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# Automorphisms And Derivations In Affine Algebraic Geometry

## March 13, 2023 - March 17, 2023

Mini-course by Professor Leonid Makar-Limanov, Wayne University, USA

### Brief description of the mini course

After this course you will know the proofs of several classical theorems of Affine Algebraic Geometry. The original proofs of these theorems were quite involved and a much longer course would be needed for their exposition.

In the first lecture we will discuss the theorems of Heinrich Jung and Rudolf Rentschler. The first one describes all invertible transformations of the plane by polynomials and the second all generalized shifts of the plane. Algebraically speaking, Jung’s theorem describes all automorphisms of the ring of polynomials with two variables and Rentschler theorem describes all subgroups of this group which are isomorphic to the group of complex numbers under addition. If we have time, we will discuss the groups of polynomial automorphisms of several other surfaces.

The second lecture is devoted to the following topic: if a cylinder is given, is it possible to recover the base of this cylinder. In general the answer is no, but we discuss two cases when this is possible. We show that if the cylinder over a curve is given then we can recover this curve (this is the theorem of Shreeram Abhyankar, Paul Eakin, and William Heinzer). If the cylinder over a surface is isomorphic to a three-dimensional space then the surface is isomorphic to a plane (this is a theorem of Takao Fujita).

Here is an algebraic translation:

If *A* is an integral domain of transcendence degree one and *A*[*x _{1}, x_{2},…, x_{n}*] is given, we can recover

*A*up to an isomorphism. If

*A*is an integral domain of transcendence degree two and

*A*[

*x*] is isomorphic to

**C**[

*y*] then

_{1},y_{2},y_{3}*A*is isomorphic to

**C**[

*z*]. The main tool used in these two lectures is

_{1},z_{2}*locally nilpotent derivations*.

In the third lecture we prove one of the most famous theorems in affine algebraic geometry, the AMS Theorem (after Abhyankar, Tsuong-tsieng Moh, Masakazu Suzuki): *any smooth “good” embedding of a line to a plane is the image of a coordinate line under an automorphism of the plane.* Algebraically, this means the following: if two polynomials *f(t)*,* g(t)*∈ **C**[*t*] generate **C**[*t*] then the smaller of the degrees of *f(t)*, *g(t)* divides the larger of the degrees of* f(t)*, *g(t)*. The main tool here is a new algorithm for finding an irreducible dependence between two polynomials in one variable.

The lectures will be delivered during three time periods, as shown below. They will take place at the St. John’s campus of Memorial University and will be broadcast via Webex. All the times are in Newfoundland Time (NST=UTC-3:30).

**Monday, March 13th: TBA**

**Tuesday, March 14th: TBA**

**Thursday, March 16th: TBA**

The lectures will be available online via Webex. The details will be given later. Contact the organizers for more information: Mikhail Kotchetov <mikhail@mun.ca>; Yuri A Bakhturin <bahturin@mun.ca>