Nakajima's (Q, T)-characters as quantum cluster variables

Nakajima introduced a t-deformation of q-characters, (q,t)-characters for short, and their twisted multiplication through the geometry of quiver varieties. The Nakajima (q, t)-characters of Kirillov-Reshetikhin modules satisfy a t-deformed T-system. The T-system is a discrete dynamical system that can be interpreted as a mutation relation in a cluster algebra in two different ways, depending on the choice of direction of evolution. In this thesis, we show that the Nakajima t-deformed T-system of type Ar forms a quantum mutation relation in a quantization of exactly one of the cluster algebra structures attached to the T-system. There are 2 main parts to our work. The bulk of the work is a combinatorial construction that proves (q, t)-characters of a certain set of Kirillov-Reshetikhin modules t-commute under Nakajima’s twisted multi- plication. We use a slightly modified version of the tableaux-sum notation for q-characters and define the notion of a block-tableau, which plays an integral role in the proof. Once t-commutativity is established, the second half of this thesis is concerned with the commutation coefficients of the given set of Kirillov-Reshetikhin modules. In particular, we show that the commutation coefficients are compatible with the cluster algebra exchange matrix and the mutation relations in the language of Berenstein-Zelevinsky

A symplectic proof of a theorem of Franks

A celebrated theorem in two-dimensional dynamics due to John Franks asserts that every area preserving homeomorphism of the sphere has either two or infinitely many periodic points. In this work we reprove Franks' theorem under the additional assumption that the map is smooth. Our proof uses only tools from symplectic topology and thus differs significantly from all previous proofs. A crucial role is played by the results of Ginzburg and Kerman concerning resonance relations for Hamiltonian diffeomorpisms.Comment: 15 pages. Minor changes. Final version to appear in Compositio Mathematic