The generalized Poincaré conjecture in higher dimensions

On the Weinstein Conjecture in Higher Dimensions

The first break-through on this conjecture was obtained by C. Viterbo, [19], showing that compact energy surfaces in R2n of contact-type have periodic orbits. Extending Gromov’s theory of pseudoholomorphic curves, [3], to symplectized contact manifolds, H. Hofer, [4], related the Weinstein conjecture to the existence of certain pseudoholomorphic curves. He showed that in dimension three the Wei...

Ricci Flow and the Poincaré Conjecture

Fuglede's Conjecture Is False in 12 and Higher Dimensions

We give an example of a set Ω ⊂ R 12 which is a finite union of unit cubes, such that L 2 (Ω) admits an orthonormal basis of exponentials { 1 |Ω| 1/2 e 2πiξ j ·x : ξ j ∈ Λ} for some discrete set Λ ⊂ R 12 , but which does not tile R 12 by translations. This answers (one direction of) a conjecture of Fuglede [1] in the negative, at least in 12 and higher dimensions.

Generalized Poincare's Conjecture in Dimensions Greater Than Four

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Generalized Duality and Singular Strings in Higher Dimensions

Deformations of gauged WZW actions are constructed for any pair (G, H) by taking different embeddings of the gauge group H ⊂ G as it acts on the left and right of the group element g. This leads to models that are dual to each other, generalizing the axial/vector duality of the two dimensional black hole manifold. The classical equations are completely solved for any pair (G, H) and in particul...