Multiplicative Galois structure of algebraic number fields

自然数与神秘有关。例如，人们早就知道，黎曼ζ函数是通过解析方法建立起来的，它“知道”质数的分布，因为它包含了这个日期。对自然数的研究导致了代数的发展，代数也有它们的ζ函数和它们所“知道”的奥秘。似乎有更多的这类结构性信息只是逐渐被揭示出来。代数数也有对称性，结果是ζ函数“知道”很多这些。特别是，它们在负整数处的值足以预测非常详细的这种联系，正如Iwasawa（1969）所推测的那样。这个经典的主猜想由Wiles（1990）和我们（2011）以更强的“等变”形式证明。详细的联系涉及算术对象，如理想类群和单位群，并在代数对象，如上同调群和k理论中给出，它们提供了解析侧和算术侧之间的联系。目前的研究更关注于更好地掌握这些代数对象，希望找到我们新认识的算术应用。

The natural numbers have associated mysteries. For example, it has long been known that the Riemann zeta function, which is built from them by analytic means, 'knows' about the distribution of the prime numbers, in the sense that this date is encoded in it.The study of natural numbers leads to algebraic numbers, which also have their zeta functions and the mysteries of what they 'know'. There seems to be much more structural information of this kind which is only gradually being unravelled.The algebraic numbers also have symmetries, and it turns out that that the zeta functions 'know' a lot about these. In particular, their values at negative integers suffice to predict a very detailed such connection, as conjectured by Iwasawa (1969). This classical Main Conjecture was proved by Wiles (1990) and in a stronger 'equivariant' form by us (2011).The detailed connections concern arithmetic objects, like ideal class groups and unit groups, and are give in terms of algebraic objects, like cohomology groups and K-theory, which provide the links between the analytic and arithmetic sides. The present research is now more concerned with getting a better grasp of these algebraic objects, in hopes of finding arithmetic applications of our new understanding.