Some integral geometric results in the simply isotropic space

The measurability with respect to the group of the motions in the three dimensional simply isotropic space I₃⁽¹⁾ of sets of geometric elements is studied and also some Crofton type formulas are given.     

Some geometric properties of Riemann's non-differentiable function

Riemann's non-differentiable function is a celebrated example of a continuous but almost nowhere differentiable function. There is strong numeric evidence that one of its complex versions represents a geometric trajectory in experiments related to the binormal flow or the vortex filament equation. In this setting, we analyse certain geometric properties of its image in $\mathbb{C}$. The objective of this note is to assert that the Hausdorff dimension of its image is no larger than 4/3 and that it has nowhere a tangent.     

Some geometric and dynamical properties of the Urysohn space

This is a survey article about the geometry and dynamical properties of the Urysohn space. Most of the results presented here are part of the author's Ph.D. thesis and were published in the articles [J. Melleray, Stabilizers of closed sets in the Urysohn space, Fund. Math. 189 (1) (2006) 53-60; J. Melleray, Compact metrizable groups are isometry groups of compact metric spaces, Proc. Amer. Math. Soc. 136 (4) (2008) 1451; J. Melleray, On the geometry of Urysohn's universal metric space, Topology Appl. 154 (2007) 384-403]; a few results are new, most notably the fact that Iso(U) is not divisible.     

Some geometric properties of the Bakry-Émery-Ricci tensor

The Bakry-Émery tensor gives an analog of the Ricci tensor for a Riemannian manifold with a smooth measure. We show that some of the topological consequences of having a positive or nonnegative Ricci tensor are also valid for the Bakry-Émery tensor. We show that the Bakry-Émery tensor is nondecreasing under a Riemannian submersion whose fiber transport preserves measures up to constants. We give some relations between the Bakry-Émery tensor and measured Gromov-Hausdorff limits.     

Specialization of integral dependence for modules

We establish the principle of specialization of integral dependence for submodules of finite colength of free modules, as part of the general algebraic-geometric theory of the Buchsbaum–Rim multiplicity. Then we apply the principle to the study of equisingularity of ICIS germs, obtaining results for Whitney’s Condition A and Thom’s Condition A_f. Notably, we describe these equisingularity conditions for analytic families in terms of various numerical invariants, which, for the most part, depend only on the members of a family, not on its total space. Oblatum 18-X-1996 & 16-X-1998 / Published online: 10 June 1999     

A categorification of integral Specht modules

We suggest a simple definition for categorification of modules over rings and illustrate it by categorifying integral Specht modules over the symmetric group and its Hecke algebra via the action of translation functors on some subcategories of category for the Lie algebra .