Mathematical Sciences: Monodromy Preserving Deformation on Z2

二维Ising模型是晶格上磁自旋通过最近邻耦合相互作用的数学模型。在物理学中有大量关于统计系统在临界点（相变）附近的行为的文献，但很少有模型可以分析地验证推测的行为。伊辛模型的重要性在于它是一个显著的例外（在二维空间中）。在那里，人们有足够的自旋相关知识来检验一些关于临界现象的更详细的猜想。这对于统计力学、量子场论和分析所依据的数学理论是很重要的。帕尔默教授是统计力学中可解模型（如伊辛模型）方面的专家。他在开发相关的无限维群论和代数几何方面走在前列，这些理论在这些物理应用中取得了巨大的成功。在之前的工作中，他引入了一组晶格模型，这些模型是对伊辛模型的数学推广。在目前的项目中，他提议证明依赖于无限自旋群先前工作的缩放假设。他还在研究将伊辛模型中的某种不变性解释为与一组有限差分算子相关的行列式束作用的规范群的表现的可能性。这可以澄清可解晶格模型中共形对称的外观。

The two dimensional Ising model is a mathematical model for magnetic spins on a lattice which interact by nearest neighbor coupling. There is a vast literature in physics concerning the behavior of statistical systems near critical points (phase transitions), but there are few models in which it is possible to verify the conjectured behavior analytically. The importance of the Ising model is that it is a notable exception (in two dimensions). There, one has sufficient knowledge of the spin correlations to examine some of the more detailed conjectures about critical phenomena. This is of importance for statistical mechanics, quantum field theory, and the mathematical theories upon which the analysis rests. Professor Palmer is an expert on solvable models, such as the Ising model, in statistical mechanics. He is in the forefront in exploiting the relevant infinite dimensional group theory and algebraic geometry that has been so successful in these physical applications. In prior work he introduced a family of lattice models which are mathematical generalizations of the Ising model. In the current project he proposes to prove scaling hypotheses that rely upon prior work on infinite spin groups. He is also investigating the possibility of interpreting a certain invariance in the Ising model as the manifestation of a gauge group acting of the determinant bundle associated with a family of finite difference operators. This could clarify the appearance of conformal symmetry in solvable lattice models.