Topological phase transitions were discovered by Berezinskii-Kosterlitz-Thouless in the 70's. They describe intriguing phase transitions for classical spins systems such as the plane rotator model (or $XY$ model). I will start by reviewing how this phase transition arises in cases such as:

• the $XY$ model (spins on $\mathbb{Z}^2$ with values in the unit circle)
• the integer-valued Gaussian Free Field (or $\mathbb{Z}$-ferromagnet)
• Abelian Yang-Mills on $\mathbb{Z}^4$

I will then connect topological phase transitions to a statistical reconstruction problem concerning the Gaussian Free Field and will show that the feasibility of the reconstruction undergoes a KT transition.

This is a joint work with Avelio Sepúlveda (Lyon) and the talk will be based mostly on the preprint: https://arxiv.org/abs/2002.12284