Abstract

This talk provides a non-technical introduction to a conjectural algebro-geometric formula for the reduced analytic stability threshold of a given Fano manifold X, introduced in my recent preprint 'Gibbs polystability of Fano manifolds, stability thresholds and symmetry breaking,' joint with Rolf Andreasson and Ludvig Svensson. This threshold is central to the existence of canonical metrics, as established by Darvas-Rubinstein's resolution of a conjecture of Tian, which states that X admits a Kähler-Einstein metric if and only if the reduced analytic stability threshold of X is strictly greater than one.

While our broader work is motivated by a probabilistic approach for constructing these metrics, this presentation will focus exclusively on the application to computing reduced analytic stability thresholds. I will demonstrate how the proof of our conjectural formula in the simplest case—the Riemann sphere—yields a sharp form of the conjecture of Tian and an improvement of the sharp logarithmic Hardy–Littlewood–Sobolev inequality on the two-sphere.

Biography

Robert Berman is a professor at Chalmers University of Technology in Sweden, where he obtained his PhD in 2006. He works primarily on analytical aspects of complex algebraic and differential geometry, with a particular focus on connections to probability, statistical mechanics, and mathematical physics.

Zoom meeting

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