- 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

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