Let a = (pq11, . . . , pqrr) be a partition and a' = (p'1q'1, . . . , p'rq'r) be its conjugate. We will prove that if qi, q'i ≥ 2 for all 1 ≤ i ≤ r, then any irreducible subvariety X of Gr(m, n) whose homology class is an integral multiple of the Schubert class [σa] of type a is a Schubert variety of type a.

We develop techniques for computing the integer valued SU(3) Casson invariant defined in [6]. Our method involves resolving the singularities in the flat moduli space using a twisting perturbation and analyzing its effect on the topology of the perturbed flat moduli space. These techniques, together with Bott-Morse theory and the splitting principle for spectral flow, are applied to calculate τ_SU(3)(Σ) for all Brieskorn homology spheres.

In the standard contact (2n + 1)-space when n > 1, we construct infinite families of pairwise non-Legendrian isotopic, Legendrian n-spheres, n-tori and surfaces which are indistinguishable using classically known invariants. When n is even, these are the first known examples of non-Legendrian isotopic, Legendrian submanifolds of (2n + 1)-space. Such constructions indicate a rich theory of Legendrian submanifolds. To distinguish our examples, we compute their contact homology which was rigorously defined in this situation in [7].

We prove the existence of a solution to the Monge-Ampère equation detHess(ø) = 1 on a cone over a thrice-punctured two-sphere. The total space of the tangent bundle is thereby a Calabi-Yau manifold with flat special Lagrangian fibers. (Each fiber can be quotiented to three-torus if the affine monodromy can be shown to lie in SL(3,Z) × R³.) Our method is through Baues and Cortés's result that a metric cone over an elliptic affine sphere has a parabolic affine sphere structure (i.e., has a Monge-Ampère solution). The elliptic affine sphere structure is determined by a semilinear PDE on CP¹ minus three points, and we prove existence of a solution using the direct method in the calculus of variations.

We introduce a new class of artinian weighted complete intersections, by abstracting the essential features of Q-cohomology algebras of equal rank homogeneous spaces of compact connected Lie groups. We prove that, on a 1-connected closed manifold M with H*(M,Q) belonging to this class, every isometry has a nontrivial invariant geodesic, for any metric on M. We use Q-surgery to construct large classes of new examples for which the above result may be applied.