Suppose that $A$ is a $\mathbb{C}$-algebra such that the polynomial ring $A[T]$ in one variable over $A$ is isomorphic to the polynomial ring $\mathbb{C}[X,Y,Z]$ in 3 variables over $\mathbb{C}$; does it follow that $A \cong \mathbb{C}[X,Y]$? It turns out that the answer is “yes”, but the proof required several decades of work by leading mathematicians. Moreover, the analogous question in dimension 3 (does $A[T] \cong \mathbb{C}[W,X,Y,Z]$ imply $A \cong \mathbb{C}[X,Y,Z]$?) remains an open problem today. This question is called the Zariski Cancellation Problem, and is a good illustration of the type of problem that interests researchers from the field called Affine Algebraic Geometry. For a second example, consider a polynomial $f \in \mathbb{C}[X,Y]$ with the property that the quotient ring $\mathbb{C}[X,Y]/(f)$ is isomorphic to $\mathbb{C}[T]$; does it follow that there exists $g \in \mathbb{C}[X,Y]$ such that the subalgebra $\mathbb{C}[f,g]$ of $\mathbb{C}[X,Y]$ is actually equal to $C[X,Y]$? The celebrated Abhyankar-Moh-Suzuki Theorem asserts that the answer is “yes”. In dimension 3, the corresponding question remains open.

As suggested by the above examples, Affine Algebraic Geometry is concerned with a set of questions revolving around the concept of polynomial ring. This subject can be studied from several angles: commutative algebra, algebraic geometry, algebraic topology, differential geometry, etc. In these lectures we adopt a purely algebraic approach. The main topic will be the theory of locally nilpotent derivations on commutative rings. We will also see how this algebraic tool can be used to investigate the questions of Affine Algebraic Geometry. In particular, we will solve some cases of the Zariski Cancellation Problem. I will occasionally point out geometric interpretations, but I will not assume prior knowledge of algebraic geometry.

Prerequisites: a basic course in commutative algebra or in algebraic geometry.

There will be five lectures of two hours each.

We will cover the following topics: Introduction to Affine Algebraic Geometry. Derivations. Locally nilpotent derivations (LNDs) on commutative rings. Solution of some cases of Zariski’s Cancellation Problem and of the problem of characterizing polynomial rings. The relation between LNDs, automorphisms and $G_a$-actions. LNDs on polynomial rings. Automorphisms of polynomial rings and the notion of variable. More open problems in Affine Algebraic Geometry.

Schedule:

• Tuesday 21/10/2025 11:00
• Thursday 23/10/2025 11:00
• Tuesday 28/10/2025 11:00
• Thursday 30/10/2025 11:00
• Friday 31/10/2025 11:00