Symmetry and rigidity unite number theory with geometry

Jiacheng Xia received his doctoral degree in mathematics from Chalmers University of Technology in 2021. Thanks to a grant from Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position with Professor Tonghai Yang at the University of Wisconsin–Madison, USA.

Number theory and geometry are among the oldest branches of mathematics. Number theory deals with questions about integers while shapes are the object of study in geometry. They developed almost separately until the last century, when modern number theory was increasingly linked to geometry, and more accurate relationships between objects of number theoretic and geometric nature were predicted. 

One theme running through the project is the interplay of symmetry and rigidity. Symmetry is a general phenomenon in mathematics and refers to the degree to which an object remains unchanged during transformations, such as rotation. Rigidity is a concept that originates in physics and can be described as an object’s lack of degrees of freedom, allowing the properties of the object to be unambiguously determined using less information than expected. Also, an object with higher symmetry often exhibits a higher degree of rigidity, and vice versa – the more rigid objects are, the more symmetry they have.

One of the project’s main aims is to study the geometric rigidity of certain objects with high symmetry, which is of interest in number theory. An appropriate tool for these calculations comprises modular forms, a kind of functions that satisfy conditions for both symmetry and rigidity. The inspiration comes from a broad research programme first developed in the early 1990s by American mathematician Stephen Kudla, which has since driven the development of arithmetic geometry.