This dilemma between 12 and 13 was the object of dispute between Isaac Newton and David Gregory in 1694. The problem of 13 balls is a typical example of that branch of combinatorial geometry that deals with packings and coverings. In packing problems the aim is to place figures of a given shape or size without overlap, as economically as possible, either inside another given figure or subject to some other restriction. The presentation will be on an intuitive basis, without recourse to any advanced theoretical considerations or abstractions. No prior knowledge of the subject is required.
Interested undergraduates are welcome to attend.
We discuss a conjecture of P. Frankl and V. Rodl concerning an extremal problem on set systems. It is known that this conjecture, if true, would provide yet another proof of the Density Theorem. We survey known results and results in progress which confirm the conjecture in certain cases. Free probability theory was introduced by Voiculescu in the context of operator algebras. Classical independence of random variables is closely connected to Cartesian products of measure spaces, i.e., tensor products of algebras of random variables on measure spaces.