La recherche

We prove that fields of meromorphic functions on Stein surfaces have cohomological dimension 2, and solve the period-index problem and Serre's conjecture II for these fields. We obtain analogous results for fields of real meromorphic functions on Stein surfaces equipped with an antiholomorphic involution. We deduce an optimal quantitative solution to Hilbert's 17th problem on analytic surfaces.

We show that any sum of squares in a field of transcendence degree 1 over Q is a sum of squares, answering a question of Pop and Pfister. We deduce this result from representation theorem, in k(C), for quadratic forms of rank at least 5 with coefficients in k, where C is a curve over a number field k.

Here are seemingly unrelated problems: computing rational homotopy groups of spheres in rational homotopy theory, purity in algebraic geometry, Koszul duality for the category of a reductive group in representation theory, splitting Drinfeld space's de Rham complex in the p-adic Langlands program, deformation quantization of Poisson manifolds in mathematical physics. And yet, all of them boil down to the same question: formality. A differential graded algebraic structure A (e.g. an associative algebra, a Lie algebra, a Pre-Calabi-Yau algebra, etc.) is formal if it is related to its homology H(A) by a zig-zag of quasi-isomorphisms preserving the algebraic structure. This thesis develops obstruction classes allowing to prove formality results. On the one hand, it incorporates aforementioned results into a single theory. On the other hand, it provides tools to study these questions in cases little studied hitherto: over any coefficient ring and for algebraic structures with several outputs: algebras encoded by properads.