﻿2021年数学学院“吉大学子全球胜任力提升计划”研究生系列短课程（15）-吉林大学数学学院

# 2021年数学学院“吉大学子全球胜任力提升计划”研究生系列短课程（15）

Abstract:  Locally nilpotent derivation (lnd) is an important algebraic tool in the research of affine algebraic geometry. In some cases where other approaches do not help lnd allow to distinguish rings and describe their groups of automorphsim. They also help to give simplified proofs of some classical results. In the lectures we will prove with the help of lnd several important theorems and discuss relevant open questions.

 授课日期Date of Lecture(Beijing time) 课程名称（讲座题目）Name (Title) of Lecture 授课时间Duration(Beijing time) 参与人数Number of Participants August 1 Introduction of   lnds 9:00-10:00am 15 August 2 Lemma on rigidity; Cancellation problem 9:00-10:00 am 15 August 3 Introduction of   AK (ML) invariant; Discussion of the Russell’s cubic. 9:00-10:00 am 15 August 4 General   properties of lnd. 9:00-10:00 am 15 August 6 A new proof of   Abhyankar-Moh- Suzuki theorem (I) 9:00-10:00 am 15 August 7 A new proof of   Abhyankar-Moh- Suzuki theorem (II) 9:00-10:00 am 15 August 8 Automorphisms of some surfaces (I) 9:00-10:00 am 15 August 9 Automorphisms   of some surfaces (II) 9:00-10:00 am 15

Lecture 1. Introduction of lnd. Several examples. Description of the group of automorphisms of a polynomial ring with two variables.

The famous Jung theorem describes the structure of the automorphisms group of C[x; y]. We will prove this theorem using lnd.

Lecture 2. Lemma on rigidity. Cancellation Theorem.

We prove the Lemma on rigidity: If a ring A does not admit (non-zero) lnd then for the polynomial ring A[t], all lnd are given by D(A)=0 and D(t) in A. After proving this Lemma we deduce from it the Cancellation Theorem of Fujita, Miyanishi and Sugie.

Lecture 3. Introduction of the AK (ML) invariant. Discussion of the Russell's cubic.

We introduce the AK (ML) invariant (the ring of absolute constants) for a ring A, denoted by AK(A) or ML(A). Using this invariant we prove that the Russell's cubic is not isomorphic to the affine space. The only known proofs of this fact are based on lnd.

Lecture 4. General properties of lnd.

We introduce general properties of ldn and discuss your questions and open questions.

Lecture 5, 6. A new proof of Abhyankar-Moh-Suzuki theorem (I),(II)

We give a new proof of the Abhyankar-Moh-Suzuki theorem using the theory of lnd.

Lecture 7, 8 Automorphisms of some surfaces (I), (II)

We talk about the automorphisms of some affine surfaces, and show how the research of lnd is related to the problem.