Mathematical Sciences: Monodromy Preserving Deformation on Z2

点击翻译按钮获取中文摘要

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.