these minors form Gröbner bases are the vexillary ones, reaching an end toward which the ladder determinantal literature had been building.

As an application of the method of Thue-Siegel, we will resolve a conjecture of Walsh to the effect that the Diophantine equation aX 4 – bY 2 = 1, for fixed positive integers a and b , possesses at most two solutions in positive integers X and Y . Since there are infinitely many pairs ( a, b ) for which two such solutions exist, this result is sharp.

We study Cuntz-Pimsner algebras naturally associated with self-similar groups (like iterated monodromy groups of expanding dynamical systems). In particular, we show how to reconstruct the Julia set of an expanding map from the Cuntz-Pimsner algebra of the associated iterated monodromy group and the gauge action on it. We compute K -theory of algebras associated with complex hyperbolic rational functions. It is proved that under some natural conditions the Cuntz-Pimsner algebra of a self-similar group is purely infinite, simple and nuclear. We also show a relation of our algebras with Ruelle algebras of the associated solenoids.

Using constructions of Voisin, we exhibit a smooth projective variety defined over a number field K and two complex embeddings of K , such that the two complex manifolds induced by these embeddings have non isomorphic cohomology algebras with real coefficients. This contrasts with the fact that the cohomology algebras with l -adic coefficients are canonically isomorphic for any prime number l , and answers a question of Grothendieck.

In this paper we consider the following problem: When are the preduals of two hyperfinite (= injective) factors ℳ and (on separable Hilbert spaces) cb-isomorphic (i.e., isomorphic as operator spaces)? We show that if ℳ is semifinite and is type III, then their preduals are not cb-isomorphic. Moreover, we construct a one-parameter family of hyperfinite type III 0 -factors with mutually non cb-isomorphic preduals, and we give a characterization of those hyperfinite factors ℳ whose preduals are cb-isomorphic to the predual of the unique hyperfinite type III 1 -factor. In contrast, Christensen and Sinclair proved in 1989 that all infinite dimensional hyperfinite factors with separable preduals are cb-isomorphic and more recently, Rosenthal, Sukochev and the first-named author proved that all hyperfinite type III λ -factors, where 0 < λ ≦ 1, have cb-isomorphic preduals.

In this paper we study the evolution of almost non-negatively curved (possibly singular) three dimensional metric spaces by Ricci flow. The non-negatively curved metric spaces which we consider arise as limits of smooth Riemannian manifolds ( M i , i g ), i ∈ ℕ, whose Ricci curvature is bigger than –1/ i , and whose diameter is less than d 0 (independent of i ) and whose volume is bigger than v 0 > 0 (independent of i ). We show for such spaces, that a solution to Ricci flow exists for a short time t ∈ (0, T ), that the solution is smooth for t > 0, and has Ricci ( g ( t )) ≧ 0 and Riem ( g ( t )) ≧ c / t for t ∈ (0, T ) (for some constant c = c ( v 0 , d 0 , n )). This allows us to classify the topological type and the differential structure of the limit manifold (in view of the theorem of Hamilton [J. Diff. Geom. 24: 153–179, 1986] on closed three manifolds with non-negative Ricci curvature).

Let A be an abelian variety defined over a number field F . Let P be a point in the Mordell-Weil group A ( F ) and H a subgroup of A ( F ). We consider the following local-global principle originated with the support problem of Erdös for the integers: the point P belongs to the group H, if for almost all primes v of F, the point P ( modulo v ) belongs to the group H ( modulo v ). We prove that the principle holds for any abelian variety A , if H is a free submodule and the point P generates a free submodule of A ( F ) over the ring End F A .