From September 24 to 28, 2025, the Autumn School on Topos and Applications—co-hosted by the School of Mathematics and Statistics at Wuhan University, the National Center for Mathematics (Tianyuan) Central China, and the Sino-French Joint Research Center—was successfully held in Lecture Hall B102 of Leijun Science and Technology Building at Wuhan University. The workshop focused on modern advancements in topos theory and its interdisciplinary applications, featuring short courses and academic lectures delivered by Professor Laurent Lafforgue, Fields Medalist and member of the French Academy of Sciences, alongside his research team. Participants included scholars, postdoctoral researchers, and graduate students from prestigious institutions such as Tsinghua University, the Academy of Mathematics and Systems Science (Chinese Academy of Sciences), Fudan University, University of Science and Technology of China, as well as international universities including the University of Oxford, Freie Universität Berlin, and École Normale Supérieure de Lyon. The event aimed to foster academic exchange and advance talent development in the field.
Professor Lafforgue is a world-renowned mathematician. He was awarded the Fields Medal in 2002 for his groundbreaking contributions to number theory and algebraic geometry—specifically for proving the Langlands conjecture for function fields of algebraic curves over fields of positive characteristic for GLnGLn. Formerly a mathematics professor at the Institut des Hautes Études Scientifiques (IHES) in France, he joined Huawei’s Paris Research Institute in 2021 to explore topos theory and its potential applications in fields such as communications and artificial intelligence.
Professor Lafforgue taught a short course titled Moduli problems, classifying toposes, invariants, and the six operations formalism during this seminar. He systematically expounded on Grothendieck’s core ideas regarding topos theory and their profound impact on modern mathematics. This theory bridges two initially distinct mathematical directions: on one hand, the geometric problem of moduli spaces, and on the other, the topological-algebraic construction of cohomological and homological invariants. The course aimed to introduce these themes and demonstrate how Grothendieck reformulated and deepened them within the highly general framework of topos theory.
In addition to Professor Lafforgue, the seminar featured short courses by Professor Olivia Caramello from the University of Insubria (Italy) and Professor Aurelien Sagnier from Sorbonne University (France), as well as invited talks by Dr. Bruno Drieux from École Polytechnique (France) and Dr. Gabriel Merlin from Université Paris-Saclay and the Grothendieck Institute (France/Italy).
During the seminar, Professor Fan Huijun, Professor Zuo Kang, and other scholars engaged in profound discussions with the keynote speaker on topics related to topos theory. The students listened attentively and actively participated in the Q&A session, sparking lively debates on technical implementations of sheaf learning and potential intersections between topos theory and large language models. This demonstrated the young scholars' keen sensitivity and enthusiasm for interdisciplinary exploration.
This seminar not only deepened the interdisciplinary integration of topos theory within mathematics but also fostered its dialogue with fields such as artificial intelligence and mathematical logic. Topos theory embodies a remarkable unifying power and abstraction in modern mathematics, and the expansion of its application boundaries will continue to invigorate innovation in fundamental sciences.
