Just at the close of the 20th century a dramatic announcement was made: the completion of a blueprint for the human genome. This milestone for mankind was the culmination of a nearly 150 year old odyssey begun with the research and dramatic discoveries of the monk Gregor Johann Mendel (1822-1884). Without question, developing this blueprint for the human genome was one of the landmarks of the history of biology.
The 20th century was a period of dramatic progress in all the sciences. However, perhaps no work was more dramatic in its implications than the birth of molecular biology. Molecular biology offered a vision of human beings understanding the complexities of life from the bottom up. Yet one of the characteristics of 20th century science was that no part of science was an island unto itself; all the sciences benefited from insights and accomplishments drawn from the others. During this exciting period mathematics proved to be a handmaiden of the progress that was being made across the board. However, the way that mathematical science has played an accelerating role in speeding up the developments in understanding the genome is part of an ongoing partnership between mathematics and biology in general. While once molecular biology was a new subject, it is now a mature subject which has bifurcated into many parts, including molecular genetics. Molecular genetics has fused with a long tradition of mathematical support for genetics in general.
The ways that mathematics has contributed to insight into the genome of humans and other species is a fascinating subject. Here I can only touch a few aspects of this interaction. I will begin with a brief history of mathematical involvement with genetics, then I will look at how the mathematical insight into the concept of distance has proved to be a boon for insight into complex biological (as well as other) problems. Sprinkled throughout I will point out some of the problems still open that are sources of further investigation. Before continuing you may wish to consult the primer of molecular biology and genetics which helps put in perspective the discussions that follow.