Explore the intricate world of Schubert calculus and quiver varieties in this 30-minute lecture by Allen Knutson, presented by the International Mathematical Union. Delve into the foundations of Schubert calculus on Grassmannians, examining incidence conditions and extensions of the problem. Uncover the puzzle rules in Schubert calculus and witness the introduction of the Yang-Baxter equation. Trace the relevant history of Yang-Baxter solutions and learn how to deduce representation theory from known Grassmannian puzzle rules. Investigate cotangent Schubert calculus, factoring diagonal inclusion, and conserved quantities in puzzles. Analyze the associativity of multiplication and explore other fusion channels. Conclude with an epilogue discussing future directions in this fascinating field of mathematics.