This series of five lectures provides a mini course to the fundamental concepts of vector bundles, characteristic classes, and the Atiyah-Singer index theorem. We will begin by exploring the basic objects, vector bundles, and developing the theory of their characteristic classes. Then to present the Atiyah-Singer index theorem, we will introduce the theory of genus and the theory of Dirac operators.
The lectures will culminate with a discussion of the Atiyah-Singer index theorem, a profound result that bridges geometry, and topology and analysis. We will present the theorem's statement, explore several of its applications, and offer insights into its proof and its broader implications across various fields of mathematics.

These lectures are intended for graduate students and undergraduates with some background in differential geometry and topology, aiming to provide a concise introduction in these important topics.

Lecture 1: Vector bundles and the theory of characteristic classes:
the concept of vector bundles; Chern-Weil theory of characteristic classes

Lecture 2: The theory of genus
bordism; definition and examples of genus

Lecture 3: The theory of Dirac operators
elliptic differential operators; Dirac operators

Lecture 4: The Atiyah-Singer index theorem
the Atiyah-Singer index theorem; some applications
Lecture 5: The Atiyah-Singer type index theorems
equivariant versions of AS index theorem; others

参加申込用URL：https://forms.gle/r9DWgLuyd3c5zJGt6

締切日：10月6日 （日）

※数学・数理科学グローバル講義Ⅱは数学・数理科学イノベーション人材育成強化コースにおける中核科目です。

※数学・数理科学グローバル講義Ⅱを履修するにはKULASISでの履修登録が必要です。
後期科目の履修登録期間は10月10日（木）・11日（金）。

※履修登録していなくても聴講（本学学生に限る）可（申込みは必要）。

数学・数理科学グローバル講義Ⅱでは４名の講師による特別講義が開講されます。
各特別講義のアブストラクト等の詳細は、当コースのホームページ
https://www.math.kyoto-u.ac.jp/ja/ktgu/coursesをご覧ください。