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Modern society has been shaped by a long tradition of mutual inspiration and enrichment between Mathematics and the Sciences. In this tradition, the Department of Applied Mathematics offers its graduate students opportunities for study in the areas of Control Theory and Dynamical Systems, Fluid Mechanics, Mathematical Medicine and Biology, Mathematical Physics, and Scientific Computation. Our students' research projects involve cutting-edge applications of mathematical theory in a broad range of fundamental and applied sciences. These applications include, for instance, cancer therapy optimization, control of shape memory alloys, fractal image processing, quantum computing, and the study of climate variability, inflationary cosmology, and nanotechnology. The Department of Applied Mathematics is one of five units that comprise the Faculty of Mathematics at the University of Waterloo, which was ranked 20th worldwide in the 2015 QS University Rankings for mathematics. Graduate students in the department benefit from our close links with the Faculties of Science and Engineering, the Centre for Mathematical Medicine, the Centre for Theoretical Neuroscience, the Institute for Quantum Computing, the Perimeter Institute for Theoretical Physics, the Waterloo Institute for Nanotechnology, the Water Institute, and the Centre for Computational Mathematics in Industry and Commerce. We offer both Master's and PhD programs. Our thesis-based Master's of Mathematics (MMath) program normally takes two years to complete. Many of the graduates of this program subsequently pursue PhD degrees, others are successful in obtaining rewarding positions in industry or government. Our PhD program generally takes four years to complete. Most of our PhD graduates find employment in university research. Others take on positions in research and development in industry, government, or commerce.<br><br>This field is centred on the subject of differential equations, which provide the basis for mathematical models in many fields including the physical sciences, engineering, the life sciences and finance. This subject is broad, ranging from the traditional ordinary and partial differential equations, to the more modern delay and stochastic differential equations. The two aspects of the subject that we emphasize are control theory and dynamical systems.<br>Firstly, control theory refers to the process of influencing the behaviour of a physical or biological system to achieve a desired goal, primarily through the use of feedback. The governing equations of the system in question are differential equations of various types. Secondly, the theory of dynamical systems deals with the qualitative analysis of solutions of differential equations on the one hand and difference equations on the other hand. The latter comprises the subfield of discrete dynamical systems, which has applications in diverse areas, for example biology and signal processing. A recent development is the notion of hybrid dynamical system, which allows the interaction of discrete events and continuous dynamics, thereby providing a natural framework for mathematical modeling of complex reactive systems or intelligent systems, in which physical processes interact with man-made automated environments.