| Algebraic statistics is the study of the algebraic varieties that correspond to discrete statistical models. Such statistical models are used throughout computational biology, for example to describe the evolution of DNA sequences. This perspective on statistics allows us to bring mathematical techniques to bear and also provides a source of new problems in mathematics.;The central focus of this thesis is the use of the language of algebraic statistics to translate between biological and statistical problems and algebraic and combinatorial mathematics. The wide range of biological and statistical problems addressed in this work come from phylogenetics, comparative genomics, virology, and the analysis of ranked data. While these problems are varied, the mathematical techniques used in this work share common roots in the field of combinatorial commutative algebra. The main mathematical theme is the use of ideals which correspond to combinatorial objects such as magic squares, trees, or posets. Biological problems suggest new families of ideals, and the study of these ideals can in some cases be useful for biology. |
