The Schedule

Mirror symmetry is a fast-moving research area at the boundary between mathematics and theoretical physics. Originated from observations in string theory, it suggests that certain geometric objects (complex Calabi-Yau manifolds) should come in pairs, in the sense that each of them has a mirror partner and these two share interesting geometric properties. In this talk, I will introduce some geometrical ideas inspired by mirror symmetry. In particular, I will go through the main steps which relate mirror symmetry to tropical and non-archimedean geometry.

The projectivisation of a very ample toric vector bundle admits a natural embedding into projective space. We describe defining equations for this embedding in a larger projective space via a more natural embedding of the vector bundle in the Cox ring of a toric variety. This is joint work with Diane Maclagan and Greg Smith.

Basically, what we describe is not “Matroid Chern Roots” but “Chern Classes of Matroid Tautological Classes”. The S and the Q defined in the BEST paper are called “Matroid Tautological Classes”, and we describe their Chern classes c_i(S) and c_i(Q). We also use our descriptions to get a nice symmetric formula for “Matroid CSM Cycles”. The Chern roots (of the Tautological Classes) are what they describe in the last part of the BEST paper, and we just use them to prove our formula for the Chern Classes of S and Q.

In this talk we define toric arrangements and its complement, an open set in an algebraic torus. We introduce the wonderful model for toric arrangements, ie a smooth projective variety that contains the complement as complement of a simple normal crossing divisor. We also discuss the cohomology ring of wonderful models. This work is due to De Concini and Gaiffi obtained in a series of recent articles.
Finally, I present the result of a work in progress with Viola Siconolfi and Lorenzo Giordani: we extend the presentation of the cohomology ring to the case of not well-connected building set.

Several important long-standing conjectures about some combinatorial objects has been solved thanks to the development of the combinatorial
Hodge theory. For instance the Heron-Rota-Welsh conjecture, which
concerns some combinatorial generalization of complement of hyperplane
arrangement called matroids, has been solved by Adiprasito, Huh and Katz in this way. To some nice matroids, one can associate complex varieties, namely some wonderful compactifications. We can then solve the conjecture in this case thanks to classical Hodge theory, which studies
the cohomology of complex varieties. Unfortunately for general matroids,
one cannot associate such a complex variety. We might then develop a
combinatorial Hodge theory. We do as if one can associate a complex
variety to the matroid, though this is not the case. This is what has
been done by Adiprasito Huh, and Katz. Their proof is very clever but it
does not give much insight into why this combinatorial Hodge theory
works in general.
Actually, to any matroid one can associate a tropical variety.
Moreover, in a joint work with Omid Amini, we developped a tropical
Hodge theory. Along other applications, this can be used to give a
geometric proof of the Heron-Rota-Welsh conjecture, as well as an
extension of the applicability of the combinatorial Hodge theory. The
heart of our proof relies on an interesting induction coming from the
tropical world, which seems to apply in many contexts.

In this talk I will present my joint work with Felipe Rinc ́on on a new cryptomorphism for valuated matroids that generalizes the usual closure operation for matroids. This definition has roots based in tropical scheme theory and relates to earlier works in l∞ nearest approximations and phylogenetics, as well as recent works in tropical principal component analysis.

I will explain a way to study orbifolds with combinatorial objects called the “relative tropicalisation”, which is a union of integral affine tori. I will indicate that there is an Artin fan picture for these objects too, where these combinatorial objects correspond to certain geometric stacks.

The width of a lattice polygon measures the smallest strip of the plane it is a subset of. A recurring situation in lattice polytope classifications is for all but finitely many of a type of polytope to have very small width. I will introduce the multi-width of a polytope as a possible way to classify infinitely many polytopes of small width and apply this to classifying small-dimensional lattice simplices. The resulting classifications display surprisingly regular structure with direct links to the Hilbert series of certain varieties.

We will introduce hyperfields as structures with multi-valued addition and understand how orderings over this domain behave. We define the notion of convexity and present properties of convex sets over hyperfields. We will then explore generalisations of several classical results in convex geometry, including Carathédory's theorem. To finish we will then discuss how separation in this setting differs from the classical setting, in particular requiring the use hemispaces rather than half spaces. This is joint work with Ben Smith (Manchester).

TBC