This project is concerned with the study analytical and dynamical aspects of generalized wave maps and the Yang-Mills equations. From the analytical standpoint we plan to address fundamental questions such as existence, uniqueness, and formation of singularities for these models. Specifically we will consider wave maps with torsion and will study how the addition of the torsion tensor affects propagation of regularity and development of singularities for these equations. From the dynamical standpoint we will start by considering the integrable versions of wave maps with torsion, as well as other integrable equations, and will study how homoclinic solutions behave under perturbation. Specifically we will study how the persistence of homoclinic orbits leads to chaotic systems. We will also attempt to quantify the difference between temporal chaos and spatial chaos, which is observed numerically in perturbations of the above models. We will test for spatial chaos by actually computing the probability distributions at two spatially distinct points and show that the mutual information function decays to zero at an exponential rate. Many physical phenomena are described by equations which have a geometric interpretation. The best known examples are Maxwell's equations and Einstein's equations for gravity. These physical models pose challenging mathematical questions whose answer will add significantly to our understanding to the nature of things. For instance, blackholes are interpreted as a certain type of singularity that occur in the equations of general relativity. However in addition to blackholes, these equations also exhibit another type of singularity called naked singularity, which are purely mathematical and do not exist in nature. In this project we will consider models and study the existence and nature of singularities, and how the geometry of the problems influence these issues. This study will help in the understanding of how , and of what type, singularities may develop in the given systems. We will also study the onset of chaos in deterministic partial differential equations, which can exhibit spatial chaos as well as temporal chaos. In such systems the precise physical information is usually lost and replaced by statistical quantities. In addition to this, the equations themselves become less useful and should be replaced by effective probablistic equations to capture the stochastic nature of deterministic spatial-temporal chaos. In this project we will develop tests for the occurrence of spatial chaos based on mutual information, which measures the amount of information transmitted spatially by the equation. We will also attempt to develop a mathematically rigorous procedure to derive from the original systems the relevant effective probabilistic models.
