Dynamical entropies are measures of the complexity of orbit structures. The topological entropy considers all the orbits, whereas the measure theoretic entropy focuses on those ``relevant" to a given invariant probability measure. The variational principle says that the topological entropy of a continuous self-map of a compact metrizable space is the supremum of the measure theoretic entropy over the set of invariant probability measures for the map.

A well known fact is that every transitive hyperbolic (Anosov) diffeomorphism has a unique invariant probability measure whose entropy equals the topological entropy. We analyze a class of deformations of Anosov diffeomorphisms containing many of the known nonhyperbolic robustly transitive diffeomorphisms. We show that these $C0$-small, but $C1$-macroscopic, deformations leave all the high entropy dynamics of the Anosov system unchanged, and that there is a partial conjugacy identifying all invariant probability measures with entropy close to the maximum for the deformation with those of the original Anosov system.

Additionally, we show that these results apply to a class of nonpartially hyperbolic, robustly transitive diffeomorphisms described by Bonatti and Viana and a class originally described by Mane. In fact these methods apply to several classes of systems which are similarly derived from Anosov, i.e., produced by an isotopy from an Anosov system.

Determining the long-term behavior of large biochemical models has proved to be a remarkably difficult problem. Yet these models exhibit several characteristics that might make them amenable to study under the right perspective. One possible approach (first suggested by Sontag and Angeli) is their decomposition in terms of so-called monotone systems, which can be thought of as systems with exclusively positive feedback. In this talk I discuss some general properties of monotone dynamical systems, especially classical and recent results regarding their generic convergence towards an equilibrium. Then I will discuss the use of monotone systems to model biochemical behaviors such as global attractivity to an equilibrium, switches and oscillations under time delays.

We will review some of the results and conjectures on dynamics of the standard map. The talk will serve as a short introduction to the subject accessible for interested graduate students.

Ergodic theory of dispersing billiards was developed in 1970s-1980s. An important part of the theory is the analysis of the structure of the sets where the billiard map is discontinuous. They were assumed to be smooth manifolds till recently, when a new pathological type of behavior of these sets was found. Thus a reconsideration of earlier arguments was needed.
I'll review the recent work which recover the ergodicity results, explain the main difficulties and some further progress.