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力学的背景下在物理学中发展起来的。70年代末，在费根鲍姆的开创性工作中，重整化思想为动力系统理论铺平了道路。重整化是现代理论物理和数学中最有力的方法之一。其主要思想是以一种简化的方式来看待一个大尺度的系统，它只考虑了大块的有效相互作用，而忽略了小尺度系统的许多细节。利用重整化的思想，通常可以建立通用的标度律来描述许多看似不同的系统的渐近行为。在统计力学中，大尺度既可以对应大的空间距离和体积，也可以对应动力系统中的大时间。值得注意的是，在许多例子中，这种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。简单不动点对应于概率论中扩散的标准标度指数。同时，对于相变附近系统或具有临界点的映射，其临界指数对应于高度非平凡重整化不动点。这种临界系统是重整化理论的重要研究对象。提出的研究项目旨在研究两种不同环境下的关键不动点。其中之一是处理临界圈图。另一个与定向聚合物理论有关。这些都是非常活跃的研究领域，我们希望对这两种情况下的普遍标度性质的理解做出重大贡献。

Renormalization theory was first developed in physics in the context of quantum field theory and statisticalmechanics. At the end of 70's in the pioneering work of M. Feigenbaum the renormalization ideas paved theirway into the theory of dynamical systems. By now renormalization is one of the most powerful methods inmodern 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.