**Ph.D. student:** Auðunn Skúta Snæbjarnarson

**Dissertation title:** Polynomial approximation on Stein manifolds and the Monge-Ampère operator

**Opponents:**

Dr. Ahmed Zeriahi, Professor at the Université Toulouse III, Paul Sabatier.

Dr. Håkan Samuelsson, Professor, Chalmers University of Technology and University of Gothenburg

**Advisor:** Dr. Ragnar Sigurðsson, Professor at the Faculty of Physical Sciences, University of Iceland.

**Doctoral committee:** Dr. Jón Ingólfur Magnússon, Professor at the Faculty of Physical Sciences, University of Iceland.

Dr. Lárus Thorlacius, Professor at the Faculty of Physical Sciences, University of Iceland.

**Chair of Ceremony:** Dr. Oddur Ingólfsson, Professor and Head of the Faculty of Physical Sciences, University of Iceland.

**Abstract:**

The first main theme of this thesis is the approximation of holomorphic functions on Stein manifolds by polynomials. In our setting a polynomial is an entire function whose absolute value is bounded from above by the exponential of a given plurisubharmonic exhaustion function on the complement of a compact set. In particular we generalize to a certain class of Stein manifolds the Bernstein-Walsh-Siciak theorem which describes the equivalence between possible holomorphic continuation of a function f defined on a compact set K in the complex space to the rapidity of the best uniform approximation of f on K by polynomials. We also generalize Winiarski's theorem which relates the growth rate of an entire function in the complex space to its best uniform approximation by polynomials on a compact set.

If a plurisubharmonic exhaustion function on a Stein manifold satisfies the homogeneous Monge-Ampere equation outside a compact subset, i.e. if it is a parabolic potential, then the polynomial spaces defined by it are finite dimensional. Therefore the problem of constructing parabolic potentials arises naturally in the theory of polynomial approximation on Stein manifolds.

The second main theme of this thesis is the complex Monge-Ampere operator. In particular we derive formulas for the Monge-Ampere measures of a particular family of plurisubharmonic exhaustion functions associated to the Lie norm of holomorphic maps.

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**About the doctoral candidate:**

Auðunn Skúta Snæbjarnarson was born in 1990, the son of Snæbjörn Friðriksson and Elsa Guðmundsdóttir. He graduated from Menntaskólinn in Akureyri the year 2010. He completed his B.Sc. degree in math from the University of Iceland in 2013 and his M.Sc. degree a year later. He started his PhD studies in the end of 2014.