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01/25/07 Colloquium
Dr. Maxim Zyskin
University of Oxford
University of Oxford
Boundary Value Problems, Integral Transforms, and Geometric Analysis
Abstract: Initial or boundary value problems for linear or integrablenonlinear PDE'sgive rise to integral transforms custom-made to solve them. These include classical transforms, Fourier, Radon, and Abel. Such transforms have many applications in PDE'sand beyond-for example in integral geometry and representation theory. My recent work is on boundary-value problems for a nematicliquid crystal in polyhedral domains with tangent boundary conditions, and applications for a new type ofliquid crystal displays, bi-stable displays. For such problems, equations describing stable configurations of liquid crystal cannot be solved explicitly, however a lot of insight on how solutions look like can be gained by a combination of geometric and analytic methods. Our results include complete topological classification of liquid crystal configurations, and minimum energy bounds in each topological class. In certain cases it can be shown that solutions are smooth, except at a discrete set of point singularities. Description of behavior of solutions near such singularities, or deriving good upper energy bounds gives rise to reduced two-dimensional equations which are integrable, and allow for explicit solutions.