Probability and Mean Field Models for Spin Glasses

自旋玻璃平均场模型的意外行为是由G. Parisi在70年代发现的。这些模型是简单的数学对象。它们表现出如此微妙的行为是值得注意的，它们严谨的研究是如此具有挑战性。PI相信，阐明这些模型将为概率论带来一个全新的方向。10年来，他一直在尝试建立研究它们的新方法。其中一个基本目标（证明给出谢林顿-柯克帕特里克模型自由能的帕里西公式）已经实现，但这一成功并不能掩盖这样一个事实，即我们的总体理解仍然相当有限。特别是，两个核心问题（被称为超对称性猜想和混沌问题）似乎一如既往地困难。它们可能与分析问题有关。PI计划研究这些非常困难的问题，但也将努力在不那么令人生畏的问题上取得渐进式进展，特别是研究新的数学规范平均场哈密顿量，它展示了一种新型的复制对称方程。至少从伽利略开始，数学和物理就在基本方面相互影响，而且这种相互影响比以往任何时候都要强烈。在70年代，物理学家发现某些合金（称为自旋玻璃）以一种非常规的方式对磁性模拟做出反应。他们引入了新的模型来解释这些行为。这些模型是简单而自然的数学对象。但是很长一段时间没有数学方法来研究它们。物理学家的早期工作使用的启发式方法不一定是完全可靠的。目前的项目是消除这种差异的长期计划的一步，通过开发新的数学工具，特别是概率论，来完全理解这些基本模型。这个项目的长期成果可能会推动概率论的发展，概率论是数学应用中最重要的分支之一。

The unexpected behavior of mean-field models for spin glasses was discovered in the seventies by G. Parisi. These models are simple mathematical objects. It is remarkable that they exhibit such a subtle behavior, and that their rigorous study is so challenging. The PI believes that elucidating these models will bring the discovery of an entire new direction in probability theory. He has been trying for 10 years to build new methods to study them. One of the fundamental objectives (the proof of the Parisi formula giving the free energy of the Sherrington-Kirkpatrick model) has been attained, but this success cannot hide the fact that overall our understanding remains rather limited. In particular, two central questions, (known as the Ultrametricity conjecture and the Chaos problem) seem as hard as ever. They are possibly related to questions of analysis. The PI plans to work on these very hard questions, but also will strive to make incremental progress on less daunting issues, in particular the study of new mathematically canonical mean field Hamiltonians that exhibit replica symmetric equations of a new type. Mathematics and Physics have influenced each other in fundamental ways since at least Galileo, and this mutual influence is stronger than ever. In the seventies, Physicists discovered that certain alloys (called spin glasses) responded in an unconventional way to magnetic simulation. They introduced new models to explain these behaviors. These models are simple and natural mathematical objects. Yet for a long time there existed no mathematical method to study them. The early work of the physicists uses heuristic methods that need not be completely reliable. The present project it a step in the long range program of eliminating this discrepancy, by developing new tools from mathematics, and in particular probability theory, to completely understand these fundamental models. The long rang outcome of this program could be a new impetus on probability theory, one of the most important branch of mathematics for applications.