The shear coordinate is a countable coordinate system to describe increasing self-maps of the unit circle, which is furthermore invariant under modular transformations. Characterizations of circle homeomorphism, quasisymmetric homeomorphisms were obtained by D.Šarić. We are interested in characterizing Weil-Petersson circle homeomorphisms using shears. This class of homeomorphisms arises from the study of Kähler geometry on the universal Teichmüller space and connects various distant fields that will be mentioned briefly.

For this, we introduce diamond shear which is the minimal combination of shears producing WP homeomorphisms. Diamond shears are closely related to log-Lambda length introduced by R. Penner. We obtain sharp results comparing the class of circle homeomorphisms with square summable diamond shears with the Weil-Petersson class and Hölder classes. We also express the Weil-Petersson metric tensor and symplectic form in terms of infinitesimal shears and diamond shears.

This talk is based on joint work with Dragomir Šarić and Catherine Wolfram. See https://arxiv.org/abs/2211.11497.

In this talk, we will show that the monomial basis is generally as good as a well-conditioned polynomial basis for interpolation, provided that the condition number of the Vandermonde matrix is smaller than the reciprocal of machine epsilon. We will also show that the monomial basis is more advantageous than other polynomial bases in a number of applications.

The Muskat equation models the interaction of two incompressible fluids with equal viscosity propagating in porous medium, governed by Darcy’s law. In this talk, we investigate the small data critical regularity theory for this equation, and in particular, the desingularization of interfaces with small moving corners. This is a joint work with Eduardo Garcia-Juarez (Universidad de Sevilla), Javier Gomez-Serrano (Brown University) and Benoit Pausader (Brown University)