published_or_final_version / abstract / Mathematics / Master / Master of Philosophy

published_or_final_version / abstract / Mathematics / Master / Master of Philosophy

A hierarchy of vector fields (master symmetries) and homogeneous nonlinear Poisson structures associated with the Toda lattice are constructed and the various connections between them are investigated. Among their properties: new brackets are generated from old ones by using Lie-derivatives in the direction of certain vector fields; the infinite sequences obtained consist of compatible Poisson brackets in which the constants of motion for the Toda lattice are in involution. The vector fields in the construction are unique up to addition of a Hamiltonian vector field. Similarly the Poisson brackets are unique up to addition of a trivial Poisson bracket. These are Poisson tensors generated by wedge products of Hamiltonian vector fields. The non-trivial brackets may also be obtained by the use of r-matrices; we give formulas and prove this for the quadratic and cubic Toda brackets. We also indicate how these results can be generalized to other (semisimple) Toda flows and we give explicit formulas for the rank 2 Lie algebra of type B₂. The main tool in this calculation is Dirac's constraint bracket formula. Finally we study nonlinear Poisson brackets associated with orbits through nilpotent conjugacy classes in gl(n, R) and formulate some conjectures. We determine the degree of the transverse Poisson structure through such nilpotent elements in gl(n, R) for n ≤ 7. This is accomplished also by the use of Dirac's bracket formula.

Generated by stochastic recursions, perpetuities encompass a vast range of discretetime financial behaviours. When focusing on the dramatic changes occurring in such processes, the analysis of threshold exceedances provides an extensive description of their underlying mechanisms. Asymptotically, an exceedance point process tends to a compound Poisson measure, highlighting a tendency to cluster. Now, the parameters of this limit law are known, but complex. Here, an empirical approach is adopted, and a class of explicit compound Poisson models developed, with a bound on the error, for the exceedance point process of a finite, multidimensional perpetuity. In a financial regulatory context, this provides a new way of examining the Value-at-Risk criterion for securities.

Thesis (M. Phil.)--University of Hong Kong, 2006. / Title proper from title frame. Also available in printed format.

Thesis (M. Phil.)--University of Hong Kong, 2006. / Title proper from title frame. Also available in printed format.

published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

No description available.

Flag varieties of reductive Lie groups and their subvarieties play a central role in representation theory. In the early 1980s, V. Deodhar introduced a decomposition of the flag variety which was then used to study the Kazdan-Lusztig polynomials. A Deodhar-type decomposition of the product of the flag variety with itself, referred to as the double flag variety, was introduced in 2007 by B. Webster and M. Yakimov, and each piece of the decomposition was shown to be coisotropic with respect to a naturally defined Poisson structure on the double flag variety. The work of Webster and Yakimov was partially motivated by the theory of cluster algebras in which Poisson structures play an important role. The Deodhar decomposition of the flag variety is better understood in terms of a cell decomposition of Bott-Samelson varieties, which are resolutions of Schubert varieties inside the flag variety. In the thesis, double Bott-Samelson varieties were introduced and cell decompositions of a Bott-Samelson variety were constructed using shuffles. When the sequences of simple reflections defining the double Bott-Samelson variety are reduced, the Deodhar-type decomposition on the double flag variety defined by Webster and Yakimov was recovered. A naturally defined Poisson structure on the double Bott-Samelson variety was also studied in the thesis, and each cell in the cell decomposition was shown to be coisotropic. For the cells that are Poisson, coordinates on the cells were also constructed and were shown to be log-canonical for the Poisson structure. / published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

Bott-Samelson varieties associated to reductive algebraic groups are much studied in representation theory and algebraic geometry. They not only provide resolutions of singularities for Schubert varieties but also have interesting geometric properties of their own. A distinguished feature of Bott-Samelson varieties is that they admit natural affine coordinate charts, which allow explicit computations of geometric quantities in coordinates. Poisson geometry dates back to 19th century mechanics, and the more recent theory of quantum groups provides a large class of Poisson structures associated to reductive algebraic groups. A holomorphic Poisson structure Π on Bott-Samelson varieties associated to complex semisimple Lie groups, referred to as the standard Poisson structure on Bott-Samelson varieties in this thesis, was introduced and studied by J. H. Lu. In particular, it was shown by Lu that the Poisson structure Π was algebraic and gave rise to an iterated Poisson polynomial algebra associated to each affine chart of the Bott-Samelson variety. The formula by Lu, however, was in terms of certain holomorphic vector fields on the Bott-Samelson variety, and it is much desirable to have explicit formulas for these vector fields in coordinates. In this thesis, the holomorphic vector fields in Lu’s formula for the Poisson structure Π were computed explicitly in coordinates in every affine chart of the Bott-Samelson variety, resulting in an explicit formula for the Poisson structure Π in coordinates. The formula revealed the explicit relations between the Poisson structure and the root system and the structure constants of the underlying Lie algebra in any basis. Using a Chevalley basis, it was shown that the Poisson structure restricted to every affine chart of the Bott-Samelson variety was defined over the integers. Consequently, one obtained a large class of iterated Poisson polynomial algebras over any field, and in particular, over fields of positive characteristic. Concrete examples were given at the end of the thesis. / published_or_final_version / Mathematics / Master / Master of Philosophy