The talk will be the occasion to take a few minutes to describe the situation of math education in France and in Europe, to motivate the interest of the lecturer in trying to bring back rigorous proofs in integration theory. The remaining 45 minutes will be devoted to explaining the basics of Henstock-Kurzweil integration theory, which, although not a response to education problems, is a modern and elementary approach of a very strong extension of the Riemann integral, providing easy access to several fundamental results of Lebesgue theory (monotone convergence theorem, existence of Lebesgue measure, etc.).

We will report on our attempts to understand noncommutative geometry in the lights of the example of quantum spheres. We will see how to produce an equivariant fundamental class and also indicate some of the limitations of isospectral deformations.

This talk is concerned with the evolution of an unbounded inviscid fluid-fluid interface subject to an axisymmetric impulse in pressure and how inertial, interfacial and gravitational forces affect that evolution. The construct was motivated by the occurrence of lung hemorrhage resulting from ultrasonic imaging and pursues the notion that bursts of ultrasound act to expel droplets that puncture the soft air-filled sacs in the lung plural surface allowing them to fill with blood. The evolution of the free surface is described by a boundary integral formulation which is integrated forward in time numerically. As the interface evolves, it is seen, depending upon the levels of gravity and surface tension, to form either axisymmetric surface jets, waves or droplets. Moreover the droplets may be spherical, inverted tear-shaped or pancake like. Also of interest is the finite time singularity which occurs when the drop pinches off; this is seen to be of the power law type with an exponent of 2/3.