Meistaraprófsfyrirlestur: Arngunnur Einarsdóttir

Ágrip: This MS thesis is a study of convergence sets of power series. In one complex variable it

is simple to describe the convergence, for if a series converges at some point not equal to the origin, then it converges in a disc or in the whole complex plane, by Abel’s theorem. In several variables a power series is said to be convergent if it converges at every point in some neighborhood of the origin, otherwise it is said to be divergent. The goal is to describe convergence sets of both convergent and divergent series. Convergence sets of convergent series in \({\mathbb C}^𝑛\) are attained using capacities and global extremal functions from potential theory. To find convergence sets of divergent series their convergence in the projective space \({\mathbb P}^{𝑛−1} ({\mathbb C})\) of \({\mathbb C}^{𝑛−1}\) is examined. The main sources of the thesis are works of Siciak, for convergent series, and Chen, Ma and Neelon, for divergent series.

Prófdómari er Alexander Rashkovskii frá Háskólanum í Stavanger, Noregi.