## Final Report Summary - REALUMAN (Real uniruled manifolds)

1) Sol manifolds do not occur in the real locus of an orientable real Del Pezzo fibration over a curve, confirming an expectation of J. Kollár. (joint work with F. Mangolte)

2) Open Gromov-Witten invariants have been defined for orientable Lagrangian submanifolds of closed symplectic manifolds of dimensions four and six, in some relative homology classes.

3) The probability of a real curve in a real projective surface to have the maximal possible number of connected components exponentially decreases as its degree grows to infinity (joint work with Damien Gayet).

4) The expected Betti numbers of a random real algebraic submanifold of a real projective manifold grows as the square root of its degree as the latter grows to infinity (joint work with Damien Gayet).

5) The expected Betti numbers of the vanishing locus of a random linear combination of the first eigenvectors of a self-adjoint elliptic pseudo-differential operator bounded from below on a closed manifold grows as the number of these eigenvectors as the latter grows to infinity (joint work with Damien Gayet).

2) Open Gromov-Witten invariants have been defined for orientable Lagrangian submanifolds of closed symplectic manifolds of dimensions four and six, in some relative homology classes.

3) The probability of a real curve in a real projective surface to have the maximal possible number of connected components exponentially decreases as its degree grows to infinity (joint work with Damien Gayet).

4) The expected Betti numbers of a random real algebraic submanifold of a real projective manifold grows as the square root of its degree as the latter grows to infinity (joint work with Damien Gayet).

5) The expected Betti numbers of the vanishing locus of a random linear combination of the first eigenvectors of a self-adjoint elliptic pseudo-differential operator bounded from below on a closed manifold grows as the number of these eigenvectors as the latter grows to infinity (joint work with Damien Gayet).

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**Record Number**: 187717 /

**Last updated on**: 2016-08-16