报告题目: Probability measures on a space, two-valued topologies and localic toposes of probability measures

报告人: Laurent Lafforgue（菲尔兹奖得主、法国科学院院士）

时间: 2023年7月2日下午3：00-5：15

地点: 东区第二教学楼2104教室

报告摘要: 

We show that the notion of probability law on a space can be reformulated in terms of that of Grothendieck topology on a certain category associated with this space. This passage is based on the law of large numbers which, in the reformulation, ceases to be a result and becomes part of the definition.

Probability theory thus appears to be immersed in topology, as soon as we extend the latter to the study of toposes, and no longer just classical topological spaces. This way of looking at probabilities is both more abstract than their classical modeling by measure theory, and closer to immediate statistical experience: indeed, the underlying category introduced in this reformulation is entirely defined in terms of countings and frequencies of incidence observed in trials.

Grothendieck's topologies that correspond to laws of probability are those that define two-valued toposes, meaning that they are expressed in the terms of a binary logic that knows only true or false.

It can be shown that the same topos-theoretic line of thought naturally extends the notion of measure to any Grothendieck site, and therefore to any geometric context as well as to any first-order theory.

报告人简介：Laurent Lafforgue是菲尔兹奖得主。曾任法国国家科学研究中心主任研究员、法国高等科学研究所数学终身教授。2000年荣获克雷研究奖（Clay Research Award），2001年荣获法国科学院颁发的雅克•赫尔布兰德奖（Jacques Herbrand Prize）。2002年在第24 届国际数学家大会上，Laurent Lafforgue因在数论和代数几何方面的突出贡献荣获菲尔兹奖（Fields Medal）。2003年荣获法国政府颁发的荣誉骑士级勋章，同年当选法国科学院院士。Laurent Lafforgue秉承同是菲尔兹奖得主 Vladimir Drinfeld证明GL2的方法，将朗兰兹猜想推广到在特征为正的代数曲线函数域的GLn上成立。Laurent Lafforgue二百多页简洁的证明可谓是天才之作，充分展现了其作为一名伟大数学家所具有的恒心和毅力。