学部・大学院区分
Undergraduate / Graduate

多・博前

時間割コード
Registration Code

3211087

科目区分
Course Category

Ａ類Ⅱ（専門科目）
Category A-2

科目名　【日本語】
Course Title

数理科学特論Ⅰ

科目名　【英語】
Course Title

Topics in Mathematical ScienceⅠ

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

春金曜日１時限
Spring Fri 1

授業形態
Course style

学科・専攻
Department / Program

Complex analytic geometry

必修・選択
Required / Selected

N/A.

授業の目的　【日本語】
Goals of the Course(JPN)

See English version.

授業の目的　【英語】
Goals of the Course

The purpose of the course is to introduce complex analytic geometry, following Clausen and Scholze. The starting point is the construction of the (presentably symmetric monoidal infinity-)category of gaseous complex vector spaces. Coordinate functions on a complex analytic space is represented as operators on this category, and the ring of holomorphic functions of the complex analytic space arises as the ring of endomorphisms of the tensor-unit. So, whereas in algebra, one associates a (presentably symmetric monoidal derived infinity-)category of modules to a commutative ring, in complex analytic geometry, the (presentably symmetric monoidal infinity-)category of (quasi-coherent) modules much be define first, and the commutative ring of holomorphic functions is then obtained from it (as the ring of endomorphisms of the tensor-unit).

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

See English version.

到達目標　【英語】
Objectives of the Course

The objective of the course is to introduce complex analytic geometry, following Clausen and Scholze. The starting point is the construction of the (presentably symmetric monoidal infinity-)category of gaseous complex vector spaces. Coordinate functions on a complex analytic space is represented as operators on this category, and the ring of holomorphic functions of the complex analytic space arises as the ring of endomorphisms of the tensor-unit. So, whereas in algebra, one associates a (presentably symmetric monoidal derived infinity-)category of modules to a commutative ring, in complex analytic geometry, the (presentably symmetric monoidal infinity-)category of (quasi-coherent) modules much be define first, and the commutative ring of holomorphic functions is then obtained from it (as the ring of endomorphisms of the tensor-unit).

授業の内容や構成
Course Content / Plan

The goal of the lectures is to introduce complex analytic geometry following the approach of Clausen and Scholze. To do so, it will be necessary to also introduce of a number of modern tools, including infinity-categories and condensed mathematics. But the focus of the course will be the application of these tools and we will learn to use them by doing. The key novelty of the Clausen-Scholze theory is the definition of a (presentably symmetric monoidal infinity-) category of quasi-coherent modules associated with a complex analytic space. Before their work, there was only a (small) category of coherent modules. In fact, the assignment of this (large infinity-)category to a complex analytic space promotes to a full-fledged six-functor formalism, which encodes all aspects of coherent cohomology of complex analytic spaces. How much of this theory we will be able to cover in the course will depend on the participants.

履修条件
Course Prerequisites

Some mathematical maturity is helpful. You should be prepared to contemplate mathematical structures that you do not fully understand at first glance.