Geometric Analysis and Topology Seminar

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The energy image density property

Speaker: Mathav Murugan, UBC

Location: Online

Videoconference link: https://nyu.zoom.us/j/96390398010

Date: Wednesday, January 21, 2026, 11 a.m.

Synopsis:

The energy image density property originates in Malliavin’s proof of Hörmander’s hypoellipticity theorem. Informally, it asserts that if the differential of a function is surjective, then the pushforward of a suitable reference measure by the function is absolutely continuous with respect to Lebesgue measure. The Bouleau--Hirsch energy image density conjecture (1986) asserts that this absolute continuity property holds for all strongly local Dirichlet forms.  Motivated by a generalization of Radmacher's theorem on metric measure spaces, Cheeger conjectured that the dimension of charts of a measurable differentiable structure on PI spaces is bounded by the Hausdorff dimension (1999). Cheeger's conjecture was resolved by De Philippis, Marchese and Rindler (2017) using the absolute continuity of pushforward of the reference measure under charts. We formulate and prove a generalization of the energy image density property that leads to a resolution of the Bouleau--Hirsch conjecture and a new proof of Cheeger's conjecture. This is joint work with Sylvester Eriksson-Bique.