APPLIED MATHEMATICS is a subject area of immense breadth. As any scientific field develops, it becomes more quantitative and requires ever more sophisticated mathematics to formulate and solve its basic problems. Nowhere is this more apparent than in the physical sciences. For example, modern statistical mechanics is full of important mathematical problems related to phase transitions and stochastic dynamical systems. Differential geometry has become the language of modern elementary particle physics. Quantum field theory in physics has led to open problems in knot theory and the mathematical field of topology. Important problems in the chemical and engineering sciences lead to difficult nonlinear differential equations that cannot be solved analytically. Often these equations model fluids. This leads to important mathematical problems related to the structure of solutions, and to numerical and theoretical issues related to the problem of computing solutions of these equations. Headway can only be made when the powerful methods of mathematical analysis are brought to bear, and numerical methods and asymptotic approximations are important. All of these mathematical issues are part of current research projects being carried out in the GGAM.
In the biological sciences, members of the GGAM are working on developing mathematical models for describing biological systems at the suborganismic, organismic and population levels. Differential equations, optimization techniques, the mathematical theory of stochastic differential equations and the theory of chaos in dynamical systems are playing fundamental roles in this area. Specific research projects at the organismic and suborganismic level include the dynamical modeling of periodic and chaotic behavior in cell biology, the control and regulation of ionic channels in excitable cells, and the modeling of intermediary metabolism in living cells. Applications of hydrodynamic theories to problems such as flow of water through fish mouths, the effect of osmotic flows on plant morphology, and medical imaging through magnetic resonance are also active areas of research. At the population level the fields of resource management, genetics, and population ecology are all represented in the GGAM. Questions here include determining the forces responsible for maintaining variability in populations, studying the dynamics of structured populations, managing renewable resources, and understanding animal behavior.
The GGAM is designed to facilitate the study of mathematical problems that are important to science. Indeed, mathematics and science go hand in hand, and there are many ways by which the scientist is led naturally to the door of the applied mathematician.