Combinatorics, Banach Spaces, Probability, Spin Glasses

小行星9703879 本研究集中于抽象概率的两个主要工具， 测量（约束复杂函数波动的通用工具）和 优化措施的理论（一个通用的工具，以找到上限和下限的 复杂函数的期望值）。调查人员将努力追查 基础理论的发展，也将试图找到新的应用， 特别是Banach空间和Hilbert空间的几何学。他还将尝试使用 在理解自旋平均场模型方面取得进展的抽象方法 眼镜，这是一个很难获得严格结果的主题。一个单独的项目 在于研究大维度离散立方体的子集，以及它们的 极值性质 真实的生活现象通常会受到我们无法控制的随机事件的影响。 因此，依赖于许多独立影响的随机变量是一个很好的模型 复杂的真实的手术的结果。令人惊讶的是，通用工具最近 被开发来严格控制这些随机变量的波动在一个广泛的 各种各样的情况，他们展示了抽象的力量和相关性 具体情况的方法。这些方法的发展和研究 新的申请将继续进行。将特别努力使用这些方法 以获得自旋玻璃模型的严格结果。这些数学模型 凝聚态非常有趣，在物理学中有相当大的兴趣， 社区

9703879 Talagrand This research centers on two main tools of abstract probability, concentration of measure (a versatile tool to bound the fluctuations of complicated functions) and the theory of majorizing measures (a versatile tool to find upper and lower bounds for the expected values of complicated functions). The investigator will try to pursue the development of the underlying theories, and will also try to find new applications, in particular to the geometry of Banach spaces and Hilbert spaces. He will also try to use abstract methods to achieve progress on the understanding of mean field models for spin glasses, a subject where rigorous results are very hard to obtain. A separate project consists in the study of the subsets of the discrete cube of large dimension, and of their extremal properties. Real life phenomenon are usually affected by random events outside our control. Random variables that depend on many independent influences are therefore a good model for the outcome of complicated real procedures. Surprisingly general tools have recently been developed to rigorously control the fluctuations of these random variables in a wide variety of situations, and they demonstrate the power and the relevance of abstract methods to concrete situations. The development of these methods and the research of new applications will be continued. A particular effort will be made to use these methods to obtain rigorous results for spin glass models. These are mathematical models for some very intriguing states of condensed matter, that are of considerable interest in the physics community.