In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts in F. Almgren, and J. Pitts, corresponding to the fundamental class of a Riemannian manifold $(M^{n+1}, g)$of positive Ricci curvature with 2 ≤ n ≤ 6. We characterize the Morse index, volume and multiplicity of this min-max hypersurface. In particular, we show that the min-max hypersurface is either orientable and of index one, or is a double cover of a non-orientable minimal hypersurface with least area among all closed embedded minimal hypersurfaces.

This note discusses the recent work of Huang-Schoen-Wang [13] on specifying total conserved quantities of a vacuum initial data set.

We introduce a new equivalence relation for complete alge- braic varieties with canonical singularities, generated by birational equivalence, by flat algebraic deformations (of varieties with canon- ical singularities), and by quasi-´etale morphisms, i.e., morphisms which are unramified in codimension 1. We denote the above equivalence by A.Q.E.D. : = Algebraic-Quasi-´ Etale- Deformation. A completely similar equivalence relation, denoted by C-Q.E.D., can be considered for compact complex spaces with canonical sin- gularities. By a recent theorem of Siu, dimension and Kodaira dimension are invariants for A.Q.E.D. of complex varieties. We address the interesting question whether conversely two al- gebraic varieties of the same dimension and with the same Ko- daira dimension are Q.E.D. - equivalent (A.Q.E.D., or at least C-Q.E.D.), the answer being positive for curves by well known results. Using Enriques’ (resp. Kodaira’s) classification we show first that the answer to the C-Q.E.D. question is positive for special algebraic surfaces (those with Kodaira dimension at most 1), resp. for compact complex surfaces with Kodaira dimension 0, 1 and even first Betti number. The appendix by S¨onke Rollenske shows that the hypothesis of even first Betti number is necessary: he proves that any sur- face which is C-Q.E.D.-equivalent to a Kodaira surface is itself a Kodaira surface. We show also that the answer to the A.Q.E.D. question is pos- itive for complex algebraic surfaces of Kodaira dimension ≤ 1. The answer to the Q.E.D. question is instead negative for sur- faces of general type: the other appendix, due to Fritz Grunewald, is devoted to showing that the (rigid) Kuga-Shavel type surfaces of general type obtained as quotients of the bidisk via discrete groups constructed from quaternion algebras belong to countably many distinct Q.E.D. equivalence classes.

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We prove that the algebra of singular cochains on a smooth manifold, equipped with the cup product, is equivalent to the A1 structure on the Lagrangian Floer cochain group associated to the zero section in the cotangent bundle. More generally, given embeddings with isomorphic normal bundles of a closed manifold B into manifolds Q1 and Q2, we construct a differential graded category from the singular cochains of these spaces, and prove that it is equivalent to the A1 category obtained by considering exact Lagrangian embeddings of Q1 and Q2 which intersect cleanly along B.