Topological methods in fair division problems

The well-known problem of fair division has many forms and arises in many real-world situations. In this talk I will consider several existence theorems for this problem. For the rent harmony and cake cutting problems and their generalizations, generalizations of Sperner's lemma related to invariants of homotopy groups will be given. A large group of fair division problems are consequences of the Borsuk–Ulam theorem (BUT), particularly the partition of measures into equal parts. Here I will present generalizations of BUT in terms of equivariant cobordisms, which simplify some existing proofs and allow one to obtain new consequences.