The first lecture will be an accessible introduction (peut-être en français) to smooth one-dimensional dynamics. After some historical examples and results, we shall consider the dynamics of some maps $f_a : x \mapsto ax(1-x)$ from the quadratic family.

In what follows, we shall investigate typical behaviour from the probabilistic viewpoint. In particular, we shall show, under certain conditions, the existence of probability measures which describe the statistical behaviour of Lebesgue almost every orbit. Time permitting, the Markov extension, together with its utility in proving results concerning continuity of statistical properties, will be presented.

The first lecture will be an accessible introduction (peut-être en français) to smooth one-dimensional dynamics. After some historical examples and results, we shall consider the dynamics of some maps $f_a : x \mapsto ax(1-x)$ from the quadratic family.

In what follows, we shall investigate typical behaviour from the probabilistic viewpoint. In particular, we shall show, under certain conditions, the existence of probability measures which describe the statistical behaviour of Lebesgue almost every orbit. Time permitting, the Markov extension, together with its utility in proving results concerning continuity of statistical properties, will be presented.

The first lecture will be an accessible introduction (peut-être en français) to smooth one-dimensional dynamics. After some historical examples and results, we shall consider the dynamics of some maps $f_a : x \mapsto ax(1-x)$ from the quadratic family.

In what follows, we shall investigate typical behaviour from the probabilistic viewpoint. In particular, we shall show, under certain conditions, the existence of probability measures which describe the statistical behaviour of Lebesgue almost every orbit. Time permitting, the Markov extension, together with its utility in proving results concerning continuity of statistical properties, will be presented.

Thermodynamic formalism originated as an approach wihin statistical mechanics in physics, in order to explain, among other things, the changes of aggregation states (phase transitions). It was introduced into dynamical systems and mathematically formalised in the 1970s by the work of Sinai, Ruelle and Bowen, mainly in the context of hyperbolic (and symbolic) systems, and found several applications, including dimension theory and fractal geometry. Gradually, also non-uniformly hyperbolic systems entered the picture, with reappearance of phase transitions. The study thermodynamic properties of interval maps is part of this, and in these lectures I will discuss them in greater detail.

Among the topics covered:
- Historic context (including the Ising model)
- Hyperbolic dynamics and the Grifftih-Ruelle Theorem
- Non-uniformly hyperbolic dynamics: Intermittent maps vs Hofbauer potential
- Interval dynamics: Collet-Eckmann maps and Feigenbaum maps and everything in between.

The lectures will be on graduate level or up.

Thermodynamic formalism originated as an approach wihin statistical mechanics in physics, in order to explain, among other things, the changes of aggregation states (phase transitions). It was introduced into dynamical systems and mathematically formalised in the 1970s by the work of Sinai, Ruelle and Bowen, mainly in the context of hyperbolic (and symbolic) systems, and found several applications, including dimension theory and fractal geometry. Gradually, also non-uniformly hyperbolic systems entered the picture, with reappearance of phase transitions. The study thermodynamic properties of interval maps is part of this, and in these lectures I will discuss them in greater detail.

Among the topics covered:
- Historic context (including the Ising model)
- Hyperbolic dynamics and the Grifftih-Ruelle Theorem
- Non-uniformly hyperbolic dynamics: Intermittent maps vs Hofbauer potential
- Interval dynamics: Collet-Eckmann maps and Feigenbaum maps and everything in between.

The lectures will be on graduate level or up.

Thermodynamic formalism originated as an approach wihin statistical mechanics in physics, in order to explain, among other things, the changes of aggregation states (phase transitions). It was introduced into dynamical systems and mathematically formalised in the 1970s by the work of Sinai, Ruelle and Bowen, mainly in the context of hyperbolic (and symbolic) systems, and found several applications, including dimension theory and fractal geometry. Gradually, also non-uniformly hyperbolic systems entered the picture, with reappearance of phase transitions. The study thermodynamic properties of interval maps is part of this, and in these lectures I will discuss them in greater detail.

Among the topics covered:
- Historic context (including the Ising model)
- Hyperbolic dynamics and the Grifftih-Ruelle Theorem
- Non-uniformly hyperbolic dynamics: Intermittent maps vs Hofbauer potential
- Interval dynamics: Collet-Eckmann maps and Feigenbaum maps and everything in between.

The lectures will be on graduate level or up.

We consider a class of Liouville equations that arise in differential geometry when prescribing the Gaussian curvature of a surface and in models of mathematical physics describing stationary Euler flows and self-dual Chern-Simons equations. We discuss methods, variational in nature, to derive general existence results from suitable improvements of the Moser-Trudinger inequality combined with Morse-theoretical methods. We will treat in particular the case with Dirac masses representing, in the above motivations, conical singularities or vortex points.