CeCANT- Prof Alex Fink
Cellular-like resolutions from Chern classes
Supervisor: Prof Alex Fink
Project Details :
This project builds on my work with Andrew Berget (in "The external activity complex of a pair of matroids"). That work resolved a conjecture from 2005 by Speyer, on the positivity of a certain statistic of matroids.
Matroids are structures which capture the combinatorics of e.g. hyperplane arrangements. The last decade has seen a surge of work using tools from algebraic geometry in matroid theory. Some matroids come from a hyperplane arrangement in a vector space from which we can build other algebraic varieties, but crucially not all matroids do, so it's surprising that algebraic geometry may be used at all.
Our proof proceeds by constructing a square-free monomial ideal whose free resolution encodes the statistic in question. When we have a hyperplane arrangement, this monomial ideal is an initial degeneration of an ideal obtained by "collapsing", in the sense of Kempf, pullbacks of tautological bundles from a Grassmannian. When there is no hyperplane arrangement, matroids let us make a parallel construction.
To extract the homological information, we intersect Chern classes of the bundles being collapsed using the fan displacement rule of Fulton and Sturmfels. Altogether these computations look as though they should assemble into a free resolution, given in the style of a cellular resolution from the fan displacement data. The first stage of the project is to confirm that this resolution in fact exists; the second stage is to find the most general hypotheses we can under which the construction works and attempt to find other applications in algebraic geometry, commutative algebra and/or combinatorics.
Funding Notes:
This project is open to candidates applying for CSC/EPSRC/Underrepresented Studentships and self-funded candidates.
Further information:
How to apply
Entry requirements
Fees and funding
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