Domains of low Gelfand-Kirillov dimension

Streszczenie autora: Let $k$ be a field and let $A$ be a finitely generated $k$-algebra that is a domain. If the algebra has finite Gelfand-Kirillov dimension, then we can invert the nonzero elements of $A$ and create a division algebra of quotients $Q(A)$. In the noncommutative case $Q(A)$ is very poorly behaved, and it is difficult to answer basic questions. We look at the following questions: - When are the subfields of $Q(A)$ finitely generated over $k$? - What is the largest transcendence degree of a subfield of $Q(A)$? - When does $Q(A)$ contain a free algebra on two generators? - What do the centralizers of elements of $Q(A)$ look like? We will explain what progress has been made on these questions.