Zariski's Finiteness Theorem and Properties of Some Rings of Invariants

Abstract: In this talk, I will present a short proof using a new idea of a
special case of Oscar Zariski's result about the finite generation in
connection with the famous Hilbert's Fourteenth Problem. This result is
useful for invariant subrings of unipotent or connected semisimple groups.
The next result I will talk about is a stronger form of one well-known
result by Andrzej Tyc. This result proves that the quotient space under a
regular $\mathbb{G}_a$-action on an affine space over the field of complex
numbers has at most rational singularities, under an assumption about the
quotient morphism. If time permits, I will also sketch the main idea of
the proof of a result which is an analogue of Masayoshi Miyanishi's result
for the ring of invariants of a $\mathbb{G}_a$-action on the polynomial
ring $R[X, Y, Z]$ for an affine Dedekind domain $R$. This proof involves
some classical topological methods.

This talk is based on joint work with R. V. Gurjar and S. R. Gurjar.