In the talk, I'll first describe a more general context of Sárközy-type problems and interesting directions in which they can be pursued. Then, I'll focus on the specific case of bounding the size of sets A s. t. A - A + 1 contains no prime. After describing the progress on the problem for integers, I'll pass on to considering an analogous question for function fields and (after a general introduction to function fields) I'll speak about my recent result in this area.

It is a classical result that a random permutation of $n$ elements has, on average, about $\log n$ cycles. We generalise this fact to all directed $d$-regular graphs on $n$ vertices by showing that, on average, a random cycle-factor of such a graph has $\mathcal{O}((n\log d)/d)$ cycles. This is tight up to the constant factor and improves the best previous bound of the form $\mathcal{O}({n/\sqrt{\log d}})$ due to Vishnoi. It also yields randomised polynomial-time algorithms for finding such a cycle-factor and for finding a tour of length $(1+\mathcal{O}((\log d)/d)) \cdot n$ if the graph is connected. The latter result makes progress on a restriction of the Traveling Salesman Problem to regular graphs, a problem studied by Vishnoi and by Feige, Ravi, and Singh. Our proof uses the language of entropy to exploit the fact that the upper and lower bounds on the number of perfect matchings in regular bipartite graphs are extremely close.

This talk is based on joint work with Micha Christoph, Nemanja Draganić, António Girão, Eoin Hurley, and Alp Müyesser.

First introduced by Fomin and Zelevinsky, cluster algebras are commutative rings that have many combinatorial properties. They have had many applications to both mathematics and physics. In this talk, I will first introduce cluster algebras and explore some of their properties. I will then move on to their applications, starting with dilogarithm identities and then moving to integrable systems and the thermodynamic Bethe ansatz (TBA). Time permitting, I will connect some of these ideas to the ODE/IM correspondence.