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

DMS 9626713 Richard Laver强无限Halpern-Lauchli猜想(关于完美树乘积的分拆关系)仍然是开放的，但在假设存在一个可测基数的情况下已经取得了进展。也就是说，在这一假设下，该猜想的博弈论版本得到了证明。将调查这些技术是否可以应用于证明整个猜想。此外，还将研究可测量的必要性。在其他涉及大基数的工作中，已经证明了一个非常大的基数假设蕴含着关于有限个左分配代数(满足定律a(Bc)=(Ab)(Ac)的代数)塔的一个组合陈述。是否需要较大的基数，将会进行研究。此外，还将研究组合结果对经典辫子群的影响的可能性。这项研究将大的基本理论与一些经典数学联系起来。“基数”是大小的数学概念，因为它适用于有限和无限集合。“大型红衣主教”指的是极大的这种尺寸，带有额外的技术特征。众所周知，大基数是如此之大，以至于无法用通常的证明方法来证明它们的存在。这使得大型基数对有限世界的影响非常令人惊讶，但它们确实存在。的确，有关于有限结构的数学断言，令人惊讶的是，它们的证明需要一个大的基数。最近，一些经典数学中的有限命题已被用大基数证明。其中一种说法与“辫子群”有关。“辫子群”是数学家们研究过的一个问题，自上世纪80年代中期S发现它与物理学的联系以来，人们进行了非常深入的研究。这个项目的研究有两个方面：确定大基数是否必要，以及追寻大基数理论对辫子群和更远的地方的迷人可能性。

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.