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4:00 pm Wednesday, February 1, 2017 Geometry-Analysis Seminar: Harmonic Maps into Cones and Singularities of Nematic Liquid CrystalsRobert 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 |

4:00 pm Wednesday, February 8, 2017 Geometry-Analysis Seminar: To Be AnnouncedMike WolfHost Department: Rice University-Mathematics Submitted by mwolf@rice.edu |

4:00 pm Wednesday, February 15, 2017 Geometry-Analysis Seminar: Chebyshev Problems on a Circular ArcBenjamin Eichinger (Johannes Kepler University Linz) in HBH 227- We consider the Chebyshev polynomials, $T_n$, on a circular arc $A_\alpha$, i.e., the monic polynomials of degree at most $n$ minimizing the sup-norm, $ \|T_n\|_{A_\alpha}. $ Thiran and Detaille found an explicit formula for the asymptotics of $\|T_n\|_{A_\alpha}$, which disproved a conjecture of Widom. We give the Szeg\H o-Widom asymptotics of the domain explicitly. That is, the limit of the properly normalized extremal functions $T_n$. Moreover, we solve a similar problem with respect to the upper envelope of a family of polynomials uniformly bounded on $A_\alpha$. Our computations show that in the proper normalization the limit of the upper envelope is the diagonal of a reproducing kernel of a certain Hilbert space of analytic functions. Similarly, we study the \textit{polynomial Ahlfors problem}, i.e., we maximize $|P'(z)|$ at a fixed point $z\in\mathbb C\setminus A_\alpha$ in the class of polynomials of degree at most $n$ that are uniformly bounded on $A_\alpha$ and vanish at the given point. The corresponding asymptotics can again be given in terms of an explicit reproducing kernel.
Submitted by damanik@rice.edu |

4:00 pm Wednesday, March 22, 2017 Geometry-Analysis Seminar: Mixing and un-mixing by incompressible flowsYao Yao (Georgia Tech.) in HBH 227- Abstract: In this talk, we consider the questions of efficient mixing and un-mixing by incompressible flows, under the constraint that the W^{1,p} Sobolev norm of flow is uniformly bounded in time. We construct some explicit flows to show that for any bounded initial density, it can be mixed to scale epsilon in time C|log(epsilon)| for 1
Host Department: Rice University-Mathematics |

4:00 pm Wednesday, March 29, 2017 Geometry-Analysis Seminar: Discrete length-volume inequalities for cubesKyle Kinneberg (Rice) in HBH 227- It has been known for some time that for any Riemannian cube, the Riemannian volume is bounded below by the product of the distances between opposite codimension-one faces. We'll discuss some discrete analogs of this fact, which concern weighted open covers of topological cubes. We'll also give some applications to lower volume bounds in metric spaces.
Submitted by kk43@rice.edu |

4:00 pm Wednesday, April 5, 2017 Geometry-Analysis Seminar: Some Combinatorics and Dynamical Systems connectionsMichael Boshernitzan (Rice University) in HBH427- Some combinatorial problems (some are open) with probability flavor will be discussed. Possible subjects: 1. We show that the collection $2^\N$ of subsets of natural numbers forms a complete space relative to the pseudometric \dist^*(U,V)=d^*(U\oplus V), U,V\subset \N, where $U\oplus V$ denotes the symmetric difference of $U, V$ and d^*(W):=\limsup_{n\to\infty}\,\frac{\big|W\cap[1,n]\big|}n, n\in\N, stands for the upper asymptotic density of $W\subset\N$. The above claim extends to some other notions ofdensity but not to the upper Banach density d_B^*(W):=\limsup_{n\to\infty},\frac{\big|W\cap[m,m+n]\big|}n, m,n\in\N. 2. We prove that every compact connected subset in $\R^2$ which does not reduce to a point must contain either a convex curve or an arithmetical progression of length 3.
Submitted by mwolf@rice.edu |