Homeomorphisms of a circle and factorization
DateTuesday, February 6, 2018 - 12:30pm
AbstractFor each n > 0 there is a one complex parameter family of homeomorphisms of the circle consisting of linear fractional transformations `conjugated by z -> z^n'. In this talk I will discuss a number of analytic issues which arise in trying to factor a general homeomorphism in terms of these basic building blocks. This is an attempt to develop a nonlinear version of Fourier series in which addition is replaced by composition.
Antipolar ordering of topological defects in active liquid crystals
DateTuesday, February 13, 2018 - 12:30pm
AbstractRecent experiments in the laboratory of Zvonimir Dogic (UC Santa Barbara) demonstrated that microtubule-motor protein assemblies can self-assemble into an active liquid crystal phase that exhibits a rich topological defect dynamics. This remarkable discovery has sparked considerable theoretical and experimental interest. I will present and validate a tensor Swift-Hohenberg PDE model for this system by merging universality ideas with the classical Landau-de Gennes theory. The resulting model agrees quantitatively with recently published data and, in particular, predicts a previously unexplained regime of antipolar order of topological defects. Our results suggest that complex nonequilibrium pattern-formation phenomena might be predictable from a few fundamental symmetry-breaking and scale-selection principles.
Characterization of Steady Solutions for the 2D Euler Equation
DateTuesday, February 20, 2018 - 12:30pm
AbstractThe motion of an ideal fluid on a 2D surface is described by the incompressible Euler equation, which can be regarded as a Hamiltonian system on coadjoint orbits of the symplectic diffeomorphisms group. Using a combinatorial description of these orbits in terms of graphs with some additional structures, we give a characterization of coadjoint orbits which may admit steady solutions of then Euler equation (steady fluid flows). It turns out that when the genus of the surface is at least one, most coadjoint orbits do not admit steady fluid flows, while the set of orbits admitting such flows is a convex polytope. This is a joint work with B.Khesin.