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ファイル更新日：2024年03月06日

教育・就職

■少人数クラスシラバス■

●L. ヘッセルホルト

学部・大学院区分 Undergraduate / Graduate	多・博前
時間割コード Registration Code
科目区分 Course Category	B類(講究) C類(実習)/Category B Category C
科目名【日本語】 Course Title	幾何学講究1 幾何学講究2 幾何学講究3 幾何学講究4 幾何学実習1 幾何学実習2 幾何学実習3 幾何学実習4
科目名【英語】 Course Title	Seminar on Geometry 1 Seminar on Geometry 2 Seminar on Geometry 3 Seminar on Geometry 4 Practical Class on Geometry 1 Practical Class on Geometry 2 Practical Class on Geometry 3 Practical Class on Geometry 4
コースナンバリングコード Course Numbering Code
担当教員【日本語】 Instructor	HESSELHOLT LARS
担当教員【英語】 Instructor	HESSELHOLT LARS
単位数 Credit	B類4単位 C類1単位
開講期・開講時間帯 Term / Day / Period
授業形態 Course style	セミナー
学科・専攻 Department / Program	多元数理科学研究科
必修・選択 Compulsory / Selected	選択必修

授業の目的【日本語】 Goals of the Course(JPN)	See English text.
授業の目的【英語】 Goals of the Course	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 p-adic geometry. Condensed mathematics holds the promise of accomplishing the same in a broader framework that, in addition to p-adic 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】 Objectives of the Course(JPN)	N/A.
到達目標【英語】 Objectives of the Course	N/A.
授業の内容や構成 Course Content / Plan	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 Course Prerequisites	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 Related Courses	Algebraic geometry, algebraic topology.
成績評価の方法と基準 Course Evaluation Method and Criteria	Evaluation is based on lectures given in class and written notes.
教科書・テキスト Textbook	https://www.math.uni-bonn.de/people/scholze/Condensed.pdf https://www.math.uni-bonn.de/people/scholze/Analytic.pdf
参考書 Reference Book	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=PLAMniZX5MiiLXPrD4mpZ-O9oiwhev-5Uq and notes from the lectures are available at https://www.math.ku.dk/english/calendar/events/condensed-mathematics
課外学習等 (授業時間外学習の指示) Study Load(Self-directed Learning Outside Course Hours)	The material is cutting-edge and somewhat demanding, so you should expect to spend a good amount of time on it.
注意事項 Notice for Students	N/A.
質問への対応方法 How to Ask Questions	Please write email to ask questions.
他学科聴講の可否 Propriety of Other department student’s attendance	It is possible.
他学科聴講の条件 Conditions for Other department student’s attendance	Please contact me ahead of time to confirm level of background knowledge.
レベル Level	3
キーワード Keyword	Condensed mathematics, analytic geometry, six-functor formalism.
履修の際のアドバイス Advice	Some familiarity with Lurie's theory of infinity-categories is helpful, since this is the modern way of doing homological algebra and derived algebraic geometry.
授業開講形態等 Lecture format, etc.	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 Additional measures for remote class (on-demand class)