Recent progress on manifolds with special holonomy. Pan-African Congress of Mathematicians 2017. Mohammed V University, Rabat. July 7, 2017
International Meeting or Conference
When Marcel Berger classified the possible holonomies of Riemannian metrics in 1954, all but a handful of the possibilities were already known to exist and those covered important cases, such as Kahler geometry (which corresponds to metrics admitting a parallel complex structure). In the following years, great advances were made, such as Calabi's construction of what are now called hyper-kohler metrics and Yau's solution of the Calabi conjecture, proving the existence of Ricci-flat Kahler metrics (which also admit a parallel holomorphic volume form) in important special cases.
This left two so-called "exceptional cases" in dimensions 7 and 8, which were not even known to exist locally until 1984. In the intervening years, these metrics have made an appearance in theoretical high-energy physics and mathematicians, particularly Dominic Joyce, have developed new techniques for constructing compact examples. In recent years, mathematicians exploring these constructions and their connections with gauge theory have made significant advances in our understanding of these mysterious but beautiful geometries.
In this talk, I will give an introduction to the subject of special holonomy followed by a survey of recent important results and the prospects for further progress and their applications.
Service Performed By
Service or Event Name
Pan-African Congress of Mathematicians 2017
Mohammed V University, Rabat
Location or Venue
Mohammed V University, Rabat, Morocco