Spencer Dembner, speaking in the Institute for Advanced Study's Special Year learning seminar, presents work on definable analytic geometry, the study of complex-analytic structures cut out within o-minimal systems. He sets up the basic objects: definable coherent sheaves, their GAGA correspondence with algebraic sheaves, and the question of whether analogues of affine schemes or Stein spaces exist where coherent cohomology vanishes entirely. Dembner shows that while such vanishing spaces are rare in the expansive structure R_an,exp, they turn out to be abundant in the smaller structure R_an, at least for curves. The talk works through the proof of this one-dimensional result and, time permitting, sketches partial progress and open problems in higher dimensions. It assumes familiarity with sheaf cohomology and o-minimality, pitched at researchers working in Hodge theory and tame geometry rather than a general audience.