Speaker: Mark Gross Title: Mirror symmetry for rational surfaces and smoothing surface singularities Speaker: Sasha Merkurjev Title: Algebraic cycles and norm varieties Abstract: Let p be a prime integer and F a field of characteristic 0.Let X be the norm variety of a symbol in K_n(F)/p, constructed in the proof of the Bloch-Kato conjecture. We prove that the function field F(X) has the following property: for any equidimensional variety Y, the change of field homomorphism CH(Y) ---> CH(Y_{F(X)}) of Chow groups with coefficients in integers localized at p is surjective in codimensions < (dim X)/(p-1). This is joint work with N. Karpenko. Speaker: Burt Totaro Title: Symmetric differentials and the fundamental group Abstract: Consider a smooth complex projective variety X. Hodge theory shows that sections of exterior powers of the cotangent bundle are related to the topology of X. What about symmetric powers of the cotangent bundle? We discuss the relation between the topology of X and its "symmetric differentials". One interest of these results is that symmetric differentials give information in the direction of "Kobayashi hyperbolicity"; for example, they limit how many rational curves X can contain. Speaker: Kirsten Wickelgren Title: 2-Nilpotent real section conjecture Abstract: Grothendieck's section conjecture predicts that the rational points of a hyperbolic curve over a number field are determined by the \'etale fundamental group of the curve. This talk is about a 2-nilpotent section conjecture over R: for a geometrically connected curve X over R such that the normalization of each component has real points, pi_0(X(R)) is determined by the maximal 2-nilpotent quotient of the fundamental group with its Galois action. It follows as a corollary that for X smooth and proper, the set of real points of X equipped with a real tangent direction is determined by the maximal 2-nilpotent quotient of Gal(C(X)) with its Gal(C/R) action. |