Etienne Moutot

Take an infinite grid and color each cell black or white. Then, walk on this grid. While walking, look around (but resist the temptation of looking for infinite clusters). Imagine you can only see the grid at a bounded distance around you. What can you say about the global coloring 
of the grid from the local information of the patterns you see around you while walking ?

In a way, it is the subject of Nivat's Conjecture. It states that if you see at most m*n rectangular patterns of size m x n, then the global coloring must be periodic. Despite the simplicity of the question, it remains open since it was formulated in 1997.

Interestingly the same statement is known to hold for a coloring of an infinite line (if you see less than n patterns of size n then the line is periodic), and proved false for a coloring of Z^3.

In this talk I will present some of the progress that have been made to approach this conjecture, and some of the particular cases where it has been proved.
The newest ones have been done using algebraic tools: by representing coloring by formal Laurent series, you can use tools from polynomial algebra to show very powerful statement about these colorings.