Maximal functions of various forms have played crucial roles in harmonic analysis. Various outstanding open problems are related to Lp boundedness (estimate) of the associated maximal functions. In this talk, we discuss Lp boundedness of maximal functions given by averages over curves.

We will discuss certain main problems concerning group actions on 1-dimensional manifolds (the circle and the interval) and perspectives for future research.

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KAI-X Distinguished lecture

Questions of parameter estimation – that is, finding the parameter values that allow a model to best fit some data – and parameter identifiability – that is, the uniqueness of such parameter values – are often considered in settings where experiments can be repeated to gain more certainty about the data. In this talk, however, I will consider parameter estimation and parameter identifiability in situations where data can only be collected from a single experiment or trajectory. Our motivation comes from medical settings, where data comes from a patient; such limitations in data also arise in finance, ecology, and climate, for example. In this setting, we can try to find the best parameters to fit our limited data. In this talk, I will introduce a novel, alternative goal, which we refer to as a qualitative inverse problem. The aim here is to analyze what information we can gain about a system from the available data even if we cannot estimate its parameter values precisely. I will discuss results that allow us to determine whether a given model has the ability to fit the data, whether its parameters are identifiable, the signs of model parameters, and/or the local dynamics around system fixed points, as well as how much measurement error can be tolerated without changing the conclusions of our analysis. I will consider various classes of model systems and will illustrate our latest results with the classic Lotka-Volterra system.

Hénon maps were introduced by Michel Hénon as a simplified model of the Poincaré section of the Lorenz model. They are among the most studied discrete-time dynamical systems that exhibit chaotic behavior. Complex Hénon maps in any dimension have been extensively studied over the last three decades, in parallel with the development of pluripotential theory. We will present the dynamical properties of these maps such as the behavior of point orbits, variety orbits, equidistribution of periodic points and fine ergodic properties of the systems. This talk is based on the work of Bedford, Fornaess, Lyubich, Sibony, Smillie, and on recent work of the speaker in collaboration with Bianchi and Sibony.