The formulation of fundamental natural laws and of technological problems in the form of rigorous mathematical models is given frequently, even prevalently, in terms of differential equations. The needs of new technologies require non-linear differential equations. An appropriate method for tackling these equations is provided by Lie group analysis.

Lie group analysis suggests a rigorous mathematical formulation of intuitive ideas of symmetry and provides constructive methods for solving non-linear differential equations analytically. Acquaintance with group analysis is important for constructing and investigating non-linear mathematical models of natural and engineering problems. Numerous physical phenomena can be investigated using Lie symmetries to unearth various group invariant solutions and conservation laws that provide significant physical insight into the problem. For example, at this point I have a project in mind related to stability and energy exchange of resonantly interacting nonlinear waves in the ocean. This problem can be considered from symmetry analysis point of vew and can be proposed as a Ph.D project for prospective graduate students.

For non-linear problems, Lie group analysis plays the same role as Fourier analysis for linear problems. Therefore, group analysis should be as familiar to the student as Fourier analysis, especially when so many real-world problems are strongly non-linear and are not tractable by means of mathematical methods taught within traditional university curricula.

Fluid flow is governed by a complicated nonlinear system of partial differential equations. In many situations of interest the flow spans a huge range of length scales, with the nonlinearity of the governing equations resulting in the transfer of energy from one length scale to another. Because of this complexity, the field of fluid mechanics has been the birth place of many important fields in mathematics. It has stimulated much work in areas such as partial differential equations, asymptotics and perturbation theory, computational methods, nonlinear waves, including solitons, instability theory, chaos, and stochastic processes. Despite these developments, turbulence in fluids remains one of the major unsolved problems in classical physics.

Earth's atmosphere and oceans exhibit complex patterns of fluid motion over a vast range of space and time scales. Moving fluids are a part of the natural world, and their dynamics impact the ocean and atmosphere as well as many other problems of science and engineering. Fluid processes are complex (the equations of motion are three-dimensional, unsteady and nonlinear) and the theory is incomplete. Many natural processes occur over too large a scale in time and space (ten orders of magnitude in length-scale variations for the flows in the ocean) so that even the most advanced computer models cannot adequately describe real systems. Environmental and Geophysical Fluid Dynamics is primarily concerned with fluid flow in rivers, lakes, oceans and the atmosphere. Such flows dominate our physical existence. Geophysical Fluid Dynamics (GFD) is traditionally the study of naturally occuring large scale fluid motions in the oceans and in the atmosphere which are affected by the Earth's rotation, but also includes extraterrestrial atmospheres and the interior of the gas giants such as Jupiter. Examples of large scale motions are the Gulf Stream in the North Atlantic Ocean and atmospheric blocking events, one of which was responsible for the great heat wave in Europe in 2003. Over the past 30 years attention has been increasingly focussed on the impact of large scale fluid flows on the environment, highlighted by fundamental issues such as global warming and long-term climate change. Smaller scale motions, such as turbulence and mixing caused by internal waves, and surface water waves have a direct impact on bio-geochemical processes in oceans and lakes and hence on human activity. Due to their ubiquitous nature, these motions are also of fundamental importance to the large scale circulation in both the oceans and atmosphere.