Mathematical Sciences: Statistical Mechanics of Disordered Systems and Conformal Field Theory

Kupiainen教授的统计力学研究项目有两个部分，一个是无序系统的物理学，另一个是共形场论在二维临界现象中的应用。对于无序系统，他将致力于发展精确的重整化群方法来研究随机环境中的随机游动、在随机场或随机交换耦合存在下的界面涨落以及自旋玻璃模型。在共形场理论方面，他将继续研究使用Wess-Zumino-Witten模型来求解各种共形理论。统计力学研究由大量相互作用的粒子组成的系统的行为。通常，粒子会排列在晶格中，最近的邻居之间会发生相互作用。基本问题是在给定关于交互的信息的情况下预测系统的总体、长期行为(平衡、相变等)。从数学上讲，即使在最简单的模型中，这也是一项非常困难的任务。当存在随机性因素时，就会出现进一步的复杂性，就像库皮亚宁教授将要进行的研究一样。

Professor Kupiainen's research project in statistical mechanics has two parts, one in the physics of disordered systems and the other in the application of conformal field theory to two-dimensional critical phenomena. In connection with disordered systems, he will endeavor to develop exact renormalization group methods for the study of random walks in random environments, of interface fluctuations in the presence of random fields or random exchange couplings, and of spin glass models. On the conformal field theory side, he will continue investigations in which the Wess-Zumino-Witten model is used to solve various conformal theories. Statistical mechanics studies the behavior of systems consisting of very large numbers of particles interacting with one another. Typically, the particles would be arrayed in a lattice and interactions would take place between nearest neighbors. The basic problem is predict the overall, long-run behavior (equilibria, phase transitions, and the like) of the system given information about the interactions. Mathematically, this is a very difficult undertaking even in the simplest models. A further complication occurs when elements of randomness are present, as in the research to be undertaken by Professor Kupiainen.