Algebraic and Combinatorial Applications in Systems and Evolutionary Biology

Date
2024
Authors
Curiel, Maize
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Gross, Elizabeth
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Mathematics
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Many real-world problems can be expressed as the solution set of polynomial equations. Under these constraints, these problems are addressed using tools from (computational) algebraic geometry, commutative algebra, and combinatorics. Inspired by questions in systems and evolutionary biology, three algebraic problems are addressed in this thesis: steady-state analysis of mass-action ordinary differential equations given by directed graphs, optimal phylogenetic consensus trees, and singularities arising from linear structural equation models given by acyclic mixed graphs. Specifically, for the first algebraic problem, we introduce a polyhedral geometry tool called a mixed volume to study the steady-state degree of partitionable binomial chemical reaction networks. For the second algebraic problem, a phylogenetic consensus tree is the optimal point of a minimization problem, or a point belonging to a tropical variety, and tools from tropical combinatorics are used to locate these optimal points. For the final algebraic problem, the main object of study is the covariance parameterization map arising from linear structural equation models given by an acyclic mixed graph. The concern is with understanding the parameters for which the covariance map fails to be locally injective.
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Mathematics
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108 pages
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