Prof. Michael Yampolsky (University of Toronto) will give a series of 4 lectures on "Computability and Computational Complexity Questions in Dynamics" at the Scuola Normale Superiore in Pisa. The lectures will take place on Thursday October 24 and 31 and November 7 and 14, at 2:30 pm in Aula Volterra, in Piazza dei Cavalieri 7. The titles and abstracts of each of the 4 lectures are given below. Please contact me if you have any questions. Stefano Marmi stefano.marmi@sns.it

1) October 24:

Computability and computational complexity in dynamics: can we trust numerical predictions of limiting behavior of orbits?

The development of the modern subject of dynamical systems went hand-in-hand with numerical modeling. However, the theoretical basis of such modeling has remained largely unexplored and offers exciting challenges. What can and cannot be computed about the behavior of a dynamical system? I will give an overview of some of the recent results and directions of study.

2) October 31:

Don’t believe your lying eyes: (non)-computability of Julia sets

The talk will be a deep dive into the question of computability and computational complexity of Julia sets. Their fractal images are among the most-computed pictures in all of mathematics. But can we trust what these computations show? The results are surprising, and involve some beautiful complex analysis. The talk will be based on my work with M. Braverman, as well as more recent results obtained with A. Dudko, C. Rojas, and others.

3) November 7:

How to lose at Monte Carlo: computing statistical behavior of dynamical systems

Introduced by Ulam and von Neumann in the 1940s, Monte Carlo technique remains the most ubiquitous tool for statistical modeling. Surprisingly, our recent work with C. Rojas shows that such modeling provably fails for some very simple dynamical systems. I will present this theorem and discuss the resulting mathematical challenges.

4) November 14:

Thurston versus Turing: algorithmic hardness of classifying finite dynamical systems

Historically, the algorithmic aspects of classification problems have motivated much of the research in geometric topology (knot theory, classification of surfaces and 3-manifolds, etc). These questions are also very fruitful in dynamics. I will discuss the joint results with N. Selinger, K. Rafi and others on algorithmic decidability of Thurston equivalence and the connection with geometrization questions in dynamics.