John Forbes Nash and his wife Alicia were tragically killed in a car crash on May 23, having just returned from a ceremony in Norway where John Nash received the prestigious Abel Prize in Mathematics (which, along with the Fields Medal, is the closest thing mathematics has to a Nobel Prize). Nash’s long struggle with mental illness, as well as his miraculous recovery, are depicted vividly in Sylvia Nasar’s book “A Beautiful Mind” and the Oscar-winning film which it inspired. In this post, I want to give a brief account of Nash’s work in game theory, for which he won the 1994 Nobel Prize in Economics. Before doing that, I should mention, however, that while this is undoubtedly Nash’s most influential work, he did many other things which from a purely mathematical point of view are much more technically difficult. Nash’s Abel Prize, for example (which he shared with Louis Nirenberg), was for his work in non-linear partial differential equations and its applications to geometric analysis, which most mathematicians consider to be Nash’s deepest contribution to mathematics. You can read about that work here. Continue reading
In my last post, I wrote about two infinite games whose analysis leads to interesting questions about subsets of the real numbers. In this post, I will talk about two more infinite games, one related to the Baire Category Theorem and one to Diophantine approximation. I’ll then talk about the role which such Diophantine approximation questions play in the theory of dynamical systems.
The Choquet game and the Baire Category Theorem
The Cantor game from Part I of this post can be used to prove that every perfect subset of is uncountable. There is a similar game which can be used to prove the Baire Category Theorem. Let be a metric space. In Choquet’s game, Alice moves first by choosing a non-empty open set in . Then Bob moves by choosing a non-empty open set . Alice then chooses a non-empty open set , and so on, yielding two decreasing sequences and of non-empty open sets with for all . Note that ; we denote this set by . Alice wins if is empty, and Bob wins if is non-empty. Continue reading