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Explore p-adic hyperbolicity properties of Shimura varieties in this mathematical lecture from the Workshop on Special Cycles and Related Topics at the Institute for Advanced Study. Examine a classical theorem by Borel stating that holomorphic maps from poly-punctured disks into Shimura varieties with torsion-free level structures extend holomorphically across punctures to minimal compactifications, leading to the conclusion that holomorphic maps from complex algebraic varieties into such varieties are algebraic. Discover the p-adic analogue of this fundamental result, with discussion covering joint research on Shimura varieties of abelian type conducted with Shankar, Zhu and Patel, as well as ongoing investigations into exceptional Shimura varieties in collaboration with Bakker, Shankar and Yao. Learn about advanced concepts in algebraic geometry, p-adic analysis, and the geometric properties of these important mathematical structures through detailed mathematical exposition and current research developments in the field.
