The André-Poincaré "probléme du scrutin" [9] can be stated as follows: In an election between two candidates A polls m votes, B polls n, m > n. If the votes are counted one by one what is the probability that A leads B throughout the counting? Many derivations and interpretations of the solution have been given and a convenient summary of methods till 1956 can be found in Feller [1]. So numerous are the generalizations of ballot problems and their applications since this date that we do not even attempt an enumeration here.
Abstract. We show that, up to Lagrangian isotopy, there is a unique Lagrangian torus inside each of the following uniruled symplectic fourmanifolds: the symplectic vector space R 4 , the projective plane CP 2 , and the monotone S 2 × S 2 . The result is proven by studying pseudoholomorphic foliations while performing the splitting construction from symplectic field theory along the Lagrangian torus. A number of other related results are also shown. Notably, the nearby Lagrangian conjecture is established for T * T 2 , i.e. it is shown that every closed exact Lagrangian submanifold in this cotangent bundle is Hamiltonian isotopic to the zero-section.
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