
THEMATIC PROGRAMS 

July 5, 2020  
Thematic Program on
the Foundations of Computational Mathematics

Monday

Swaminathan Sethu Raman 
PAST TALKS  
Friday

Andy Hammerlindl 
Monday

Chris Conidis 
Monday

Johanna Franklin Effective randomness There are three primary approaches to formalizing a notion of effective randomness for infinite binary sequences: one based on incompressibility, one based on unpredictability, and one that is measuretheoretic. I will describe all these approaches and show how they can be made to be equivalent. Then I will present different ways in which an infinite binary sequence can be said to be very nonrandom and discuss the properties that these sequences must have. 
Monday

Jonathan Hauenstein 
Monday

Cristóbal Rojas The basic idea of algorithmic randomness is that an individual algorithmic random sequence should satisfy all the ``effective'' probability laws. There are several different possible definitions, depending on the kind of the considered probability laws and their ``degree of effectivity''. A very natural source of probability laws is the ergodic theory of dynamical systems. In this talk we present some recent results relating algorithmic randomness to the statistical properties of the trajectories of points in ergodic dynamical systems. We do this in the framework of general computable probability spaces (the effective version of the spaces where usual ergodic theory takes place). First we extend algorithmic randomness to this setting and develop some useful tools. Then, we introduce a ``dynamical'' notion of randomness: typicality. Roughly, a point is typical for some ergodic dynamics if it follows the statistical behavior of the system (given by Birkhoff's pointwise ergodic theorem) with respect to every bounded continuous function used to follow its trajectory (or equivalently, every computable function). The main result is the following characterization: in any computable probability space, a point is Schnorr random if and only if it is typical for every mixing computable dynamical system. 
Monday 
Chin How Jeffrey Pang 
Monday 
Michael Coons 
Monday 
Workshop on Computational Differential Geometry, Topology, and Dynamics NO SEMINAR 
Monday

Alexander Grigo The study of mechanical systems has a several centuries long history. During the previous century there have been major results concerning the stability (KAM theory) and instability (hyperbolicity) in such dynamical systems. A particularly nice class of ``mechanical'' systems for which rigorous results are known are geodesic flows on compact manifold without boundary, and billiards. I will discuss several results on stable and random dynamics in billiards and related systems. The talk will not require any prior knowledge on billiards. 