Mini-Workshop: Singularities in $\mathrm G_2$-geometry

All currently known construction methods of smooth compact -manifolds have been tied to certain singular -spaces, which in Joyce’s original construction are -orbifolds and in Kovalev’s twisted connected sum construction are complete G2-manifolds with cylindrical ends. By a slight abuse of terminology we also refer to the latter as singular -spaces, and in fact both construction methods may be viewed as desingularization procedures. In turn, singular -spaces comprise a (conjecturally large) part of the boundary of the moduli space of smooth compact -manifolds, and so their deformation theory is of considerable interest. Furthermore, singular -spaces are also important in theoretical physics. Namely, in order to have realistic low-energy physics in M-theory, one needs compact singular -spaces with both codimension 4 and 7 singularities according to Acharya and Witten. However, the existence of such singular -spaces is unknown at present. The aim of this workshop was to bring reserachers from special holonomy geometry, geometric analysis and theoretical physics together to exchange ideas on these questions.

Duke Scholars

Author Mark Haskins Mathematics