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Simons Lectures/Einstein Chair Lectures STONY BROOK MATHEMATICS AND CUNY GRADUATE CENTER Simons Lectures/Einstein Chair Lectures YEVSEY NISNEVICH Courant Institute Adeles, Grothendieck topologies & G-bundles for semisimple groups G OPEN TO ALL- EXPERTS AND NON EXPERTS 1) Preliminaries on schemes: Noetherian schemes, closed, non-closed and generic points, the local rings and the residue fields of points, the structure sheaf. The Krull dimension of schemes. The functor of points Y -> X(Y) of a scheme X with the values in another (variable) scheme Y and the Grothendieck-Yoneda embedding. Fiber product of schemes, fibers of a morphism of schemes f: X -> Y. 2) Some important classes of morphisms of schemes: unramified, flat, etale and smooth morphisms. Regular and singular points on a scheme. 3) Grothendieck topologies: the general definition and some functorial properties. The main examples: the Zariski, Nisnevich, etale and faithfully flat quasi-compact topologies on the category of schemes and its subcategories (in the increasing order). The concepts of points and local rings of points for Grothendieck topologies. Sheaves on a Grothendieck topology and a fiber of a sheaf over a point on a topology. Henselian rings and their main properties, Henselization of a ring. The description of local rings and fibers of sheaves over points for the first 3 topologies listed above and appearance of Henselian rings for the Nisnevich and etale topologies. 4) Cohomology of sheaves of abelian groups on a Grothendieck topology. Some functorial properties. Cohomological dimensions of the main 3 topologies. 5) The principal homogeneous spaces for a sheaf of (possibly) non-abelian groups G on a topology \tau, locally trivial in this topology (= \tau G-torsors). Cohomology of non-abelian sheaves on a topology \tau and non-abelian exact sequences for a sheaf of subgroups H of G. 6) Group schemes - definition and the main examples: additive G_a, multiplicative G_m, algebraic tori, full linear group GL_m over an arbitrary base scheme. Semisimple and reductive group schemes over a base scheme. 7) Adeles for schemes of dimension 1 with values in a group scheme G. Henselian adeles. The Nisnevich topology as a mean for a geometrization of adeles. Nisnevich cohomology and adelic class groups. 8) The main vanishing theorems for the class groups and the Nisnevich and etale cohomology of semisimple group schemes over general Dedekind (= regular 1-dimensional) rings. Cohomological expressions for these invariants in the non-simply connected case. Finiteness theorems for Dedekind rings with finitely generated divisor class groups. 9) Applications of the vanishing theorems: proof of the conjecture of Grothendieck and Serre on the Zariski local triviality of rationally trivial etale G-torsors over regular base of dimension 1 and 2 and for Henselian local regular rings of an arbitrary dimension. 10) Further applications: extensions of the Uniformization Theorem for the moduli space of G-vector bundles over a smooth irreducible projective curve C/k onto arbitrary fields k and general semisimple group schemes G over C. It was previously known for an algebraically closed field k and a constant semisimple group scheme G/k only and it was due mainly to Drinfeld-Simpson (1995). STONY BROOK MATHEMATICS 2:00 PM & 3:30 PM Tuesday, October 12th & Friday, October 15th (math common room 4-125) Monday, October 18th & Friday, October 22nd (math room P-131) CUNY GRAD CENTER Wednesdays, October 13th & 20th (room 6417) 3:30 PM & 5:00 PM (Two parts each day and independent at each site) .
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