**Michele D'Adderio** (Università di Pisa, Italy)

**Title:** q,t-combinatorics and sandpiles

**Abstract:** In 1987 Bak, Tang and Wiesenfeld introduced their famous sandpile model as a first system showing self-organized criticality. In 1988 Macdonald introduced his famous symmetric polynomials. Each of these two discoveries produced a huge amount of research that is still developing intensely today. But until recently, these two lines of research went on without any relevant interaction. In this talk we show how the combinatorics generated by these two important mathematical objects come together in a surprising way, proving that a synergy between these two topics is inevitable.

**Nathalie Aubrun **(Université Paris-Saclay, France)

**Title:** Snake tilings, skeletons susbhift and self-avoiding walk

**Abstract: **Wang tiles are squares with coloured edges that can be placed side by side as long as two neighbouring tiles have the same colour on their common side. Given a finite set of such tiles, whether it is possible to create an infinite tiling of the plane is an undecidable problem, known as the domino problem. This problem has also been studied for about fifteen years for groups of finite type other than Z^2. In this talk I will focus on a variant of this problem, the snake domino tiling. Instead of looking for a tiling of the whole plane (or the whole group), we only look for a tiling of an infinite path. This problem remains undecidable in the Euclidean plane Z^2. I will present recent progress on the snake tiling problem for groups, and make the connection with self-avoiding paths. (Joint work with Nicolas Bitar)

**Anna Ben-Hamou** (Sorbonne Université, France)

**Title:** Cutoff for permuted Markov chains

**Abstract:**** **For a given finite Markov chain with uniform stationary distribution, and a given permutation on the state-space, we consider the Markov chain which alternates between random jumps according to the initial chain, and deterministic jumps according to the permutation. In this framework, Chatterjee and Diaconis (2020) showed that when the permutation satisfies some expansion condition with respect to the chain, then the mixing time is logarithmic, and that this expansion condition is satisfied by almost all permutations. We will see that the mixing time can even be characterized much more precisely: for almost all permutations, the permuted chain has cutoff, at a time which only depends on the entropic rate of the initial chain.

**Craig S. Kaplan **(University of Waterloo, Canada)

**Title:** Computing tiling properties of polyforms

**Abstract:** Polyforms—shapes constructed by gluing together copies of cells in an underlying grid—are a convenient experimental tool with which to probe problems in tiling theory. Unlike shapes more generally, they can be enumerated exhaustively, and are amenable to analysis using discrete computation. Furthermore, polyforms appear to be quite expressive in terms of the range of tiling-theoretic behaviours they can exhibit. I discuss the computation of isohedral numbers and Heesch numbers, both of which are connected to a variety of unsolved problems in tiling theory, and the connection of these problems to the world's first aperiodic monotiles, discovered in 2023.

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**Cyril Nicaud **(Université Gustave Eiffel, Marne-La-Vallée, France)

**Title:** Random automata

**Abstract:** In this talk, we will survey several results concerning random finite state automata, including random generation and algorithm analysis. We will place special focus on the subset construction, the standard algorithm for building a deterministic automaton equivalent to non-deterministic one.

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