In this talk we consider particular solutions of the Boltzmann equation which have the form $f (x,v,t) = g (v − M (t)x,t)$ where $M (t) = A(I + tA)^{−1}$ with the matrix $A$ describing a shear flow or a dilatation or a combination of both. These solutions are known as equidispersive solutions. We will show that, for different choices for the matrix A and for different homogeneities of the collision kernel, we obtain different long time asymptotics for the corresponding equidispersive solutions. In particular we will focus on the case of simple shear flow and prove rigorously the existence of self-similar solutions with exponentially increasing internal energy.

Stallings' theorem states that a finitely generated group splits over a finite subgroup if and only if it has more than one end. As a consequence of this, group splittings over finite subgroups are invariant under quasi-isometry. I will discuss a generalisation of Stallings' theorem which shows that under suitable hypotheses, group splittings over classes of infinite groups, namely coarse $PD_n$ groups, are also invariant under quasi-isometry.

A central tool in the construction of moduli spaces throughout algebraic geometry and beyond, geometric invariant theory (GIT) aims to sensibly answer the question, "How can we quotient an algebraic variety by a group action?" In this talk I will explain some basics of GIT and indicate how it can be used to build moduli spaces, before exploring one of its salient features: the non-canonicity of the quotient. I will show how the dependence on an additional parameter, a choice of so-called 'linearisation', leads to a rich 'wall crossing' picture, giving different interrelated models of the quotient. Time permitting, I will also speak about recent developments in non-reductive GIT, and joint work extending this dependence to the non-reductive setting.

Spectra provide a way of understanding cohomology theories in terms of homotopy theory. Spectra are a bit like CW-complexes, they have homotopy groups which may be used to characterize homotopy equivalences. However, a spectrum has homotopy groups in negative degrees, too, and they are abelian groups in all degrees. We will discuss spectra representing ordinary cohomology, bordism, and K-theory.

Numbers like the special values of the gamma and the Bessel functions or the Euler-Mascheroni constant are not expected to be periods in the usual sense of algebraic geometry. However, they can be regarded as coefficients of the comparison isomorphism between two cohomology theories associated to pairs consisting of an algebraic variety and a regular function: the de Rham cohomology of a connection with irregular singularities, and the so-called “rapid decay cohomology”. Following ideas of Kontsevich and Nori, I will explain how this point of view allows one to construct a Tannakian category of exponential motives over a subfield of the complex numbers. The upshot is that one can attach to exponential periods a Galois group that conjecturally governs all algebraic relations between them. Classical results and conjectures in transcendence theory may be reinterpreted in this way. No prior knowledge of motives will be assumed, and I will focus on examples rather than on the more abstract aspects of the theory. This is a joint work with P. Jossen (ETH Zürich).