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Explore the intersection of o-minimal structures and complex geometry in this mathematical seminar examining definable analytic spaces and their coherent sheaves. Learn how o-minimal structures determine collections of "tame" or "definable" subsets of ℝⁿ and discover the fragment of complex geometry present within these structures, including which holomorphic functions are definable and which spaces are cut out by definable holomorphic functions. Understand the growing importance of definable analytic spaces in recent mathematical research, particularly their applications in Hodge theory, and see how definable analytic geometry emerges as an interesting subject parallel to algebraic or analytic geometry while interpolating between them. Examine the robust notion of definable coherent sheaf and its expected properties, including the GAGA theorem relating definable and algebraic sheaves. Investigate the central question of whether definable coherent sheaves satisfy vanishing theorems and whether spaces analogous to affine schemes or Stein spaces exist where all coherent cohomology vanishes. Discover findings showing that while such spaces appear rare in the usual structure ℝₐₙ,ₑₓₚ, they are abundant in the smaller structure ℝₐₙ, at least in dimension one, with discussion of the situation in higher dimensions and partial results as time permits.
