Abstract

In simple Lie groups, except the series PU(1,n) with n>1, either lattices are all arithmetic, or mathematicians constructed infinitely many nonarithmetic lattices. So far there are only finitely many nonarithmetic lattices constructed for PU(1,2) and PU(1, 3) and no examples for n>3. One important construction is via monodromy of hypergeometric functions. The discreteness and arithmeticity of those groups are classified by Deligne and Mostow. Thurston also obtained similar results via flat conic metrics. However, the classification of those lattices up to conjugation and finite index (commensurability) is not completed. When n=1, it is the commensurabilities of hyperbolic triangles. The cases of n=2 are almost resolved by Deligne-Mostow and Sauter's commensurability pairs, and commensurability invariants by Kappes-Möller and McMullen. Our approach relies on the study of some higher dimensional Calabi-Yau type varieties instead of complex reflection groups. We obtain some relations and commensurability indices for higher n and also give new proofs for existing pairs in n=2.

Speaker

I am an assistant professor at YMSC and BIMSA starting from Nov. 2020. I was a Hans Rademacher Instructor of Mathematics at University of Pennsylvania during June 2018-July 2020. I graduated from Harvard University in 2018, under the supervision of Shing-Tung Yau.

My research is in algebraic geometry and differential geometry. I am interested in the geometry and arithmetic properties of Calabi-Yau varieties, especially the differential systems arising from Calabi-Yau families. I am also interested in moduli of K3 surfaces and cubic fourfolds.