Mathematical Sciences: Statistical Mechanics and Quantum Field Theory

这个数学物理项目与物理学家以启发式方式使用的重整化群技术的严格分析有关。（基本的物理问题是确定具有已知短程相互作用的大粒子系统的宏观性质。）其中一些工作涉及到使用计算机来帮助寻找定理的证明，这些定理需要复杂的估计量的方案。重整化群技术已被证明是统计力学中一个强有力的工具，特别是在临界现象理论中。同样的技术在构造量子场论的现代方法中也起着主导作用。目前这两个领域的大部分研究都是关于将现有方法从相对简单的系统扩展到更现实的系统。科赫教授将继续研究与临界现象理论相关的模型。他的第二个重点是对无序系统的分析，如Hopfield模型和Sherrington - Kirkpatrick模型，它们在一定范围的参数下表现出自旋玻璃行为。

This project in mathematical physics has to do with the rigorous analysis of renormalization group techniques used in a heuristic fashion by physicists. ( The basic physical problem is to determine macroscopic properties of large systems of particles with known short range interactions. ) Some of the work involves a novel use of the computer to assist in finding proofs of theorems that require intricate schemes for estimating quantities. Renormalization group techniques have proved to be a powerful tool in statistical mechanics, especially in the theory of critical phenomena. The same techniques also play a dominant role in the modern approach to constructive quantum field theory. Much of the current research in these two areas is concerned with the extension of present methods from relatively simple to more realistic systems. Professor Koch will continue his investigation of models relevant to the theory of critical phenomena. His second main emphasis is an analysis of disordered systems such as the Hopfield model and the Sherrington - Kirkpatrick model, which exhibit spin glass behavior for a certain range of parameters.