The Lie algebra isomorphism between $\mathrm{su}\text{(1,1)×}\mathrm{su}\text{(1,1)}$ and o(2,2) is used to obtain a list of subalgebras of the latter. The resulting list of 32 subalgebras is then examined on a case by case basis to see if each can be the Lie algebra of the holonomy group of a neutral metric in four dimensions. The conclusions, taken in conjunction with previously known results, furnish a classification of such Lie subalgebras of o(2,2), with only one case remaining unresolved.

Fractional Stochastic Differential Equations are becoming more popular in the literature as they can model phenomena in financial data that typical Stochastic Differential Equations models cannot. In the formulation considered here, the Hurst parameter, H, controls the Fraction of Differentiation, which needs to be estimated from the data. Fortunately, the covariance structure among observations in time is easily expressed in terms of the Hurst parameter which means that a likelihood is easily defined. This work derives the Maximum Likelihood Estimator for H, which shows that it is biased and is not a consistent estimator. Simulation data used to understand the bias of the estimator is used to create an empirical bias correction function and a bias-corrected estimator is proposed and studied. Via simulation, the bias-corrected estimator is shown to be minimally biased and its simulation-based standard error is created, which is then used to create a 95% confidence interval for H. A simulation study shows that the 95% confidence intervals have decent coverage probabilities for large n. This method is then applied to the S&P500 and VIX data before and after the 2008 financial crisis.

In this investigation, we present symmetry algebras of the canonical geodesic equations of the indecomposable solvable Lie groups of dimension five, confined to algebras $A_{5,7}^{abc}$ to $A_{18}^a$ . For each algebra, the related system of geodesics is provided. Moreover, a basis for the associated Lie algebra of the symmetry vector fields, as well as the corresponding nonzero brackets, are constructed and categorized.

A problem that is frequently encountered in a variety of mathematical contexts is to find the common invariant subspaces of a single or of a set of matrices. A new method is proposed that gives a definitive answer to this problem. The key idea consists of finding common eigenvectors for exterior powers of the matrices concerned. A convenient formulation of the Plücker relations is then used to ensure that these eigenvectors actually correspond to subspaces or provide the initial constraints for eigenvectors involving parameters. A procedure for computing the divisors of a totally decomposable vector is also provided. Several examples are given for which the calculations are too tedious to do by hand and are performed by coding the conditions found into Maple. Our main motivation lies in Lie symmetry, where the invariant subspaces of the adjoint representations for the Lie symmetry algebra of a differential equation must be known explicitly and comprehensively in order to determine all the ideals of the Lie symmetry algebra.

All the simple and then semisimple subalgebras of gl(4,R) are found. Each such semisimple subalgebra acts by commutator on gl(4,R). In each case the invariant subspaces are found and the results are used to determine all possible subalgebras of gl(4,R) that are not solvable.

We obtain minimal dimension matrix representations for each of the Lie algebras of dimensions five, six, seven and eight obtained by Turkowski that have a non-trivial Levi decomposition. The key technique involves using the invariant subspaces associated to a particular representation of a semi-simple Lie algebra to help in the construction of the radical in the putative Levi decomposition.