无处不在的随机事件
现代世界是由不断流动的随机事件组成的,从商业物流到凝聚态物理学,一切事物都受随机性的影响。随机性能够以多种方式出现,一种最常见的,也可以说是最重要的随机性类型,是由“高斯过程”产生的。
“高斯分布”有时也被称为“正态分布”或“钟形曲线”,在我们的世界中以惊人的频率出现:比如婴儿出生时的体重、学生在学校的考试成绩、运动员的退役年龄……这些都是完全遵循高斯分布的随机事件。
Michel Talagrand是随机过程领域的专家,他的大部分工作与理解和利用高斯分布有关。1952年,Talagrand出生于法国。他在1977年获得巴黎第六大学的数学博士学位,并在美国俄亥俄州立大学工作了几年,是获奖无数的法国科学院的成员。
Michel Talagrand。(图/Peter Bagde / Typos1 / Abel Prize 2024)
今天,挪威科学和文学学院决定将2024年阿贝尔奖授予Michel Talagrand,以表彰他在概率论和随机过程方面所做的工作。阿贝尔奖有时也被称为数学界的诺贝尔奖。
三个领域
对随机现象的透彻理解,对当今世界显然是至关重要的。例如,我们的天气预报和大型语言模型都是基于随机算法而构建的。Talagrand的众多突破性发现的共同主题,都有助于我们理解并研究我们身边的随机过程。
这次,Talagrand的获奖与他具体在以下三个领域作出的贡献有关。
随机过程的上确界
随机过程可以产生一系列随机值,而“上确界”是这些值的集合中预期的最大值。如果冲击海滩的海浪高度是一个随机过程,那么了解明年冲击海滩的最大海浪可能有多高就将很有用。
Talagrand的大部分工作都与随机过程的几何有关。他通过发展通用链式(generic chaining)理论,为高斯过程的上确界提供了清晰的上界和下界。
测量的集中性
概率论的一个关键结论是大数定律,即归一化的独立随机变量的和,会向着其均值收敛。因此,这个归一化的和是集中的,或者说是自平均的。数学家们逐渐意识到,这种集中无处不在,许多被定义为大量独立随机变量的函数的随机变量,似乎以高概率接近它们的均值。
换言之,当一个过程依赖于一系列不同的随机来源时,不同的随机因素不仅不会变得更加复杂,反而有可能相互补偿并产生可预测性更高的结果。对此,Talagrand对这一与直觉相反的现象给出了精确的定量估计,为适用于高维统计问题的非渐近独立性理论奠定了基础。
自旋玻璃态
抛开抽象的概率论,“自旋玻璃态”是一种特殊的物质形式,其中的原子可以自我排列。长时间以来,许多物理学家一直着迷于这种无序系统所表现出的复杂行为。
1980年,Giorgio Parisi提出了该类型中的一个最简单模型的自由能表达式。数学家Francesco Guerra严谨地证明了这个公式是自由能的上界。在一篇开创性的文章中,Talagrand用他的统计学和概率论知识,证明了自由能的互补下界,完成了让Parisi荣获2021年诺贝尔物理学奖的工作的证明,也为自旋玻璃态的数学理论的发展及其在统计学习中的应用奠定了基础。
高产的数学家
Talagrand是一位多产的数学家,他的工作改变了概率论、泛函分析和统计学。他的研究特点是渴望在最基本的层面上理解有趣的问题,并在此过程中建立新的数学理论。
阿贝尔奖委员会主席Helge Holden教授说:“Talagrand是一位杰出的数学家,也是一位出色的问题解决者。他对我们理解随机过程,尤其是高斯过程做出了深远的贡献。他的工作重塑了概率论的几个领域。此外,他证明了著名的自旋玻璃态的自由能帕里西公式,这是一个惊人的成就。”
#创作团队:
整理:原原
排版:雯雯
#参考来源:
https://abelprize.no/
#图片来源:
封面图&首图:Peter Badge / Typos1 / Abel Prize 2024
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