Triangular 𝐺ₐ actions on 𝐂⁴

Abstract

All proper rational actions of the additive group on complex affine three space admit equivariant trivializations with quotient isomorphic to complex two space. An example of an additive group action on complex seven space with a nonfmitely generated ring of invariants is presented.

For a finitely generated group G, we study deformation spaces of metric G-trees, which are analogues of the Teichmuller spaces of surfaces for group actions on trees. Deformation spaces of metric G-trees generalize Culler-Vogtmann’s Outer space, the deformation space of free actions on trees, which has proven to be immensely useful in the study of Out(Fn), the outer automorphism group of the free group of rank n ≥ 2. Let D be a deformation space of metric G-trees. The group of positive real numbers R>0 acts on D by scaling the metrics on the trees and we define the projectivized deformation space as the quotient PD = D/R>0. The outer auto- morphism group Out(G) contains a certain subgroup OutD(G) that acts on D and PD by precomposing the G-actions on the trees. In Chapter 1, we present a complete argument that under certain assumptions the projectivized deformation space PD is a model for the classifying space of OutD(G) for a family of subgroups. In Chapter 2, we introduce an asymmetric pseudometric on PD that generalizes the asymmetric Lipschitz metric on Outer space and is an analogue of the Thurston metric on Teichmuller space. Making use of the Lipschitz metric on PD, we prove existence of train track representatives for irreducible automorphisms of virtually free groups and nonelementary generalized Baumslag-Solitar groups that contain no solvable Baumslag-Solitar group BS(1, n) with n ≥ 2. In Chapter 3, we define the higher holomorphs Aut(G, k), k ∈ N, which are “higher-pointed” variants of the automorphism group Aut(G). Following the construction of the spine of Outer space, we construct a family of simplicial complexes S(PD, k), k ∈ N on which certain subgroups AutD(G, k) ≤ Aut(G, k) act and we show that these complexes are always contractible.

A classical result in differential geometry states that for a free and proper Lie group action, the quotient map to the orbit space induces an isomorphism between the de Rham complex of differential forms on the orbit space and the basic differential forms on the original manifold. In this paper, this result is generalized to the case of a proper Lie groupoid, in which the orbit space is equipped with the quotient diffeological structure. As an application of this, we obtain a de Rham theorem for the de Rham complex on the orbit space.KeywordsDiffeologyLie groupoidde Rham complexBasic formsLinearizationMathematics Subject Classification (2010)58H0522A22

We have extended our graphical processing unit (GPU)-accelerated direct configuration interaction program to multiple devices, reducing iteration times for configuration spaces of 165 million determinants to only 3 s using NVIDIA P100 GPUs. Similar improvements in the one- and two-particle reduced density matrix formation allow for fast analytical energy gradients and electronic properties. Our parallel algorithm enables the calculation of arbitrarily large configuration spaces (limited only by available system memory), with iteration times of 13 min for an active space of 18 electrons in 18 orbitals (2.4 billion determinants) using six consumer grade NVIDIA 1080Ti GPUs. These advances enable routine molecular dynamics simulations, geometry optimizations, and absorption spectrum calculations for molecules with large configuration spaces, a task that has heretofore required massive computational effort. In this work, we demonstrate the utility of our program by generating the absorption spectrum for diphenyl acetylene at the floating occupation molecular orbital complete active space configuration interaction level of theory. Several active spaces were investigated to assess the dependence of spectral features on orbital space dimension.

In this paper we prove that an isometry between orbit spaces of two proper isometric actions is smooth if it preserves the codimension of the orbits or if the orbit spaces have no boundary. In other words, we generalize Myers–Steenrod’s theorem for orbit spaces. These results are proved in the more general context of singular Riemannian foliations.

Introduction. 1. Hilbert's fifth problem and proper actions of Lie groups S. Illman. 2. Equivariant algebraic vector bundles over representations - a survey M. Masuda. 3. G-manifolds and G-vector bundles in algebraic, semialgebraic, and definable categories T. Kawakami. 4. Geometry of Finite Topological and equivariant finite topological spaces S. Kono, F. Ushitaki. 5. On the theory of homotopy representations. A survey I. Nagasaki. 6. Manifolds as fixed point sets of smooth compact Lie group actions K. Pawalowski. 7. Surgery and homotopy theory in study of the transformation groups Y. Kitada. 8. Kervaire's obstructions of free actions of finite cyclic groups on homotopy spheres Y. Kitada. 9. The Burnside ring revisited M. Morimoto. 10. Multiplicative stabilization and transformation groups S. Kwasik, R. Schultz. 11. Symmetries on manifolds, deformations and rational homotopy: a survey M. Raussen. 12. Rigidity of codimension one locally free actions of solvable Lie groups N. Tsuchiya, A. Yamakawa. 13. Smooth actions of non-compact semi-simple Lie groups F. Uchida, K. Mukoyama. 14. Hamiltonian group actions and equivariant indices T. Takakura. 15. Controlled methods in equivariant topology, a survey E. Pedersen. Index.

1. In view of the recent work on topological groups it is natural to consider the situation which arises when such groups act as transformation groups on various types of spaces. Such a study is begun here from the point of view of almost periodic transformation groups, the definition of which is suggested by von Neumann's paper on almost periodic functions in a group.t Compact topological transformation groups are a special case of almost periodic transformation groups, at least for a rather wide class of spaces. The paper concerns itself chiefly with the nature of the minimal closed invariant sets of such groups. There are some results for general spaces but the main results are for Euclidean spaces and more particularly for threedimensional Euclidean spaces. One of the most interesting theorems states that if a compact one-dimensional group acts on three-space in such a way that its orbits are uniformly bounded in diameter, then every point of the space is fixed under the group, so that if such a group is to act in a nontrivial manner the diameters of its orbits must be unbounded. Under some restrictions a similar theorem is proved for one-parameter almost periodic groups. Furthermore it is shown that for this latter class of groups, many of the orbits must actually be simple closed curves if they have one-dimensional closures. 2. The group considered here will be denoted by G. It will be subjected to various conditions as the occasion demands but it will always be Abelian. In case it is the group of real numbers, it will be spoken of as a one-parameter group; in case it is the real numbers reduced modulo one, it will be spoken of as the circle group. The space on which the group acts will be denoted by R. It will be specialized in various ways, but in any case it will always be a locally compact metric space. If x and y are two points of R, the distance between them will be denoted by d(x, y).

The differential geometry of circle systems has received a large amount of attention of late. The subject has been approached from two quite distinct sides. On the one hand we have Koenigs, Cosserat, Moore, Bompiani and others who fix their attention on what we may call the descriptive differential properties of circles. The totality of circles in three-dimensional space may be represented by an S6, Iying in an Ss, and this variety may be studied by the now familiar methods of projective diBerential geometry. The theorems so reached are invariant under the twenty-four parameter group of sphere transformations. The other class of writers, wherein we may include Bianchi, Tzitzeica, Guichard and Eisenhart, have confined themselves largely to congruences (two-parameter systems) of circles, frequently to normal congruences. The methods employed have been the general ones of differential geometry and the center of interest has been rather more in the axes of the circles than the circles themselves. It has seemed for some time to the present writer that the last word on these subjects had not by any means been written, and that by a different method of approach not only might we obtain simpler proofs of known theorems but discover a number of new theorems as well. The most interesting properties of circles are those which are invariant under inversion, or under the tenparameter group of conformal transformations of space; the best approach to this group is through the use of pentaspherical coordinates. It is true that some of the writers mentioned above have made use of these coordinates, and still more has been done by Darboux in his Theorie des Surfaces, yet the possibilities of these coordinates have been by no means exhausted, and in the present paper they are more systematically applied to problems in differential circle geometry than has been the case in the past. A circle may be regarded in two different aspects, either as a locus of points, or an envelop of spheres. The two points of view are, of course, closely related, but the change of emphasis leads naturally to rather different sets of theorems. The first section of the present article is devoted to preliminary formulse for points and spheres in pentaspherical coordinates, and certain

Essential results about actions of compact Lie groups on connected manifolds are generalized to proper actions of arbitrary groups on connected cohomology manifolds. Slices are replaced by certain fiber bundle structures on orbit neighborhoods. The group dimension is shown to be effectively finite. The orbits of maximal dimension form a dense open connected subset. If some orbit has codimension at most $2$, then the group is effectively a Lie group.

In [8] counting complexity classes #PR and #PC in the Blum-Shub-Smale (BSS) setting of computations over the real and complex numbers, respectively, were introduced. One of the main results of [8] is that the problem to compute the Euler characteristic of a semialgebraic set is complete in the class FPR#PR. In this paper, we prove that the corresponding result is true over C, namely that the computation of the Euler characteristic of an affine or projective complex variety is complete in the class FPC#PC. We also obtain a corresponding completeness result for the Turing model.

We revisit the theory of Kac–Moody symmetric spaces (of "non-compact type") in order to include the affine case, complex numbers, and Galois descent to almost split real Kac–Moody symmetric spaces.

This paper concerns some of the conditions satisfied by additive group actions on complex affine space which can be written locally as a translation of a variable. Assume X is the affine variety C n , Ga = (C, +), and σ : Ga × X → X is the action defined by a group monomorphism G a → Aut C X. If σ is locally trivial, then the action satisfies what is termed a “GICO” condition. It will be shown that a large class of Ga -actions on C 4, that is, fixed-point free, “twin-triangular” actions with finitely-generated rings of invariants, are at least GICO. Remaining questions are discussed.

We study the geometry of the stratification induced by an affine hyperplane arrangement \mathcal H on the quotient of a complex affine space by the action of a discrete group preserving \mathcal H . We give conditions ensuring normality or normality in codimension 1 of strata. We apply these results to retrieve the list of the categorical quotients of closures of Jordan classes and of sheets in a complex simple algebraic group G that are normal. In the simply-connected case, we show that normality of such a quotient is equivalent to its smoothness.

Essential modern transportation systems together with the high demand for sustainable pavements under applied vehicular loading have led to a great deal of research worldwide of concrete pavements. Despite progressive knowledge of concrete pavement behaviour under applied loads, concrete pavements are still subject to deterioration due to crack initiation and propagation, indicating the need for further research. Cracks can be related to fatigue of concrete or erosion of materials in sub-layers. Transverse joints in concrete pavements are the locations where most pavement distress appears, leading to deterioration of the riding quality and featuring high maintenance cost. The state of stresses in the concrete surrounding dowel bars, in dowel jointed concrete pavements, are a major factor that contribute to transverse joint distress. A three-dimensional (3D) finite element model is developed in this study for analysing a dowel-jointed concrete pavement. The effects of different pavement and joint related parameters on the load transfer characteristics of a joint have been evaluated using the 3D finite element model. The numerical results from FE modelling are validated with classical analytical solutions of shear and moment along the dowel. Five loading cases are applied in the model to replicate realistic vehicular loadings approaching and leaving the joint. Group action of the dowel bar system has also been examined.