December 2016 January 2017 |

4:00 pm Wednesday, February 1, 2017 Geometry-Analysis Seminar: Harmonic Maps into Cones and Singularities of Nematic Liquid Crystalsby Robert Hardt (Rice University) in HBH 227- Abstract. In 1985, J. Ericksen derived a model for uniaxial liquid crystals to allow for disclinations (i.e. line defects or curve singularities). It involved not only a unit director vectorfield on a region of R^3 but also a scalar order parmeter quantifying the expected inner product between this vector and the molecular orientation. FH.Lin, in several papers, related this model, for certain material constants, to harmonic maps to a metric cone over S^2 . He showed that a minimizer would be continuous everywhere but would have higher regularity fail on the singular defect set . The optimal partial regularity result of R.Hardt-FH.Lin in 1993, for this model, improved this to regularity away from isolated points. This result unfortunately still excluded line singularities. This paper accordingly also introduced a modified model involving maps to a metric cone over RP^2, the real projective plane. Here the nontrivial homotopy leads to the optimal estimate of the singular set being 1 dimensional. In 2010, J. Ball and A.Zarnescu discussed a derivation from the de Gennes Q tensor and interesting orientability questions involving RP^2 . In a recent work with FH.Lin and O. Alper, we see that the singular set with the RP^2 cone model necessarily consists of Holder continuous curves.
Submitted by hardt@rice.edu |