3:00pm - 5:00pmNew developments in matroid theory
Chair(s): Alex FInk (Queen Mary), Ivan Martino (Northeastern University, United States of America), Luca Moci (Bologna)
The interactions between Matroid Theory, Algebra, Geometry, and Topology have long been deep and fruitful. Pertinent examples of such interactions include breakthrough results such as the g-Theorem of Billera, Lee and Stanley (1979); the proof that complements of finite complex reflection arrangements are aspherical by Bessis (2014); and, very recently, the proof of Rota's log-concavity conjecture by Adiprasito, Huh, and Katz (2015).
The proposed mini-symposia will focus on the new exciting development in Matroid Theory such as the role played by Bergman fans in tropical geometry, several results on matroids over a commutative ring and over an hyperfield, and the new improvement in valuated matroids and about toric arrangements. We plan to bring together researchers with diverse expertise, mostly from Europe but also from US and Japan. We are going to include a number of postdocs and junior mathematicians.
(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)
Characterizing quotients of positroids
Anastasia Chavez
UC Berkeley
We characterize quotients of specific families of positroids. Positroids are a special class of representable matroids introduced by Postnikov in the study of the nonnegative part of the Grassmannian. Postnikov defined several combinatorial objects that index positroids. In this paper, we make use of two of these objects to combinatorially characterize when certain positroids are quotients. Furthermore, we conjecture a general rule for quotients among arbitrary positroids on the same ground set.
Algebraic matroids and flocks
Rudi Pendavingh
TU Eindhoven
Let $K$ be a field, $E$ a finite set, and $Xsubseteq K^E$ an algebraic variety. Then $M(X)$ is the matroid on ground set $E$ in which a set $Fsubseteq E$ is independent if and only if the projection ${(x_i: i in F): xin X}$ is a dominant subset of $K^F$. In general it is difficult to decide if a given matroid $M$ is algebraic over $K$, that is if $M=M(X)$ for some variety $Xsubseteq K^E$.
We have recently found that if the field $K$ has positive characteristic $p$, then the variety $X$ determines further structure on $M(X)$ which comprises information on the generic tangent spaces of $X$ as well as a family of closely related varieties. This additional structure can either be cast as a matroid over a hyperfield, or as a {em flock}, which is essentially a labelling of the cells of a tropical linear space by linear subspaces of $K^E$.
We show how this gives useful necessary conditions on the algebraicity of matroids.
This is joint work with Guus Bollen and Jan Draisma.
Tropical Ideals
Jeffrey Herschel Giansiracusa
Swansea
The scheme-theoretic approach to tropical geometry has motivated the study of tropical ideals, which are sequences of (valuated) matroids $M_i$ on the monomials of a polynomial ring that form an ideal in the sense that $x_j M_i subset M_{i+1}$. While the class of arbitrary ideals can behave very badly, tropical ideals exhibit many nice properties, while also presenting many new features, challenges, and mysteries. There are realizable tropical ideals, meaning that they are formed by tropicalizing classical ideals, and there are non-realizable tropical ideals. Three interesting questions are:
1. What invariants of a classical ideal are encoded in its associated tropical ideal?
2. How does the tropicalization of an ideal change as the ideal changes (moving within the Hilbert scheme)?
3. How can one construct non-realizable tropical ideals?
In this talk I will discuss examples, progress on each of these questions.