授業の目的【日本語】

See English text. 
授業の目的【英語】

Scholze became an instant mathematical superstar with his 2011 thesis that introduced "Perfectoid Spaces." His theory has vastly expanded the reach of geometrical and analytical methods in padic geometry. Condensed mathematics holds the promise of accomplishing the same in a broader framework that, in addition to padic geometry, includes complex geometry and analysis. Such a unification is entirely new and may well supply the geometrical underpinning to make it possible to attack some of the most important conjectures in mathematics. The goal of the course is to study this new framework, as it is being developed. 
到達目標【日本語】

N/A. 
到達目標【英語】

N/A. 
授業の内容や構成

In many situations, topology and algebra do not interact well. For example, the category of topological abelian groups is not an abelian category. To wit, the identity map from the real numbers with the discrete topology to the real numbers with the usual topology does not have a kernel or a cokernel. Condensed mathematics is a replacement of the notion of a topological space that, on the one hand, does not loose any information, and, on the other hand, interacts as well with algebra as one could possibly hope. By definition, a condensed set is a sheaf of sets on the category of profinite sets with finite mutually surjective families of maps as coverings. A topological space X gives rise to a condensed set that to a profinite set S assigns the set of continuous maps from S to X. However, there are many more condensed sets, including the "missing" kernel and cokernel of the map above. Moreover, condensed sets form a topos, so condensed sets behave just like sets, except that the axiom of choice does not hold in general. So we can define condensed abelian groups, condensed rings, etc. as usual. The category of condensed abelian groups is an abelian category, so there are no problems doing homological algebra therein. In fact, among abelian categories, the category of condensed abelian groups is particularly easy to work with, since it has enough compact projective generators. 
履修条件

Some knowledge of homological algebra is necessary, and knowledge of some algebraic topology and algebra geometry would be helpful. 
関連する科目

Algebraic geometry, algebraic topology. 
成績評価の方法と基準

Evaluation is based on lectures given in class and written notes. 
教科書・テキスト

https://www.math.unibonn.de/people/scholze/Condensed.pdf https://www.math.unibonn.de/people/scholze/Analytic.pdf 
参考書

Clausen and Scholze gave twenty lectures about condensed mathematics at an online masterclass organized by the University of Copenhagen in November 2020. The lectures are available at https://www.youtube.com/playlist?list=PLAMniZX5MiiLXPrD4mpZO9oiwhev5Uq and notes from the lectures are available at https://www.math.ku.dk/english/calendar/events/condensedmathematics 
課外学習等 (授業時間外学習の指示)

The material is cuttingedge and somewhat demanding, so you should expect to spend a good amount of time on it. 
注意事項

N/A. 
質問への対応方法

Please write email to ask questions. 
他学科聴講の可否

It is possible. 
他学科聴講の条件

Please contact me ahead of time to confirm level of background knowledge. 
レベル

3 
キーワード

Condensed mathematics, analytic geometry, sixfunctor formalism. 
履修の際のアドバイス

Some familiarity with Lurie's theory of infinitycategories is helpful, since this is the modern way of doing homological algebra and derived algebraic geometry. 
授業開講形態等

Participants take turns to give lectures and prepare lecture notes. The notes should be helpful for all participants. 
遠隔授業(オンデマンド型)で行う場合の追加措置

