Chemical Reaction Network Theory. My students and I are interested in complex chemical systems in which several reactions occur simultaneously. Real systems are almost always of this kind, so it becomes important to understand reactors with complicated chemistry in a systematic way.
Complex chemistry gives rise to intricate systems of nonlinear equations that don’t lend themselves to analytic solution. What’s more, increased complexity in the governing equations can give rise to complicated new phenomena that simple systems don’t admit. Even in the isothermal setting normally studied in biology there can, for example, be unstable steady states, multiple steady states, sustained composition oscillations, and wild, chaotic dynamics—possibilities we need to take into account.
Since each new network of chemical reactions gives rise to its own complicated system of differential equations, it becomes apparent that, in the absence of an overarching theory, we would be forced to study complex chemical systems on a case-by-case basis, and each new case would be fraught with terrible analytical difficulties. What’s needed is a way of looking at things from a broader and more general perspective.
That’s what Chemical Reaction Network Theory tries to do. The aim of the theory is to tie aspects of reaction network structure in a precise way to the variety of qualitative behaviors that might be engendered. A lot of progress has been made along these lines, but there is also much that remains unknown. For more on chemical reaction network theory, see the annotated bibliography.
Other Areas of Study. Although my attention now is largely focused on chemical reaction network theory, with particular reference to biology, I maintain an interest in two other areas with which I was intensely occupied in the past. One of these is mathematical foundations of classical thermodynamics. Another is a general theory of reactor-separator design, which has some ties, at least in spirit, with classical thermodynamics and with reaction network theory. In particular, I am interested in understanding theoretical limits to what can be achieved, consistent with certain design constraints, over all possible steady-state designs (even unimagined ones) that are consistent with those constraints. Articles about both topics can be found in the annotated bibliography.