We give a survey of recent result regarding scaling limits of systems from statistical mechanics, as well as the universality of the behaviour of such systems in so-called cross-over regimes. It transpires that some of these universal objects are described by singular stochastic PDEs. We then give a survey of the recently developed theory of regularity structures which allows to build these objects and to describe some of their properties. We place particular emphasis on the renormalisation procedure required to give meaning to these equations.

These are expanded notes of the 20th Takagi Lectures held at The University of Tokyo on November 4, 2017.

中文翻译：

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抛物线随机偏微分方程的重整化

我们对统计力学中系统的缩放限制以及此类系统在所谓的交叉机制中行为的普遍性的最新结果进行了调查。事实证明，其中一些通用对象是由奇异随机偏微分方程描述的。然后，我们对最近发展的规则结构理论进行了调查，该理论允许构建这些对象并描述它们的一些属性。我们特别强调赋予这些方程意义所需的重整化过程。

这些是 2017 年 11 月 4 日在东京大学举行的第 20 届高木讲座的扩展笔记。