On the conormal bundle of a G-stable subvariety

Let G be a reductive algebraic group and X a smooth G-variety. For a smooth locally closed G-stable subvariety M⊂X, we prove that the G-complexity of the (co)normal bundle of M is equal to the G-complexity of X. In particular, if X is spherical, then all (co)normal bundles are again spherical G-varieties. If X is a G-module with finitely many orbits, the closures of the conormal bundles of the orbits coincide with the irreducible components of the commuting variety. We describe properties of these closures for the representations associated with short gradings of simple Lie algebras. Received: 22 April 1998     

Undecidability results for restricted universally quantified formulae of set theory

The problem of establishing whether there are sets satisfying a formula in the first order set theoretic language ??_? based on =,?, which involves only restricted quantifiers and has an equivalent ??-prenex form ((??)₀-formula), is neither decidable nor semidecidable. In fact, given any ω-model of ZF – FA, where FA denotes the Foundation Axiom, the set of existential closures of (??)₀-formulae true in the model is a productive set. Undecidability arises even when dealing with restricted universal quantifiers only, provided a predicate is_a_pair(x), meaning that x is a pair of distinct sets, is added to ??_?. If satisfiability refers to ω-models of ZF – FA in which a form of Boffa's antifoundation axiom holds, then semidecidability fails as well; in fact, given any such model, the set of existential closures of formulae involving only restricted quantifiers and the predicate is_a_pair which are true in it, is a productive set. These results are all proved by making use of appropriate codings of Turing machine computations in the set theoretic language.     

Groups whose cyclic subgroups have bounded permutability properties

It follows from classical results of Neumann and Macdonald that a group G has finite commuator subgroup if and only if either the normalizers of cyclic subgroups of G have boundedly finite indices or cyclic subgroups of G have bounded indices in their normal closures. In this paper, groups with a similar condition are considered, when normality is replaced by permutability.      

Closures of SL(2)-orbits in projective spaces

LetB be a Borel-subgroup ofSL(2,C). In this note we describe the closures of arbitrarySL(2,C)-andB-orbits in the projective space of regularSL(2,C)-modules by means of simple combinatorial data. We give a criterion to detect whether the number of orbits in an orbit-closure is finite or not.     

Verallgemeinerte topogene Ordnungen

Topogenous orders in the sense of Császár are a common generalization of proximity and topology. ech closures are a generalization of the topological closure operators in the sense of Kuratowski. We show that the topogenous orders as well as the ech closures are special cases of the so called compressed operators. Moreover, the now defined categoryCOM (in germanBAL) of compress spaces and compress faithful maps is a properly fibred topological category in the sense of Herrlich which is weakly cartesian closed, that means the product map of two quotient maps inCOM is a quotient map inCOM. Therefore by results of L. D. Nel it is possible to construct a cartesian closed properly fibred topological category in whichCOM can be nicely embedded. Further it turns out that the compressed operators be in a natural connexion with the uniform convergence structures in the sense of Cook and Fischer and in addition with the limit structures in the sense of Fischer. For principal ideal uniform convergence structures we prove that they are precompact and complete iff the properly constructed compressed operator is compact.     

Derived equivalences of selfinjective algebras preserve singularities

We prove that the derived equivalences (more generally the stable equivalences of Morita type) of finite dimensional selfinjective algebras over algebraically closed fields preserve the types of singularities in the orbit closures of module varieties. As an application, we obtain that the orbit closures in the module varieties of the Brauer tree algebras are normal and Cohen-Macaulay. Mathematics Subject Classification (2000):14B05, 14L30, 16D50, 16G20     

Minimal singularities for representations of Dynkin quivers

We develop some reduction techniques for the study of singularities in orbit closures of finite dimensional modules. This enables us to classify all singularities occurring in minimal degenerations of representations of Dynkin quivers. They are all smoothly equivalent to the singularity at the zero-matrix inside thep×q-matrices of rank at most one.     

Determination of Spherical Area Measures by Means of Dilation Volumes

For a large class of compact sets X in ℝ^d which can be represented as finite unions of sets of positive reach, the spherical area measure of order d — 1 is determined by the volumes of dilations of X with infinitesimal multiples of all rotations of a suitable convex test set. For spherical area measures of lower orders, a similar result is obtained by approximating the set X by the closures of the complements of close parallel bodies.     

Normality of orbit closures for Dynkin quivers¶of type ?n

It is known that the orbit closures for the representations of the equioriented Dynkin quivers ? _n are normal and Cohen–Macaulay varieties with rational singularities. In the paper we prove the same for the Dynkin quivers ? _n with arbitrary orientation. Received: 25 October 2000 / Revised version: 28 February 2001     

Semigroup closures of finite rank symmetric inverse semigroups

We introduce the notion of semigroup with a tight ideal series and investigate their closures in semitopological semigroups, particularly inverse semigroups with continuous inversion. As a corollary we show that the symmetric inverse semigroup of finite transformations I _λ ⁿ of the rank ≤ n is algebraically closed in the class of (semi)topological inverse semigroups with continuous inversion. We also derive related results about the nonexistence of (partial) compactifications of classes of semigroups that we consider.     

H-fields and their Liouville extensions

We introduce H-fields as ordered differential fields of a certain kind. Hardy fields extending , as well as the field of logarithmic-exponential series over are H-fields. We study Liouville extensions in the category of H-fields, as a step towards a model theory of H-fields. The main result is that an H-field has at most two Liouville closures. Received: 15 November 1999 / in final form: 19 December 2000 / Published online: 1 February 2002     

Pandemic simulation of antivirals + school closures: buying time until strain-specific vaccine is available

A strain-specific vaccine is unlikely to be available in the early phases of a potential H5N1 avian influenza pandemic. It could be months and at the current production rate may not provide timely protection to the population. Intervention strategies that control the spread of infection will be necessary in this situation, such as the use of the US stockpile of antiviral medication coupled with a 6-month school closure. The agent-based simulation model, EpiSimS, was used to assess the impact of this intervention strategy followed by three different vaccine approaches: (1) 2-dose, 80% effective, (2) 1-dose, 30% effective, and (3) 1 dose, 80% effective. Simulations show that the combination of antivirals, school closures, and a strain-specific vaccine can reduce morbidity and mortality while in effect. A significant second infection wave can occur with current vaccine technology once school closures are relaxed, though an ideal vaccine is able to contain it. In our simulations, worker absenteeism increases in all cases mostly attributed to household adults staying home with children due to the school closures.

Gradient flows with wildly embedded closures of separatrices

We show that for any n ≥ 4 there exists an n-dimensional closed manifold M ⁿ on which one can define a Morse-Smale gradient flow f ^t with two nodes and two saddles such that the closure of the separatrix of some saddle of f ^t is a wildly embedded sphere of codimension 2. We also prove that the closures of separatrices of a flow with three equilibrium points are always embedded in a locally flat way.     

Classification of hermitian forms and semisimple groups over fields of virtual cohomological dimension one

Summary Letk be a perfect field with cdk(i)≤1. It has recently been proved by the author that homogeneous spaces under connected linear groups overk satisfy a Hasse principle with respect to the real closures ofk. Using this result we classify the semisimple algebraic groups overk and, in particular, characterize the anisotropic ones. Similarly we classify the various types of hermitian forms over skew fields overk and exhibit to what extent weak or strong local-global principles hold. In the case wherek is the function field of a smooth projective curveX over ℝ, we also cover the local-global questions vis-à-vis the completions ofk at the points ofX.     

On Frobenius splitting of orbit closures of spherical subgroups in flag varieties

Let H denote a spherical subgroup within a semisimple algebraic group G. In this paper we study the closures of the finitely many H-orbits in the flag variety of G. Using the language of Frobenius splitting we provide a criterion for these closures to have nice geometric and cohomological properties. We then show how the criterion applies to the spherical subgroups of minimal rank studied by N. Ressayre. Finally, we also provide applications of the criterion to orbit closures which are not multiplicity-free in the sense defined by M. Brion.     

Reductions and special parts of closures

We provide an axiomatic framework for working with a wide variety of closure operations on ideals and submodules in commutative algebra, including notions of reduction, independence, spread, and special parts of closures. This framework is applied to tight, Frobenius, and integral closures. Applications are given to evolutions and special Briançon–Skoda theorems.     

Gelfand�CTsetlin algebras and cohomology rings of Laumon spaces

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GL _n . We calculate the equivariant cohomology rings of the Laumon moduli spaces in terms of Gelfand–Tsetlin subalgebra of U(gl _n ) and formulate a conjectural answer for the small quantum cohomology rings in terms of certain commutative shift of argument subalgebras of U(gl _n ).     

On closures of orbits and arithmetic of quaternions

ForG=PGL₂(ℚ _p )×PGL₂ ℚ we study the closures of orbits under the maximal split Cartan subgroup ofG in homogeneous spacesΓ\G. We show that if a closure of an orbit contains a closed orbit then the orbit is either dense or closed. We show the relation of this to divisibility properties of integral quaternions and other lattices. Sponsored in part by the Edmund Landau Center for Research in Mathematical Analysis supported by the Minerva Foundation (Germany). Research at MSRI supported by NSF grant DMS8505550.