Enumerative problems in statistics, combinatorics and topology

Algebraic geometry has made great advances in the last two centuries. A particular role was played by enumerative geometry, where correct setting of moduli spaces found applications beyond mathematics. In my talk I would like to present a new work on applications of enumerative geometry providing a unified approach to fundamental invariants in algebraic statistics, combinatorics and topology. Achieving our results would not be possible without the fundamental work of De Concini, Huh, Laksov, Lascoux, Pragacz, Procesi and Sturmfels. The talk is based on joint works with Conner, Dinu, Manivel, Monin, Seynnaeve, Vodicka and Wisniewski.