Ralph Morrison
Williams College
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Publication
Featured researches published by Ralph Morrison.
Research in the Mathematical Sciences | 2015
Sarah B. Brodsky; Michael Joswig; Ralph Morrison; Bernd Sturmfels
We study the moduli space of metric graphs that arise from tropical plane curves. There are far fewer such graphs than tropicalizations of classical plane curves. For fixed genus g, our moduli space is a stacky fan whose cones are indexed by regular unimodular triangulations of Newton polygons with g interior lattice points. It has dimension 2g+1 unless g≤3 or g=7. We compute these spaces explicitly for g≤5.
Mathematische Zeitschrift | 2016
Matthew Baker; Yoav Len; Ralph Morrison; Nathan Pflueger; Qingchun Ren
We study smooth tropical plane quartic curves and show that they satisfy certain properties analogous to (but also different from) smooth plane quartics in algebraic geometry. For example, we show that every such curve admits either infinitely many or exactly 7 bitangent lines. We also prove that a smooth tropical plane quartic curve cannot be hyperelliptic.
Journal of Symbolic Computation | 2015
Ralph Morrison; Qingchun Ren
Mumford showed that Schottky subgroups of PGL ( 2 , K ) give rise to certain curves, now called Mumford curves, over a non-archimedean field K. Such curves are foundational to subjects dealing with non-archimedean varieties, including Berkovich theory and tropical geometry. We develop and implement numerical algorithms for Mumford curves over the field of p-adic numbers. A crucial and difficult step is finding a good set of generators for a Schottky group, a problem solved in this paper. This result allows us to design and implement algorithms for tasks such as: approximating the period matrices of the Jacobians of Mumford curves; computing the Berkovich skeleta of their analytifications; and approximating points in canonical embeddings. We also discuss specific methods and future work for hyperelliptic Mumford curves.
arXiv: Geometric Topology | 2010
Colin Adams; Rachel Hudson; Ralph Morrison; William George; Laura Starkston; Samuel J. Taylor; Olga Turanova
In this paper, we introduce two new invariants that are closely related to Milnor’s curvature-torsion invariant. The first, a particularly natural invariant called the spiral index of a knot, captures the number of local maxima in a knot projection that is free of inflection points. This invariant is sandwiched between the bridge and braid index of a knot, and captures more subtle properties. The second invariant, the projective superbridge index, provides a method of counting the greatest number of local maxima that occur in a given projection. In addition to investigating the relationships among these invariants, we use them to classify all those knots for which Milnor’s curvature-torsion invariant is 6π.
Journal of Number Theory | 2011
Duc Khiem Huynh; Steven J. Miller; Ralph Morrison
Collectanea Mathematica | 2015
Ralph Morrison
Linear Algebra and its Applications | 2016
Ralph Morrison; Ngoc Mai Tran
arXiv: Number Theory | 2010
Steven J. Miller; Ralph Morrison
arXiv: Combinatorics | 2018
Ivan Aidun; Frances Dean; Ralph Morrison; Teresa Yu; Julie Yuan
arXiv: Combinatorics | 2018
Ivan Aidun; Frances Dean; Ralph Morrison; Teresa Yu; Julie Yuan