A mathematical model for the growth of two coupled mathematical specialties, differential geometry and topology, is analyzed. The key variable is the number of theorems in use in each specialty. Obsolescences of theorems-in-use due to replacement by more general theorems introduces non-linear terms of the differential equations. The stability of stationary solutions is investigated. The phase portrait shows that the number of theorems in low-dimensional topology relative to those in differential geometry is increasing. The model is qualitatively consistent with the growth of publications in these two specialties, but does not give quantitative predictions, partly because we do not use an explicit solutions as a function of time and partly because only two specialties are used. The methods of analysis and some of the concepts can be extended to the development of more general and realistic models for the growth of specialties.
Peer review plays an important role in maintaining the quality of science. Selection of peers is at the heart of the process by which science advances. Editors and others responsible for selecting a group of peers often rely on their position in a network by which experts in a field are linked to one another by bonds of common interest and recognized expertise. In this paper, we report one aspect of a study aimed at characterizing the structure of this network: the asymmetry of the fraction of experts receiving varying numbers of nominations as experts by peers. The distribution of such nominations is very skew, and we have found that a law of cumulative advantage provides the best theoretical approximation for the distribution of nominations, expecially when the overall pool of data is broken down into well-defined specialties.