Points And Curves In The Monster Tower

Cartan introduced the method of prolongation which can be applied either to manifolds with distributions (Pfaffian systems) or integral curves to these distributions. Repeated application of prolongation to the plane endowed with its tangent bundle yields the Monster tower, a sequence of manifolds, each a circle bundle over the previous one, each endowed with a rank $2$ distribution. In an earlier paper (2001), the authors proved that the problem of classifying points in the Monster tower up to symmetry is the same as the problem of classifying Goursat distribution flags up to local diffeomorphism. The first level of the Monster tower is a three-dimensional contact manifold and its integral curves are Legendrian curves. The philosophy driving the current work is that all questions regarding the Monster tower (and hence regarding Goursat distribution germs) can be reduced to problems regarding Legendrian curve singularities.

Let $A$ be a finite abelian group. The author sets up an algebraic framework for studying $A$-equivariant complex-orientable cohomology theories in terms of a suitable kind of equivariant formal group. He computes the equivariant cohomology of many spaces in these terms, including projective bundles (and associated Gysin maps), Thom spaces, and infinite Grassmannians.

"This volume is a collection of papers presented at the 11th International Workshop on Real and Complex Singularities, held July 26-30, 2010, in Sao Carlos, Brazil, in honor of David Mond's 60th birthday. This volume reflects the high level of the conference discussing the most recent results and applications of singularity theory. Articles in the first part cover pure singularity theory: invariants, classification theory, and Milnor fibres. Articles in the second part cover singularities in topology and differential geometry, as well as algebraic geometry and bifurcation theory: Artin-Greenberg function of a plane curve singularity, metric theory of singularities, symplectic singularities, cobordisms of fold maps, Goursat distributions, sections of analytic varieties, Vassiliev invariants, projections of hypersurfaces, and linearity of the Jacobian ideal."--P. [4] of cover.

The modern theory of singularities provides a unifying theme that runs through fields of mathematics as diverse as homological algebra and Hamiltonian systems. It is also an important point of reference in the development of a large part of contemporary algebra, geometry and analysis. Presented by internationally recognized experts, the collection of articles in this volume yields a significant cross-section of these developments. The wide range of surveys includes an authoritative treatment of the deformation theory of isolated complex singularities by prize-winning researcher K Miyajima. Graduate students and even ambitious undergraduates in mathematics will find many research ideas in this volume and non-experts in mathematics can have an overview of some classic and fundamental results in singularity theory. The explanations are detailed enough to capture the interest of the curious reader, and complete enough to provide the necessary background material needed to go further into the subject and explore the research literature.

"July 2011, volume 212, number 996 (first of 4 numbers)."

Let G be a simple algebraic group defined over an algebraically closed field k whose characteristic is either 0 or a good prime for G, and let uEG be unipotent. The authors study the centralizer CG(u), especially its centre Z(CG(u)). They calculate the Lie algebra of Z(CG(u)), in particular determining its dimension; they prove a succession of theorems of increasing generality, the last of which provides a formula for dim Z(CG(u)) in terms of the labelled diagram associated to the conjugacy class containing u.

The authors prove that if $F$ is a finitely generated free group and $\phi$ is an automorphism of $F$ then $F\rtimes_\phi\mathbb Z$ satisfies a quadratic isoperimetric inequality. The authors' proof of this theorem rests on a direct study of the geometry of van Kampen diagrams over the natural presentations of free-by-cylic groups. The main focus of this study is on the dynamics of the time flow of $t$-corridors, where $t$ is the generator of the $\mathbb Z$ factor in $F\rtimes_\phi\mathbb Z$ and a $t$-corridor is a chain of 2-cells extending across a van Kampen diagram with adjacent 2-cells abutting along an edge labelled $t$. The authors prove that the length of $t$-corridors in any least-area diagram is bounded by a constant times the perimeter of the diagram, where the constant depends only on $\phi$. The authors' proof that such a constant exists involves a detailed analysis of the ways in which the length of a word $w\in F$ can grow and shrink as one replaces $w$ by a sequence of words $w_m$, where $w_m$ is obtained from $\phi(w_{m-1})$ by various cancellation processes. In order to make this analysis feasible, the authors develop a refinement of the improved relative train track technology due to Bestvina, Feighn and Handel.

The author develops a non-abelian version of $p$-adic Hodge Theory for varieties (possibly open with ``nice compactification'') with good reduction. This theory yields in particular a comparison between smooth $p$-adic sheaves and $F$-isocrystals on the level of certain Tannakian categories, $p$-adic Hodge theory for relative Malcev completions of fundamental groups and their Lie algebras, and gives information about the action of Galois on fundamental groups.

"Volume 205, number 962 (first of 5 numbers)."

For a given plane domain, the author adds a constant multiple of the Dirac measure at a point in the domain and makes a new domain called a quadrature domain. The quadrature domain is characterized as a domain such that the integral of a harmonic and integrable function over the domain equals the integral of the function over the given domain plus the integral of the function with respect to the added measure. The family of quadrature domains can be modeled as the Hele-Shaw flow with a free-boundary problem. The given domain is regarded as the initial domain and the support point of the Dirac measure as the injection point of the flow.