# Modularity of Calabi-Yau Varieties

Speaker:

Noriko Yui
Date:

Thu, Apr 25, 2019
Location:

PIMS, University of Saskatchewan
Conference:

Hugh C. Morris Lecture Abstract:

Let X be a Calabi-Yau variety of dimension d. We will confine ourselves to Calabi-Yauvarieties of small dimensions, e.g., d < 3. Dimension one Calabi–Yaus are elliptic curves, those of dimension two are K3 surfaces, and dimension three ones are Calabi-Yau threefolds. Geometry and physics are both very much in evidence on Calabi-Yau varieties over the field of complex numbers.
Today I will focus on Calabi-Yau varieties defined over the field Q of rational numbers (or number fields), and will discuss the modularity/automorphy of Calabi-Yau varieties in the framework of the Langlands Philosophy.
In the last twenty-five years, we have witnessed tremendous advances on the modularity question for Calabi-Yau varieties. All these results rest on the modularity of the two-dimensional Galois representations associated to them. In this lecture, I will present these fascinating results. If time permits, I will discuss a future direction for the realization of the Langlands Philosophy, in particular, for Calabi-Yau threefolds.