# Isotropic curvature, macroscopic dimension Filling radius of contractible loops and fundamental group

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Gabriele La Nave, University of Illinois, Urbana-Champaign
Fine Hall 314

I will discuss the proof of a conjecture of Gromov's to the effect that manifolds with uniformly positive isotropic curvature (and bounded geometry) are macroscopically 1-dimensional on the scale of the isotropic curvature. One of the main techniques involved is modeled on Donaldson's version of H\"ormander technique to produce (almost) holomorphic sections. We use this to produce  sections of the restriction of the complexified tangent bundle of M to a stable embedded minimal disk which destabilize the disk if the distance grows too much. As a consequence we prove that compact manifolds with positive isotropic curvature have virtually free fundamental groups."