The ERC CoG Foursurf is concerned with the subject of enumerative geometry: a discipline within geometry dating back to the Ancient Greeks. A typical problem from enumerative geometry is the determination of the number of circles tangent to three given circles in the plane (Apollonius's problem). It is also a field with many exciting modern aspects due to unexpected connections to theoretical physics (specifically, supersymmetric quantum field theory and string theory).
Traditionally, enumerative geometry was mostly concerned with counting curves. E.g. Apollonius's problem asks for the number of circles tangent to three given circles. Another 19th century example is the statement that a cubic surface, a geometric shape defined by an equation of degree three in three variables, contains precisely 27 lines (when viewed over the complex numbers and in projective space). Building on some recent developments in the field, this ERC CoG is about the next chapter in enumerative geometry: counting surfaces.
Specifically, this project explores the counting of surfaces on so-called Calabi-Yau fourfolds --- a type of space with four complex dimensions and many symmetries. One reason for studying this subject is because it connects to numerous other fields in mathematics such as Hodge theory, combinatorics, modular forms, representation theory as well as theoretical physics (specifically, supersymmetric quantum field theory and string theory). The impact of this project is on creating new connections of this kind. This will reveal hidden relations between previously unrelated problems in mathematics and physics and may help us to solve some of those problems.
