Cycles and Spectra

Some recent work on spaces of algebraic cycles is surveyed. The main focus is on spaces of real and quaternionic cycles and their relation to equivariant Eilenberg- MacLane spaces.Dedicated to IMPA on the occasion of its 50^th anniversary     

Equivariant Cyclic Cohomology of H-Algebras

We define an equivariant K ₀-theory for Yetter–Drinfeld algebras over a Hopf algebra with an invertible antipode. We then show that this definition can be generalized to all Hopf-module algebras. We show that there exists a pairing, generalizing Connes pairing, between this theory and a suitably defined Hopf algebra equivariant cyclic cohomology theory.     

Rigidity of Coxeter groups

Let be a Coxeter group acting properly discontinuously and cocompactly on manifolds and such that the fixed point sets of finite subgroups are contractible. Let be a -homotopy equivalence which restricts to a -homeomorphism on the boundary. Under an assumption on the three dimensional fixed point sets, we show that then is -homotopic to a -homeomorphism.     

Morse theory for the Yang-Mills functional via equivariant homotopy theory

In this paper we show the existence of non-minimal critical points of the Yang-Mills functional over a certain family of 4-manifolds with generic -invariant metrics using Morse and homotopy theoretic methods. These manifolds are acted on fixed point freely by the Lie group with quotient a compact Riemann surface of even genus. We use a version of invariant Morse theory for the Yang-Mills functional used by Parker in A Morse theory for equivariant Yang-Mills, Duke Math. J. 66-2 (1992), 337-356 and Råde in Compactness theorems for invariant connections, submitted for publication.     

Estimation of a covariance matrix with location: Asymptotic formulas and optimal B-robust estimators

Applying the non-singular affine transformations AZ + μ to a spherically symmetrically distributed variate Z generates the covariance-location model, indexed by the parameters AA^T and μ, consisting of so-called elliptical distributions. We develop an algebraic machinery that simplifies the derivation of influence functions and asymptotic variance-covariance matrices for equivariant estimators of Σ and μ and reveals a natural structure of Σ. In addition, optimal B-robust estimators are derived.     

Estimating Equations with Nuisance Parameters: Theory and Applications

In a variety of statistical problems the estimate _n of a parameter is defined as the root of a generalized estimating equation G_n(_nn)=0 where _n is an estimate of a nuisance parameter . We give sufficient conditions for the asymptotic normality of #x0398;_n defined in this way and derive their asymptotic distribution. A circumstance under which the asymptotic distribution of #x0398;_n will not be influenced by that of _n) is noted. As an example, we consider a covariance structure analysis in which both the population mean and the population fourth-order moment are nuisance parameters. Applications to pseudo maximum likelihood, generalized least squares with estimated weights, and M-estimation with an estimated scale parameter are discussed briefly.     

On Injective and Cofibrant Equivariant Minimal Models

Let X be a G-connected nilpotent simplicial set, where G is a finite Hamiltonian group. We construct a cofibrant equivariant minimal model of X with the strong homotopy type of the injective minimal model of X defined by Triantafillou.     

Projectively Equivariant Symbol Calculus for Bidifferential Operators

We prove the existence and uniqueness of a projectively equivariant symbol map, which is an isomorphism between the space of bidifferential operators acting on tensor densities over ⁿ and that of their symbols, when both are considered as modules over an imbedding of sl(n+1, ) into polynomial vector fields. The coefficients of the bidifferential operators are densities of an arbitrary weight. We obtain the result for all values of this weight, except for a set of critical ones which does not contain 0. In the case of second-order operators, we give explicit formulas and examine in detail the critical values.     

Equivariant bifurcation theory and symmetry breaking

A study is made of the failure of the Maximal Isotropy Subgroup Conjecture for the Weyl group seriesW(D) _k . As part of the investigation, a general genericity and stability theorem is proved for bifurcation diagrams in equivariant bifurcation theory. As well, a concept of determinacy for equivariant bifurcation theory is introduced and it is shown that, for all compact Lie groupsG and absolutely irreducibleG-representationsV, G-equivariant bifurcation problems onV are finitely determined.     

On the Hamiltonian Hopf Bifurcations in the 3D Hénon–Heiles Family

An axially symmetric perturbed isotropic harmonic oscillator undergoes several bifurcations as the parameter adjusting the relative strength of the two terms in the cubic potential is varied. We show that three of these bifurcations are Hamiltonian Hopf bifurcations. To this end we analyse an appropriately chosen normal form. It turns out that the linear behaviour is not that of a typical Hamiltonian Hopf bifurcation as the eigen-values completely vanish at the bifurcation. However, the nonlinear structure is that of a Hamiltonian Hopf bifurcation. The result is obtained by formulating geometric criteria involving the normalized Hamiltonian and the reduced phase space.