Evolutionary theory is attractive for the same reasons that theoretical physics is: there is the hope of solving some of the mysteries of existence. Evolution takes on additional meaning because the mysteries are not just about “stuff”, but about life itself. Evolution is a fertile field for emergent phenomena, both within the products of evolution – the organism – and in the dynamics of evolutionary change itself. Mathematics is the discipline that enables one to explain these emergent phenomena. By “explain” I mean to show relationships between things that no one suspected or understood.

Adaptation and diversification are spectacular emergent phenomena in and of themselves. But evolutionary dynamics are rich enough to host other, higher order phenomena. I have focused on identifying and explaining phenomena in evolution beyond adaptation and diversification. These include the evolution of inheritance mechanisms (i.e. the maintenance and expansion of information in the organism and species), the evolution of evolvability, and the evolution of modularity in the genotype-phenotype map.

These phenomena appear in evolutionary computation as well. Evolutionary algorithms are computer programs that solve problems by evolving the answers in analogy to Darwinian evolution with artificial or natural selection. I have introduced the evolution of evolvability and Price’s equation into the evolutionary computation literature. The theory of evolutionary computation has many attractive open questions.

When the analysis of these questions becomes mathematically difficult, one has two options. The first option is computer simulation – numerical results can provide illustrations of phenomena. But as Murray Gell-Mann astutely said, “What is the point of studying a complex system that we don’t understand by making a complex computer model that we don’t understand?” The second option is to hunker down and find the mathematical results one needs or produce them oneself. I have pursued all of these options.

For some of these problems, the mathematic analysis leads one to matrix theory. The student of complex systems hears at the beginning that nonlinear dynamics are the fountain for complexity. A closeup lens on nonlinear dynamics often reveals eigenvalues and eigenvectors as the key objects in analyzing the larger system. Eigenvalues and eigenvectors are themselves emergent properties of linear operators. Much of my research has therefore focused on understanding that “first emergent property”, the spectral properties of matrices and linear operators. Some biological questions have results from the linear algebra literature there waiting for them; others require new mathematics to answer. I pursue such results as the logic of the phenomenon in question requires.

The process of organic evolution has produced spectacular results and left many mysteries. Nothing is more pleasing that to shine a little light on some of these mysteries.

Learn more at http://dynamics.org/Altenberg/

Contact Lee at altenber@hawaii.edu