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  6. CSM classes of tropical varieties

CSM classes of tropical varieties

Supervisor: Dr Felipe Rincon

Project description:

Tropical algebraic geometry is a relatively new field of mathematics concerned with understanding combinatorially defined polyhedral objects that encode interesting behaviour of algebraic varieties. A recent development in the field is the introduction of tropical Chern-Schwartz-MacPherson (CSM) classes for tropical varieties, by Rincón and collaborators.

CSM classes of tropical varieties have very rich combinatorial and geometric properties. For instance, they were used in a central way by Ardila, Dehnam, and Huh in their renowned proof that the h-polynomial of the broken circuit complex of any matroid has log-concave coefficients. In fact, partly because of this work, June Huh was awarded the very prestigious Fields Medal in 2022. 

Another example of the nice properties of CSM classes is that Speyer’s g-polynomial, a mysterious matroid invariant, can be expressed as certain intersection numbers of CSM cycles. This opens the door to the use of CSM classes as a tool for understanding the g-polynomial better, and as a result attempt to prove the 20-year-old f-vector conjecture.

The proposal of this PhD project is to push the study of CSM classes further. As part of his current EPSRC New Investigator Award, the applicant and his PDRA have discovered a new expression for CSM classes in the Chow ring of a matroid, which can be used to gain a better understanding of them.

Concrete problems to tackle in this PhD studentship are:

  • Investigate tropical instances of the Hirzebruch-Riemann-Roch theorem, such as Noether’s formula for tropical surfaces and generalisations to higher dimensional varieties.
  • Gain a better understanding of Chern numbers of matroids, which encode the possible intersection numbers that arise when intersecting various CSM classes.
  • Investigate connections to positivity questions, such as Speyer’s f-vector conjecture.

These problems have the nice advantage of providing the PhD student with a very solid and broad background in combinatorics and algebraic geometry. In addition, they can be tackled from different points of view, including computational approaches. 

Further information:

How to apply

Entry requirements

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