Agnes at Brown schedule and abstracts

Agnes at Brown will be held October 26-28, 2012 at Brown University. Funding is provided by the National Science Foundation and Brown University.

• Dmitry Arinkin (University of Wisconsin at Madison), Singular support of coherent sheaves: The notion of singular support plays an important role in the theory of constructible (and perverse) sheaves and holonomic D-modules. Informally, the singular support measures `non-smoothness' of the object. For instance, the singular support of a constructible sheaf is zero if and only if the sheaf is locally constant.

• The goal of this talk is to sketch an analogous theory for coherent sheaves. It turns out that the singular support could be non-trivial only if the variety is singular. I will define singular support for coherent sheaves on a singular variety that is locally a complete intersection. The singular support measures `imperfection' of a coherent sheaf: it equals zero if and only if the coherent sheaf has finite Tor dimension (i.e., the sheaf is perfect).

• Frédéric Campana (Université Nancy), The geometry of orbifold pairs and birational classification: An obifold pair (X,D) consists of a complex projective variety together with an effective Q-divisor analogous to a ramification divisor. Orbifold pairs interpolate between projective and quasi-projective varieties, and naturally arise from multiple fibres of fibrations, which they permit to virtually eliminate by ramified base-change. Their geometry can be defined in complete analogy with the particular case of projective manifolds (ie: when D=0). Only in this enlarged category can arbitrary (quasi)-projective manifolds be decomposed in objects for which the canonical bundle is either ample, numerically trivial, or anti-ample.

• Melody Chan (Harvard University), From algebraic to tropical curves: I will give an expository talk introducing tropical geometry via the the study of tropical curves, and linear series on them, in relation to algebraic curves. In particular, I will highlight recent developments in the subject by O. Amini, M. Baker, L. Caporaso, and others.

• Daniel Huybrechts (Universität Bonn), Cycles on K3 surfaces: I will survey recent progress on the Chow group of K3 surfaces (due to O'Grady, Voisin, Kemeny and myself). On the one hand, we can now prove that symplectic automorphisms of finite order act trivially on the Chow group of degree zero cycles (as predicted by the Bloch-Beilinson philosophy) and on the other hand the role of the Beauville-Voisin ring has become clearer.

• Hee Oh (Brown University), Ergodic theory and rational points on homogeneous varieties

• Andrei Okounkov (Columbia University), The M-theory index

• Cathy O'Neil (New York), How math is used outside academia: We will explore some common mathematical models, their use, their transparency, and their impact in the fields of finance, economics, medicine, education, and on the internet.

• Christian Schnell (Stony Brook University), From the generic vanishing theorem to D-modules: In the late 1980s, Green and Lazarsfeld studied the cohomology of topologically trivial line bundles on compact Kaehler manifolds. Among other things, they proved the "generic vanishing theorem": the cohomology of a generic such line bundle vanishes below a certain degree that only depends on the manifold. Recently, Mihnea Popa and I discovered that behind those results lies a certain class of D-modules on abelian varieties; in the talk, I am going to explain why.

• Angelo Vistoli (Scuola Normale Superiore, Pisa), The fundamental gerbe of a curve and the section conjecture: I will report on joint work with Niels Borne. I will explain how to formulate Grothendieck's section conjecture in terms of Deligne's relative fundamental groupoid of a curve. Then I'll discuss our work about the Nori fundamental gerbe, and show how this leads naturally to a formulation of the section conjecture in positive characteristic.