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I will discuss the relation between non-compact spin chains and the zero-range processes introduced by Sasamoto-Wadati, Povolotsky and Barraquand-Corwin. The main difference compared to the prime examples of integrable particle processes, namely the SSEP and the ASEP, is that for the models discussed in this talk several particles can occupy one and the same site. Guided by the desire to maintain the integrable structure, I will introduce boundary conditions for these models that are obtained from the boundary Yang-Baxter equation. This allows to define analogues of the open SSEP and ASEP with boundary reservoirs. Some recent exact results concerning these types of integrable non-equilibrium zero-range processes will be discussed.

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.

The notion of congruence (modulo an integer q) was formalised by C. F. Gauss in his Disquisitiones arithmeticae. This is a basic yet fundamental concept in all aspects of number theory. Indeed congruences allow to evaluate and compare integers in way considerably richer than the archimedean order alone permits.

In analytic number theory, several outstanding question -starting with Dirichlet’s theorem on primes in arithmetic progressions- reduce to the of measuring whether some classical arithmetic function (say the characteristic function of prime numbers) correlate with suitable q periodic functions for instance Gauss sums, Jacobi sums or Kloosterman sums. It turns out that these functions, when the modulus q is a prime (to which one can reduce via the Chinese Reminder Theorem) can be recognised as « trace functions». The study of trace functions was initiated by A. Weil in the 1940’s and was pursued by A. Grothendieck in the second half of the century with his refoundation of alge...

The theory of degree growth and algebraic entropy plays a crucial role in the field of discrete integrable systems. However, a general method for calculating degree growth for lattice equations (partial difference equations) is not yet known. In this talk, I will propose a new method to rigorously compute the exact degree of each iterate for lattice equations. The strategy is to extend Halburd's method, which is a novel approach to computing the exact degree of each iterate for mappings (ordinary difference equations) from the singularity structure, to lattice equations.

First, I will illustrate, without rigorous discussion, how to calculate degrees for lattice equations using the lattice version of Halburd's method and discuss what problems we need to solve to make the method rigorous. Then, I will provide a framework to ensure that all calculations are accurate and rigorous. If time permits, I would also like to discuss how to detect the singularity structure of a lattice equati...

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.