A classification of hyperpolar and cohomogeneity one actions

An isometric action of a compact Lie group on a Riemannian manifold is called hyperpolar if there exists a closed, connected submanifold that is flat in the induced metric and meets all orbits orthogonally. In this article, a classification of hyperpolar actions on the irreducible Riemannian symmetric spaces of compact type is given. Since on these symmetric spaces actions of cohomogeneity one are hyperpolar, i.e. normal geodesics are closed, we obtain a classification of the homogeneous hypersurfaces in these spaces by computing the cohomogeneity for all hyperpolar actions. This result implies a classification of the cohomogeneity one actions on compact strongly isotropy irreducible homogeneous spaces.

     

Properly discontinuous actions of subgroups in amenable algebraic groups and its application to affine motions

This note will concern properly discontinuous actions of subgroups in real algebraic groups on contractible manifolds. Let (π,X,ρ) be such an action, where ρ:π→ Diff(X) is a homomorphism. We assume that ? extends to a smooth action of a real algebraic group G containing π. If such π has a nontrivial radical (i.e., unique maximal normal solvable subgroup), then we can apply the method of Seifert construction [14],[17] to yield that the quotient π\X supports the structure of an injective Seifert fibering with typical (resp. exceptional) fiber diffeomorphic to a solv (resp. infrasolv)-manifold (when π acts freely). When G is an amenable algebraic group, we can say about a uniqueness property for such actions. Namely, let (π_i, X_i, ρ_i) be actions as above (i= 1,2). Then, given an isomorphism f of π₁ onto ?₂, there is a diffeomorphism h: X₁→X₂ such that h(ρ₁(r)x)=ρ₂(f(r)h(x).As an application, we try to decide the structure of affine motions of some euclidean space $\text{R}$ⁿ. First we verify the conjecture of [17, 4 5], i.e., a compact complete affinely flat manifold admits a maximal toral action if its fundamental group has a nontrivial center. Second, a compact complete affinity flat manifold whose fundamental group is virtually polycyclic supports the structure of an infrasolvmanifold. This structure varies depending on its solvable kernel (if it is abelian or nilpotent, it must be a euclidean space form or an infranilmanifold respectively). If a group of the affine group A(n) acts properly discontinuously and with compact quotient of $\text{R}$ⁿ, then it is called an affine crystallographic group. Finally, we can say so far as to a uniqueness property that two virtually polycyclic affine crystallographic groups are conjugate inside Diff($\text{R}$ⁿ) if they are isomorphic (cf.[8]).     

A group theoretic criterion for property FA

We give group-theoretic conditions on a set of generators of a group which imply that admits no non-trivial action on a tree. The criterion applies to several interesting classes of groups, including automorphism groups of most free groups and mapping class groups of most surfaces.

     

KILLING VECTOR FIELDS OF CONSTANT LENGTH ON LOCALLY SYMMETRIC RIEMANNIAN MANIFOLDS

In this paper nontrivial Killing vector fields of constant length and the corresponding ows on smooth complete Riemannian manifolds are investigated. It is proved that such a ow on symmetric space is free or induced by a free isometric action of the circle S ¹. Examples of unit Killing vector fields generated by almost free but not free actions of S ¹ on locally symmetric Riemannian spaces are found; among them are homogeneous (nonsimply connected) Riemannian manifolds of constant positive sectional curvature and locally Euclidean spaces. Some unsolved questions are formulated. DOI: .     

Anti-Markovnikov stereoselective addition of bis(trimethylsilyl)octadiene to ketal obtained by unprecedented retro-Claisen condensation of 3-hydroxy-2,4-pentanedione bis-ketal

In the course of the Lewis acid-mediated cycloaddition of 1,8-bis(trimethylsilyl)-2,6-octadiene to bis-ketals, we have observed an unprecedented retro-Claisen condensation from the ketalisation of a 3-hydroxy-2,4-pentadione giving rise to a substituted 2,2,3-trimethoxybutane and an anti-Markovnikov stereoselective cycloaddition of the 1,8-bis(trimethylsilyl)-2,6-octadiene to this latter ketal.     

Higher order generalized Euler characteristics and generating series

For a complex quasi-projective manifold with a finite group action, we define higher order generalized Euler characteristics with values in the Grothendieck ring of complex quasi-projective varieties extended by the rational powers of the class of the affine line. We compute the generating series of generalized Euler characteristics of a fixed order of the Cartesian products of the manifold with the wreath product actions on them.     

A generalization of Dirichlet approximation theorem for the affine actions on real line

In this paper, we obtain strong density results for the orbits of real numbers under the action of the semigroup generated by the affine transformations T₀(x)=x/a and T₁(x)=bx+1, where a,b>1. These density results are formulated as generalizations of the Dirichlet approximation theorem and improve the results of Bergelson, Misiurewicz, and Senti. We show that for any x,u>0 there are infinitely many elements γ in the semigroup generated by T₀ and T₁ such that |γ(x)−u|<C(t1/|γ|−1), where C and t are constants independent of γ, and |γ| is the length of γ as a word in the semigroup. Finally, we discuss the problem of approximating an arbitrary real number by the ratios of prime numbers and the ratios of logarithms of prime numbers.     

Twisted equivariant K-theory for proper actions of discrete groups

We will make a construction of twisted equivariant K-theory for proper actions of discrete groups by using ideas of Lück and Oliver (Topology 40:585–616, 2001) to expand a construction of Adem and Ruan (Comm. Math. Phys. 237:533–556, 2003).     

A selection of open problems

This is a collection of open problems which touch on Neil Hindman's mathematics and were collected in conjunction with the Conference on Ramsey Theory and Topological Algebra in his honor.     

A priori orienteering with time windows and stochastic wait times at customers

In the pharmaceutical industry, sales representatives visit doctors to inform them of their products and encourage them to become an active prescriber. On a daily basis, pharmaceutical sales representatives must decide which doctors to visit and the order to visit them. This situation motivates a problem we more generally refer to as a stochastic orienteering problem with time windows (SOPTW), in which a time window is associated with each customer and an uncertain wait time at a customer results from a queue of competing sales representatives. We develop a priori routes with the objective of maximizing expected sales. We operationalize the sales representative’s execution of the a priori route with relevant recourse actions and derive an analytical formula to compute the expected sales from an a priori tour. We tailor a variable neighborhood search heuristic to solve the problem. We demonstrate the value of modeling uncertainty by comparing the solutions to our model to solutions of a deterministic version using expected values of the associated random variables. We also compute an empirical upper bound on our solutions by solving deterministic instances corresponding to perfect information.