Bursting oscillations in nerve cells have been the focus of a great deal of attention by mathematicians. These are typically studied by taking advantage of multiple time-scales in the system under study to perform a singular perturbation analysis. Bursting also occurs in hormone-secreting pituitary cells, but is characterized by fast bursts with small electrical impulses. Although the separation of time-scales is not as clear, singular perturbation analysis is still the key to understand the bursting mechanism. In particular, we will show that canards are responsible for the observed oscillatory behaviour.

Suspension/colloid transport in porous media is a basic phenomenon in environmental, petroleum and chemical engineering. Suspension of particles moves through porous media and particles are captured by straining or attraction. We revise the classical equations for particle mass balance and particle capture kinetics and show its non-realistic behaviour in cases of large dispersion and of flow-free filtration. In order to resolve the paradoxes, the pore-scale model is derived. The model can be transformed to Boltzmann equation with particle distribution over pores. Introduction of sink-source terms into Boltzmann equation results in much more simple calculations if compared with the traditional Chapman-Enskog averaging procedure. Technique of projecting operators in Hilbert space of Fourier images is used. The projection subspace is constructed in a way to avoid dependency of averaged equations on sink-source terms. The averaging results in explicit expressions for particle flux and capture rate. The particle flux expression describes the effect of advective particle velocity decrease if compared with the carrier water velocity due to preferential capture of "slow" particles in small pores. The capture rate kinetics describes capture from either advective or diffusive fluxes. The equations derived exhibit positive advection velocity for any dispersion and particle capture in immobile fluid that resolves the above-mentioned paradox. Finally, we discuss validation of the model for propagation of contaminants in aquifers, for filtration, for potable water production by artesian wells, for formation damage in oilfields.

Histograms are convenient and easy-to-use tools for estimating the shape of data, but they have serious problems which are magnified for multivariate data. We combine classic histograms with shape estimation by polynomials. The new relatives, `polynomial histograms', have surprisingly nice mathematical properties, which we will explore in this talk. We also show how they can be used for real data of 10-20 dimensions to analyse and understand the shape of these data.

Noncommutative geometry (in the sense of Alain Connes) involves replacing a conventional space by a "space" in which the algebra of functions is noncommutative. The simplest truly non-trivial noncommutative manifold is the noncommutative 2-torus, whose algebra of functions is also called the irrational rotation algebra. I will discuss a number of recent results on geometric analysis on the noncommutative torus, including the study of nonlinear noncommutative elliptic PDEs (such as the noncommutative harmonic map equation) and noncommutative complex analysis (with noncommutative elliptic functions).