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School of Mathematical Sciences

Matroid Chern classes

Supervisor: Dr Alex Fink

Project description:

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 five to ten years have seen a surge of work proving inequalities for matroid invariants by interpreting these invariants using algebraic geometry, some of which have stood as conjectures since the 1970s.

A 2020 paper of Berget, Eur, Spink and Tseng provides formulas for a number of matroid invariants, like the Tutte polynomial and CSM classes, in terms of combinatorial versions of the tautological vector bundles on the Grassmannian. Small cases suggest that these bundles satisfy many more inequalities than are proved for them there. Namely, the theorem of Fulton and Lazarsfeld appears to have an analogue: perhaps Schur functions of the Chern classes are nonnegative for every matroid. If true in full generality this would be a more comprehensive treatment of these matroid positivity results than has come before.

The objective of this project is to produce combinatorial, ideally enumerative, and/or geometric interpretations of some of these Schur functions with an eye to proving at least special cases of the Fulton–Lazarsfeld theorem. Among the further tools which could be brought to bear are tropical geometry, commutative algebra, and representation theory of the general linear or symmetric groups.

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