Gaussian graphical models are semi-algebraic subsets of the cone of positive definite covariance matrices. Submatrices with low rank correspond to generalizations of conditional independenceâ€¦ (More)

Postnikov constructed a decomposition of a totally nonnegative Grassmannian (Grkn)â‰¥0 into positroid cells. We provide combinatorial formulas that allow one to decide which cell a given point inâ€¦ (More)

For a planar directed graph G, Postnikovâ€™s boundary measurement map sends positive weight functions on the edges of G onto the appropriate totally nonnegative Grassmann cell. We establish an explicitâ€¦ (More)

Postnikov constructed a decomposition of a totally nonnega tive Grassmannian(Grkn)â‰¥0 into positroid cells. We provide combinatorial formulas that allow one to decide whi ch cell a given pointâ€¦ (More)

For a planar directed graph G, Postnikovâ€™s boundary measurement map sends positive weight functions on the edges of G onto the appropriate totally nonnegative Grassmann cell. Assuming, without muchâ€¦ (More)

Notes taken from Kelli Talaskaâ€™s H113 (Honors Introduction to Abstract Algebra) at UC Berkeley. These notes were taken live during the lecture with TEX; particular thanks to the authors of latexmk.â€¦ (More)

This dissertation is dedicated to the memory of my mother, Kathleen Marie Harter. ii ACKNOWLEDGEMENTS I have come this far largely because I have had a great deal of support, encouragement , andâ€¦ (More)

Deodhar introduced his decomposition of partial flag varieties as a tool for understanding Kazhdan-Lusztig polynomials. The Deodhar decomposition of the Grassmannian is also useful in the context ofâ€¦ (More)

Let G be a perfectly oriented planar graph. Postnikovâ€™s boundary measurement construction provides a rational map from the set of positive weight functions on the edges ofG onto the appropriateâ€¦ (More)