报告题目：Lectures on affine algebraic geometry
报 告 人：Leonid Makar-Limanov，Wayne State University
报告地点：Zoom 会议 ID： 923 1259 5196 Passcode: 844680
校内联系人：孙晓松 sunxs@jlu.edu.cn
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.
报告人简介：Leonid Makar-Limanov，美国韦恩州立大学数学系教授，仿射代数几何领域的国际知名专家，在仿射簇自同构的结构和局部幂零导子理论方面取得了很有影响的学术成果，在J. Eur. Math. Soc., Israel J. Math., J. Algebraic Geom., J. Algebra等国际重要学术期刊发表论文70余篇.
