Supervisor: Prof Alex Fink
Project Details :
Matroids are combinatorial structures which capture the notion of “dependence” common to situations like cycles in graphs and linear dependence in vector spaces. The last decade has seen a surge of work using tools from algebraic geometry to prove inequalities about matroids, in some cases solving long-standing conjectures. Some matroids come from 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.
In a collaboration in progress, Andy Berget and I have solved a conjecture about matroids from 2009. We construct a simplicial complex, or from another point of view a square-free monomial ideal, by intersecting two families of tropical varieties; for matroids from vector spaces it can also be constructed as an initial ideal, and gains an interpretation from homology of line bundles. We show that David Speyer's g-invariant is encoded in homological information from this complex; the conjecture was that a coefficient of the g-invariant was positive, and this coefficient can be written as a sum of dimensions of homology of links.
This procedure of building a complex from tropical intersections and extracting homological information should have applications in other settings in algebraic or tropical geometry. The PhD project is to work out the details of one or more of these.
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