Three-dimensional geometry and topology. Volume 1, by William P. Thurston, (edited by Sylvio Levy). Princeton Mathematical Series, 35. £29.95. 1997. ISBN 0-691-08304-5 (Princeton University Press).

The notion of Anosov representations has been introduced by Labourie in his study of the Hitchin component for SL(n,R). Subsequently, Anosov representations have been studied mainly for surface groups, in particular in the context of higher Teichmuller spaces, and for lattices in SO(1,n). In this article we extend the notion of Anosov representations to representations of arbitrary word hyperbolic groups and start the systematic study of their geometric properties. In particular, given an Anosov representation Γ→G we explicitly construct open subsets of compact G-spaces, on which Γ acts properly discontinuously and with compact quotient. As a consequence we show that higher Teichmuller spaces parametrize locally homogeneous geometric structures on compact manifolds. We also obtain applications regarding (non-standard) compact Clifford–Klein forms and compactifications of locally symmetric spaces of infinite volume.

Anosov representations: domains of discontinuity and applications

Basics of spin geometry.- Explicit computations of spectra.- Lower eigenvalue estimates on closed manifolds.- Lower eigenvalue estimates on compact manifolds with boundary.- Upper eigenvalue bounds on closed manifolds.- Prescription of eigenvalues on closed manifolds.- The Dirac spectrum on non-compact manifolds.- Other topics related with the Dirac spectrum.

The Dirac Spectrum

The notion of Anosov representations has been introduced by Labourie in his study of the Hitchin component for SL(n,R). Subsequently, Anosov representations have been studied mainly for surface groups, in particular in the context of higher Teichmueller spaces, and for lattices in SO(1,n). In this article we extend the notion of Anosov representations to representations of arbitrary word hyperbolic groups and start the systematic study of their geometric properties. In particular, given an Anosov representation of $\Gamma$ into G we explicitly construct open subsets of compact G-spaces, on which $\Gamma$ acts properly discontinuously and with compact quotient. 
As a consequence we show that higher Teichmueller spaces parametrize locally homogeneous geometric structures on compact manifolds. We also obtain applications regarding (non-standard) compact Clifford-Klein forms and compactifications of locally symmetric spaces of infinite volume.

Anosov representations: Domains of discontinuity and applications

Seifert fibred homology 3-spheres and the Yang-Mills equations on Riemann surfaces with marked points

The index of a holomorphic flow with an isolated singularity

In complex hyperbolic geometry we consider an open set biholomorphic to an open ball in Cn, and we equip it with a particular metric that makes it have constant negative holomorphic curvature. This is analogous to but different from the real hyperbolic space. In the complex case, the sectional curvature is constant on complex lines, but it changes when we consider real 2-planes which are not complex lines.

Complex Hyperbolic Geometry

Euler obstruction and indices of vector fields

In this article we study the singularities defined by real analytic maps 

$$ \left( {\mathbb{R}^m ,0} \right) \to \left( {\mathbb{R}^2 ,0} \right) $$

with an isolated critical point at the origin, having a Milnor fibration. It is known [14] that if such a map has rank 2 on a punctured neighbourhood of the origin, then one has a fibre bundle φ : S m−1 − → S 1, where K is the link. In this case we say that f satisfies the Milnor condition at 0 ∈ ℝ m . However, the map φ may not be the obvious map $ \frac{f} {{\parallel f\parallel }} $ as in the complex case [14, 9]. If f satisfies the Milnor condition at 0 ∈ ℝ m and for every sufficiently small sphere around the origin the map $ \frac{f} {{\parallel f\parallel }} $ defines a fibre bundle, then we say that f satisfies the strong Milnor condition at 0 ∈ ℝ m . In this article we first use well known results of various authors to translate “the Milnor condition” into a problem of finite determinacy of map germs, and we study the stability of these singularities under perturbations by higher order terms. We then complete the classification, started in [20, 21] of certain families of singularities that satisfy the (strong) Milnor condition. The simplest of these are the singularities in ℝ2 n ≅ ℂ n of the form $\{ \sum _{i = 1}^nz_i^{{a_i}}z_i^{ - {b_i}} = 0, {a_i} > {b_i} \geqslant 1\}$ We prove that these are topologically equivalent (but not analytically equivalent!) to Brieskorn-Pham singularities.

On Real Singularities with a Milnor Fibration

Let V be a compact local complete intersection defined as the zero set of a section of a holomorphic vector bundle over the ambient space. For each connected component S of the singular set Sing(V) of V, we define the Milnor class μ * (V,S) in the homology of S. The difference between the Schwartz-MacPherson class and the Fulton-Johnson class of V is shown to be equal to the sum of μ * (V,S) over the connected components S of Sing(V). This is done by proving Poincare-Hopf type theorems for these classes with respect to suitable tangent frames. The 0-degree component μ 0 (V, S) coincides with the Milnor numbers already defined by various authors in particular situations. We also give an explicit formula for μ * (V, S) when S is a non-singular component and V satisfies the Whitney condition along S.

/pdf/milnor-classes-of-local-complete-intersections-4l14ea73is.pdf

José Seade

Papers

The index of a holomorphic flow with an isolated singularity

Complex Hyperbolic Geometry

Euler obstruction and indices of vector fields

On Real Singularities with a Milnor Fibration

Milnor classes of local complete intersections