This Members' Colloquium talk explores how Selmer groups function as a cohomological tool to simplify the process of finding solutions to diophantine equations by reducing them to more manageable field and modular arithmetic. Learn about practical applications of this approach, with special focus on the recent breakthrough in resolving Hilbert's tenth problem over infinite finitely generated rings. Discover why there cannot exist an algorithm to determine whether polynomial equations have solutions over these rings. The lecture will be delivered by Ari Shnidman from the Institute for Advanced Study on March 31, 2025, at 1:00pm in Simonyi 101, with remote access options available.