Doctoral Defense in Mathematics - Daniel Amankwah

Doctoral candidate:
Daniel Amankwah

Title of thesis:
Scaling limits of weighted random tree-like planar maps

Opponents:
Dr. Cyril Marzouk, Assistant Professor at the Applied Mathematics Department, École polytechnique, France
Dr. Bénédicte Haas, Professor at the Department of Mathematics, Université Sorbonne Paris Nord, France

Advisor:
Dr. Sigurður Örn Stefánsson, Professor at the Faculty of Physical Science, University of Iceland

Other members of the doctoral committee:
Dr. Benedikt Steinar Magnússon, Associate Professor at the Faculty of Physical Science, University of Iceland
Dr. Benedikt Stufler, Associate Professor at the Institute of Discrete Mathematics and Geometry, Vienna University of Technology, Austria

Chair of Ceremony:
Dr. Birgir Hrafnkelsson, Professor and Head of the Faculty of Physical Sciences, University of Iceland

Abstract:
The topic for this thesis lies in the scope of random planar maps. We investigate scaling limits of some models of discrete planar maps which are by construction “tree-like”. Specifically, we consider some models of “Halin-like” maps and series-parallel maps. The main aim is to study how some fundamental limiting objects appear as scaling limits of these models, hence affirming their universality properties. The Brownian Continuum Random tree (CRT), originally introduced by David Aldous is one of such limiting objects and has been known to be the limit of various different discrete models of uniformly sampled and weighted planar maps. The stable looptrees, due to Curien and Kortchemski is also of significant interest in this thesis. They are known to arise as scaling limit of models of tree-like maps for the case when each face in the maps is assigned a heavy tailed weight so that a typical face is in the domain of attraction of a stable law. One motivation for this thesis is to understand how these convergences generically happen for maps characterized via exclusion of minors. To our knowledge, series-parallel maps known to be characterized by not containing $K_4$ as a minor is the largest known collection of maps, defined by the exclusion of minors, which admits the CRT as a scaling limit.

About the doctoral candidate:
Daniel Amankwah was born and raised in Accra, Ghana. He earned a Bachelor's degree in Mathematics from the University of Ghana in 2012, followed by a Master's degree in Mathematics from the African Institute for Mathematical Sciences (AIMS) in Senegal in 2014, and a Master's degree in Physics from Aix-Marseille University in France in 2017. Before commencing his PhD in Mathematics at the University of Iceland, he served as a lecturer at the University of Ghana. He currently resides in Reykjavík with his wife and their two daughters.