Talks

Two-dimensional integrable lattice models that can be described in terms of (non-intersecting, possibly osculating) paths with suitable boundary conditions display the arctic phenomenon: the emergence of a sharp phase boundary between ordered cristalline phases (typically near the boundaries) and disordered liquid phases (away from them). We show how the so-called tangent method can be applied to models such as the 6 Vertex model or its triangular lattice variation the 20 Vertex model, to predict exact arctic curves. A number of companion combinatorial results are obtained, relating these problems to tiling problems of associated domains of the plane.

A system of hard rigid rods of length $k \gg1$ on hypercubic lattices in dimensions $d \geq2$, is known to undergo two phase transitions when chemical potential is increased: from a low-density phase to an intermediate density nematic phase, and on further increase to a high-density phase with no nematic order. I will present non-rigorous arguments to support the conjecture that for large $k$, the second phase transition is a first-order transition with a discontinuity in density in all dimensions greater than $1$. The chemical potential at the transition is $\approx A k \ln k$ for large $k$, and that the density of uncovered sites drops from a value $\approx B (\ln k)/ k^2$ , to a value of order $\exp(−ck)$, where $c$ is some constant, across the transition. We conjecture that these results are asymptotically exact, and $A = B= 1$, in all dimensions $d ≥ 2$.

In this series of lectures, I will give an introduction to the theory of moments of L-functions. I will focus on important examples, such as the moments of the Riemann zeta function and Dirichlet L-functions, as well as some GL_2 families. I will also present some of the important tools for understanding moments, as well as applications of moments.

We study the structure of singularities in the discrete Korteweg–deVries equation and its modified sibling. Four different types of singularities are identified. The first type corresponds to localised, ‘confined’, singularities. Two other types of singularities are of infinite extent and consist of oblique lines. The fourth type of singularity corresponds to horizontal strips where the product of the values on vertically adjacent points is equal to 1. Due to its orientation this singularity can, in fact, interact with the other types. This type of singularity was dubbed ‘taishi’. The taishi can interact with singularities of the other two families, giving rise to very rich and quite intricate singularity structures. Nonetheless, these interactions can be described in a compact way through the formulation of a symbolic representation of the dynamics. We give an interpretation of this symbolic representation in terms of a box & ball system related to the ultradiscrete KdV equation.

We introduce two properties that characterise integrable discrete systems: singularity confinement and low growth. The latter is quantified through the dynamical degree, a quantity that is equal to 1 for integrable systems and larger than 1 for non-integrable ones. We show how the structure of singularities conditions the growth properties of a given system. We introduce the full deautonomisation discrete integrability criterion and illustrate its application through concrete examples. Starting from the results of R. Halburd we show how one can obtain the dynamical degree of a given mapping based on its singularity structure. The notion of Diophantine approximation is introduced as a practical way to obtain the dynamical degree. We show how one can obtain the degree growth of a given birational mapping in an algorithmic way using only the information on its singularities. Several examples of second-order mappings are presented and we show how our approach can be extended to higher-orde...

We review the theory of exponential sums due to Weyl and van der Corput and consider several applications. If time permits, we also look at the theory of p-adic exponent pairs, as developed by Milićević.