AbstractThe original version of Sullivan's rational surgery realization theorem provides necessary and sufficient conditions for a prescribed rational cohomology ring to be realized by a simply-connected smooth closed manifold. We will present a version of the theorem for almost complex manifolds. It has been shown there exist closed smooth manifolds M^n of Betti number b_i=0 except b_0=b_{n/2}...
AbstractThe geometric P=W conjecture is a conjectural description of the asymptotic behaviour of acelebrated correspondence in non-abelian Hodge theory. in a joint work with Enrica Mazzon andMatthew Stevenson, we establish the full geometric conjecture for compact Riemann surfaces ofgenus one, and obtain partial results in arbitrary genus: this is the first non-trivial evidence of theconiecture...