学部・大学院区分
Undergraduate / Graduate

多・博前

時間割コード
Registration Code

3212208

科目区分
Course Category

Ｂ類（講究）
Category B

科目名　【日本語】
Course Title

幾何学講究４

科目名　【英語】
Course Title

Seminar 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
Credits

開講期・開講時間帯
Term / Day / Period

秋集中その他その他
Intensive(Fall) Other Other

授業形態
Course style

学科・専攻
Department / Program

多元数理科学研究科

必修・選択
Required / Selected

授業の目的　【日本語】
Goals of the Course(JPN)

The aim of the course is to study the "Condensed Mathematics" developed by Dustin Clausen and Peter Scholze, following the notes from Scholze's course 2019 course in Bonn.

授業の目的　【英語】
Goals of the Course

到達目標　【日本語】
Objectives of the Course(JPN))

cholze 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　【英語】
Objectives of the Course

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.