We develop a p-adic version of the so-called Grothendieck-Teichmüller theory (which studies $\text{Gal}\left(\overline{Q}/Q\right)$ by means of its action on profinite braid groups or mapping class groups). For every place v of $\overline{Q}$, we give some geometrico-combinatorial descriptions of the local Galois group $\text{Gal}({\overline{Q}}_v/Q_v)$ inside $\text{Gal}\left(\overline{Q}/Q\right)$. We also show that $\text{Gal}({\overline{Q}}_p/Q_p)$ is the automorphism group of an appropriate ${\pi }_1$-functor in p-adic geometry.

It is shown that if X is a random variable whose density satisfies a Poincaré inequality, and Y is an independent copy of X, then the entropy of (X + Y)/√2 is greater than that of X by a fixed fraction of the entropy gap between X and the Gaussian of the same variance. The argument uses a new formula for the Fisher information of a marginal, which can be viewed as a local, reverse form of the Brunn-Minkowski inequality (in its functional form due to A. Prékopa and L. Leindler).

The main result asserts the existence of noncontractible periodic orbits for compactly supported time-dependent Hamiltonian systems on the unit cotangent bundle of the torus or of a negatively curved manifold whenever the generating Hamiltonian is sufficiently large over the zero section. The proof is based on Floer homology and on the notion of a relative symplectic capacity. Applications include results about propagation properties of sequential Hamiltonian systems, periodic orbits on hypersurfaces, Hamiltonian circle actions, and smooth Lagrangian skeletons in Stein manifolds.

This paper introduces a general method for relating characteristic classes to singularities of a bundle map. The method is based on the notion of geometric atomicity. This is a property of bundle maps $\alpha :E\to F$ which universally guarantees the existence of certain limits arising in the theory of singular connections. Under this hypothesis, each characteristic form $\text{Φ}$ of E or F satisfies an equation of the form

$$\text{Φ}=L+dT$$,

where L is an explicit localization of Φ along the singularities of α and T is a canonical form with locally integrable coefficients. The method is constructive and leads to explicit calculations. For normal maps (those transversal to the universal singularity sets) it retrieves classical formulas of R. MacPherson at the level of forms and currents; see Part I ([HL4]). It also produces such formulas for direct sum and tensor product mappings. These are new even at the topological level. The condition of geometric atomicity is quite broad and holds in essentially every case of interest, including all real analytic bundle maps. An important aspect of the theory is that it applies even in cases of "excess dimension," that is, where the the singularity sets of α have dimensions greater than those of the generic map. The method yields explicit number of examples are worked out detail.

We obtain a classification of Borel measurable, GL(n) covariant, symmetric-matrix-valued valuations on the space of n-dimensional convex polytopes. The only ones turn out to be the moment matrix corresponding to the classical Legendre ellipsoid and the matrix corresponding to the ellipsoid recently discovered by E. Lutwak, D. Yang, and G. Zhang.

Consider an analytic function f which has a Siegel disk properly contained in the domain of holomorphy. We prove that if the rotation number is of bounded type, then f has a critical point in the boundary of the Siegel disk.