Conférencier: Shuo Li (UQAM): TBA

A square is a word of the form $uu$, where $u$ is a finite word. The problem of determining the number of distinct squares in a finite word was initially explored by Fraenkel and Simpson in 1998. They proved that the number of distinct squares, denoted as $\Squares{w}$, in a finite word $w$ of length $n$ is upper bounded by $2n$ and conjectured that $\Squares{w}$ is no larger than $n$. In this talk, we review some old and new findings concerning the square-counting problem and prove that $\Squares{w} \leq n-\Theta(\log_2(n))$.