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

强无限Halpern-Lauchli猜想（关于完美树的积的划分关系）仍然是开放的，但是在假设可测基数存在的情况下已经取得了进展。也就是说，这个猜想的博弈论版本已经在这个假设下得到了证明。我们将研究这些技术是否可以用来证明整个猜想。同时，还将研究测量的必要性。在其他涉及大基数的工作中，已经证明了一个非常大基数的假设暗示了一个关于有限左分配代数(满足定律a(bc) = （ab)（ac）的代数）的组合命题。将研究是否需要大型枢机。本文还将探讨组合结果对经典辫状群的影响。本研究将大基数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。“大型基数机”是指具有额外技术特性的超大型基数机。众所周知，大基数是如此之大，以至于它们的存在无法用通常的证明方法来证明。这使得大基数对有限世界的影响非常令人惊讶，但事实确实如此。的确，有一些关于有限结构的数学断言，令人惊讶的是，它们需要一个大基数来证明。最近用大基数证明了一些经典数学中的有限命题。其中一种说法与“辫群”有关（这是数学家们研究的一个主题，自20世纪80年代中期发现它与物理学的联系以来，他们进行了非常深入的研究）。这个项目的研究是双重的：确定大基数是否必要，并追求大基数理论对编织群体和其他群体的影响的迷人可能性。

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.