Mathematical Sciences: Problems in Large Cardinals, Forcing,and Combinatorics

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DMS 9626713 Richard Laver The strong infinite Halpern-Lauchli Conjecture (a partition relation about a product of perfect trees) is still open, but progress has been made assuming the existence of a measurable cardinal. Namely, a game theoretic version of the conjecture has been proved under that assumption. It will be investigated whether these techniques can be applied to prove the entire conjecture. Also, the necessity of the measurable will be studied. In other work involving large cardinals, it has been proved that a very large cardinal assumption implies a combinatorial statement about a tower of finite left distributive algebras (algebras satisfying the law a(bc) = (ab)(ac)). It will be studied whether the large cardinals are needed. Also the possibility of implications of the combinatorial result for the classical braid groups will be investigated. This research links large cardinal theory with some classical mathematics. A "cardinal" is the mathematical notion of size as it applies to finite and infinite sets. A "large cardinal" is an extremely large such size, with additional technical features. It is known that large cardinals are so large that their existence can't be proved by the usual methods of proof. This makes it very surprising that large cardinals have implications for the finite world, but they do. Indeed, there are mathematical assertions about finite structures which, surprisingly, need a large cardinal for their proof. Recently some finite statements which are in the vein of classical mathematics have been proved using large cardinals. One of the statements is related to the "braid groups" (a subject which as been studied by mathematicians, with very intensive investigations since the mid-1980's upon discovery of its connections with physics). The research in this project is twofold: to determine if the large cardinal is necessary, and to pursue the fascinating possibility of implications of large cardinal theory for the braid groups and beyond.