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A matrix is said to be totally positive, if all its minors are positive. The notion of total positivity plays an important role in different areas of mathematics. Lusztig has generalized this notion to all split real Lie groups. He also introduced a notion of positivity in the corresponding flag varieties, that played a key role in the work of Fock and Goncharov on higher Teichm\”uller spaces. 

 

In this talk I will introduce a new notion of positivity in flag varieties that includes but generalizes Lusztig’s definition. I will discuss some applications of this new notion to higher Teichm\”uller spaces, as well as the geometric and dynamical properties of the corresponding representations of surface groups.