Renormalization for Circle Maps with Singularities and for Directed Polymers

重整化理论最初是在量子场论和统计学的背景下在物理学中发展起来的。 力学70年代末，M.费根鲍姆重整化思想为他们 进入动力系统理论。到目前为止，重整化是最强大的方法之一， 现代理论物理学和数学。其主要思想是以一种简化的方式来看待大尺度系统，这种方式只考虑大块体的有效相互作用，而忽略了小尺度系统的许多细节。利用重整化的思想，人们通常可以建立描述许多看似不同的系统的渐近行为的普适标度律。大尺度可以对应于统计力学中的大空间距离和体积，也可以对应于动力系统中的大时间。值得注意的是，在许多例子中，这样的普遍行为可以被严格地研究。粗略地说，它可以用重整化的不动点来描述。简单的不动点对应于概率论中扩散的标准标度指数。同时临界指数系统附近的相变，或与临界点映射对应的高度非平凡重整化不动点。这种临界系统对于重整化理论是非常有意义的。 拟议的研究项目旨在研究两种不同环境中的临界不动点。其中之一涉及临界圆图。另一个是定向聚合物理论。这些都是非常活跃的研究领域，我们希望在这两种情况下的普遍标度性质的理解作出重大贡献。

Renormalization theory was first developed in physics in the context of quantum field theory and statistical mechanics. At the end of 70's in the pioneering work of M. Feigenbaum the renormalization ideas paved their way into the theory of dynamical systems. By now renormalization is one of the most powerful methods in modern theoretical physics and mathematics. The main idea is to look at a system at large scales in a simplified way which takes into account only effective interaction of large blocks and ignores many fine details of a system at small scales. Using the idea of renormalization one can often establish universal scaling laws which describe asymptotic behaviour of many seemingly different systems. The large scale can correspond to either large spatial distances and volumes in statistical mechanics, or to large time in the context of dynamical systems. Remarkably, in many examples such universal behaviour can be studied rigorously. Roughly speaking, it can be described in terms of fixed points of renormalization. Simple fixed points correspond to standard scaling exponents as in the case of diffusion in probability theory. At the same time critical exponents for systems near phase transitions, or for maps with critical points correspond to highly non-trivial renormalization fixed points. Such critical systems are of great interest for renormalization theory. The proposed research project aims to study critical fixed points in two different settings. One of them deals with critical circle maps. The other one is related to the theory of directed polymers. These are very active research areas, and we hope to contribute significantly to the understanding of universal scaling properties in both cases.