The University of Wisconsin-Madison received a $125,380 Project Grant award from the National Science Foundation Division of Mathematical Sciences under the Mathematical and Physical Sciences federal grant program (CFDA 47.049). The grant supports research into algorithmic and computational questions in algebraic geometry and number theory as they relate to finiteness theorems. Specifically, the Principal Investigator will study algorithmic approaches to the Shafarevich conjecture determining Abelian varieties over a fixed number field and the Riemann-Hilbert correspondence for finding algebraic differential equations with given monodromy representations. These areas in turn relate to problems in finding branched covers of curves and studying variations of Hodge structure, motivated by connections to Faltings's theorem on rational solutions to polynomial equations. The work will investigate several difficult computational problems to further algorithmic solutions in diverse fields of mathematics, reflecting the NSF's mission to strengthen the scientific enterprise.