Lattice-Statistical Models and Applications

9614170吴这个项目是关于统计物理中晶格模型的理论研究，重点是精确的结果和应用。虽然大多数精确的结果仅限于2维，但该项目建议通过围绕2维解决方案来研究3维系统。具体计划包括：分析二维量子自旋系统与三维经典量子自旋系统之间的关系，进一步研究关于Potts模型配分函数零点分布的某些猜想，以及三维定向渗流问题。其他项目包括：晶体表面粗糙的研究，三元聚合物溶液中的凝胶-溶胶转变，整数限制分割的数学问题，以及纽结和扭结理论中新的链接不变量的推导。这位PI因其对材料相关问题的更多数学方面的贡献而广为人知。在对材料的基本性质和相变的研究中，这种PI做出了贡献，使问题简化到可以得到数学上精确和严格的解的程度。然后，通过建立在确切结果的基础上，可以理解更现实的问题。目前的大部分赠款支持在某些简化模型上大量(3维)搜索精确解，其中解是已知的薄膜(2-d)。***

9614170 Wu This project is concerned with theoretical studies of lattice models in statistical physics with emphasis on exact results and applications. While most exact results are limited to 2-dimensions, this project proposes to study the 3-dimensional systems by building around the 2-d solution. Specific plans include: analysis of the relationship between a quantum spin system in 2-dimension and a classical one in 3-d, further work on certain conjectures on the distribution of zeroes of the Potts model partition function and directed percolation problem in 3-d. Additional projects involve: investigation of roughening of crystal surfaces, gel-sol transition in a ternary polymer solution, the mathematical problem of restricted partitions of an integer and derivation of a new link invariant in the theory of knots and kinks. %%% This PI is widely known for his contributions to more mathematical aspects of the materials related problems. In research on the fundamental properties of materials and phase transitions, this PI has made contributions where the problem has been simplified to the extent that a mathematically exact and rigorous solution is possible. The more realistic problems then can be understood by building up on the exact results. Much of the present grant supports search for exact solutions in bulk (3-dimensional) on certain simplified models where the solution is known for a thin film (in 2-d). ***