Events
Department of Mathematics and Statistics
Texas Tech University
N/A
N/A
First we shall present some basic facts about condenser capacity,
analytic capacity and holomorphic motions.
Then we will discuss superharmonicity properties of those
capacities under holomorphic motions and, given a condenser, we will
describe the holomorphic motions for which harmonicity occurs. Finally, we will examine a
special holomorphic motion and characterize the condensers for which
harmonicity occurs.
In this talk we will formulate the existence of almost-Clifford structures on smooth manifolds of appropriate dimension in terms of a Kuranishi-Kodaira-Spencer theory, obtaining local structural equations analogous to Cauchy-Riemann conditions. Globally, the satisfaction of these structural equations have obstructions detected precisely by (higher) prolongations of the corresponding G-structures, governed by differential-graded Lie algebras. These obstructions can be compared to the well-understood almost-complex and almost-quaternionic cases (classical Kodaira-Spencer vs. twistor theory). We present also the obstruction for the second complex Clifford algebra, known as the bicomplex numbers, describing an existence result for integrable almost-bicomplex structures, and two compatible double-complexes of differential forms (one elliptic, one non-elliptic) which has its own cohomology and notion of spectral sequence. We draw attention to the previously unobserved similarities between this formalism and work on the “generalized geometry” of Hitchin, Gualtieri, Cavalcanti, and others, suggesting applications of the bicomplex differential geometry to problems in T-duality. If time allows we may suggest a related spectral sequence for almost-quaternionic geometry based on the work of Widdows.https://community.amstat.org/cas/new-item/new-item9  | Wednesday
Oct. 30
3:00 PM
Math 111
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| Algebra and Number Theory
No Seminar
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A new numerical approach based on the minimization of the local truncation error is suggested for the solution of partial differential equations; see [1]. Similar to the finite difference method, the form and the width of the stencil equations are assumed in advance. A discrete system of equations includes regular uniform stencils for internal points and non-uniform stencils for the points close to the boundary. The unknown coefficients of the discrete system are calculated by the minimization of the order of the local truncation error. The main advantages of the new approach are a high accuracy and the simplicity of the formation of a discrete (semi-discrete) system for irregular domains. For the regular uniform stencils, the stencil coefficients can be found analytically. For non-uniform cut stencils, the stencil coefficients are numerically calculated by the solution of a small system of linear algebraic equations (20-100 algebraic equations). In contrast to the finite elements, there is no necessity to calculate by integration the elemental mass and stiffness matrices that is time consuming for high-order elements. As a mesh, the grid points of a uniform Cartesian mesh as well as the points of the intersection of the boundary of a complex domain with the horizontal, vertical and diagonal lines of the uniform Cartesian mesh are used; i.e., in contrast to the finite element meshes, a trivial mesh is used with the new approach. Changing the width of the stencil equations, different high-order numerical techniques can be developed. Currently the new technique is applied to the solution of the wave, heat, Helmholtz and Laplace equations. The theoretical and numerical results show that for the width of the stencil equations equivalent to that for the linear quadrilateral finite elements, the new technique yields the fourth-order of accuracy for the numerical results on irregular domains for the considered partial differential equations (it is much more accurate compared with the linear and high-order finite elements at the same number of degrees of freedom). 3-D numerical examples on irregular domains show that at accuracy of 5%, the new approach reduces the number of degrees of freedom by a factor of greater than 1000 compared to that for the linear finite elements with similar stencils. This leads to a huge reduction in computation time for the new approach at a given accuracy. This reduction in computation time will be even greater if a higher accuracy is needed; e.g., 1% or less.