A nonsmooth Morse-Sard theorem for subanalytic functions

According to the Morse-Sard theorem, any sufficiently smooth function on a Euclidean space remains constant along any arc of critical points. We prove here a theorem of Morse-Sard type suitable as a tool in variational analysis: we broaden the definition of a critical point to the standard notion in nonsmooth optimization, while we restrict the functions under consideration to be semialgebraic or subanalytic. We make no assumption of subdifferential regularity. ?ojasiewicz-type inequalities for nonsmooth functions follow quickly from tools of the kind we develop, leading to convergence theory for subgradient dynamical systems.     

The simplicity of certain groups of subanalytic diffeomorphisms

Let M be a connected real analytic manifold. We denote by , 1r<∞, the group of subanalytic C^r diffeomorphisms of M which are isotopic to the identity via a compactly supported subanalytic C^r isotopy. We show that satisfies Epstein's axioms. This implies that the commutator subgroup of is simple. Moreover, we show that the commutator subgroup of is dense in . As a corollary we obtain that is topologically simple.     

Infinitesimal orbit type theorem for normalizable actions

Let α:G×M→M$\alpha :G\times M\to M$ be a differentiable action of a Lie group G$G$ on a connected differentiable manifold M$M$. Assume that the local tube theorem holds at every x∈M$x\in M$. Then the following is proved: there is a maximal infinitesimal orbit type, called the principal, such that the union of the orbits belonging to this infinitesimal type is an open and dense set in M$M$.     

A preparation theorem for Weierstrass systems

It is shown that Lion and Rolin's preparation theorem for globally subanalytic functions holds for the collection of definable functions in any expansion of the real ordered field by a Weierstrass system.

     

Equivariant class group. II. Enriched descent theorem

We prove a version of Grothendieck’s descent theorem on an ‘enriched’ principal fiber bundle, a principal fiber bundle with an action of a larger group scheme. Using this, we prove the isomorphisms of the equivariant Picard and the class groups arising from such a principal fiber bundle.     

Equivariant Hopf bifurcation for neutral functional differential equations

In this paper we employ an equivariant Lyapunov-Schmidt procedure to give a clearer understanding of the one-to-one correspondence of the periodic solutions of a system of neutral functional differential equations with the zeros of the reduced bifurcation map, and then set up equivariant Hopf bifurcation theory. In the process we derive criteria for the existence and direction of branches of bifurcating periodic solutions in terms of the original system, avoiding the process of center manifold reduction.

     

Equivariant entire cyclic cohomology: I. Finite groups

We extend the framework of entire cyclic cohomology to the equivariant context.Supported in part by the Department of Energy under Grant DE-FG02-88ER25065.     

A subanalytic triangulation theorem for real analytic orbifolds

Let X be a real analytic orbifold. Then each stratum of X is a subanalytic subset of X. We show that X has a unique subanalytic triangulation compatible with the strata of X. We also show that every Cr-orbifold, 1?r?∞, has a real analytic structure. This allows us to triangulate differentiable orbifolds. The results generalize the subanalytic triangulation theorems previously known for quotient orbifolds.     

The Equivariant Analytic Index for Proper Groupoid Actions

The paper constructs the analytic index for an elliptic pseudodifferential family of L^m-operators invariant under the proper action of a continuous family groupoid on a G-compact, C⁰ G-space.Mathematical Subject Classification (1991): 19K35, 19K56, 22A22, 46L80, 46L87, 58B34, 58J40; Secondary: 19L47, 55N15, 58B15, 58H05.     

紧黎曼对称空间到Grassmann流形的等变等距极小浸入

在本文中，我们利用李群及其表示理论作为主要工具，　讨论了紧黎曼对称空间到Ｇｒａｓｓｍａｎｎ　流形的等变等距极小浸入问题．     

The Reduction of Equivariant Dynamics to the Orbit Space for Compact Group Actions

Symmetry introduces degeneracies in dynamical systems, as well as in bifurcation problems. An “obvious” idea in order to remove these degeneracies is to project the dynamics onto the quotient space obtained by identifying points in phase space which lie in the same group orbits (the so-called orbit space). Unfortunately, several difficulties arise when one tries to implement this idea. First, the orbit space is not, in general a manifold. Second, how does one explicitely realize the orbit space, and how does one compute and analyze the projected dynamics? In this paper I will describe the methods which have been developped in order to answer these questions, and I will show on three examples how they apply. We shall see that, although not always suitable to treat equivariant dynamics, these methods sometimes lead to insightful reductions.     

Equivariant Kasparov Theory and Generalized Homomorphisms

Let G be a locally compact group. We describe elements of KK ^G (A, B) by equivariant homomorphisms, following Cuntz's treatment in the non-equivariant case. This yields another proof for the universal property of KK ^G : It is the universal split exact stable homotopy functor. To describe a Kasparov triple (, , F) for A, B by an equivariant homomorphism, we have to arrange for the Fredholm operator F to be equivariant. This can be done if A is of the form ; and more generally if the group action on A is proper in the sense of Exel and Rieffel.     

Equivariant cohomology and localization for Lie algebroids

Let M be a manifold carrying the action of a Lie group G, and let A be a Lie algebroid on M equipped with a compatible infinitesimal G-action. Using these data, we construct an equivariant cohomology of A and prove a related localization formula for the case of compact G. By way of application, we prove an analog of the Bott formula.     

The affine deformation space of a rank two Schottky group: a picture gallery

The Margulis invariant of an affine hyperbolic element measures the signed Lorentzian displacement along the unique closed geodesic in its class. Given a group of affine transformations of Minkowski spacetime whose linear part is Schottky, the Margulis invariant is a useful tool in determining properness of its action. This paper, based on a presentation given at CGG IV Oostende 2005, describes the affine deformation space of a rank two Schottky group. Several illustrations are included, contrasting the difference between the case of a three-holed sphere and that of a one-holed torus. Paper based on a presentation given at CGG IV Oostende 2005. Attendance of the conference and the subsequent writing of this paper was made possible by an NSERC Discovery Grant.     

The Lagrange inversion theorem in the smooth case

The classical Lagrange inversion theorem is a concrete, explicit form of the implicit function theorem for real analytic functions. An explicit construction shows that the formula is not true for all merely smooth functions. The authors modify the Lagrange formula by replacing the smooth function by its Maclaurin polynomials. The resulting modified Lagrange series is, in analogy to the Maclaurin polynomials, an approximation to the solution function accurate to o(x^N) as x→0.     

Integration of locally exponential Lie algebras of vector fields

A locally convex Lie algebra is said to be locally exponential if it belongs to some local Lie group in canonical coordinates. In this note we give criteria for locally exponential Lie algebras of vector fields on an infinite-dimensional manifold to integrate to global Lie group actions. Moreover, we show that all necessary conditions are satisfied if the manifold is finite-dimensional connected and σ-compact, which leads to a generalization of Palais’ Integrability Theorem.      

Partial category actions on sets and topological spaces

We introduce (continuous) partial category actions on sets (topological spaces) and show that each such action admits a universal globalization. Thereby, we obtain a simultaneous generalization of corresponding results for groups, by Abadie, and Kellendonk and Lawson, and for monoids, by Megrelishvili and Schröder. We apply this result to the special case of partial groupoid actions where we obtain a sharpening of a result by Gilbert, concerning ordered groupoids, in the sense that mediating functions between universal globalizations always are injective.     

Expansiveness of algebraic actions on connected groups

We study endomorphism actions of a discrete semigroup on a connected group . We give a necessary and sufficient condition for expansiveness of such actions provided is either a Lie group or a solenoid.

     

A non-fixed point theorem for Hamiltonian lie group actions

We prove that, under certain conditions, if a compact connected Lie group acts effectively on a closed manifold, then there is no fixed point. Because two of the main conditions are satisfied by any Hamiltonian action on a closed symplectic manifold, the theorem applies nicely to such actions. The method of proof, however, is cohomological; and so the result applies more generally.