Online talk: Iain Moffatt

Mon, June 1 3pm EST (8pm BST, 7am Tue NZST)
Iain Moffatt, Royal Holloway University
The Tutte polynomial of a delta-matroid and the world of graph polynomials
Zoom [email rose.mccarty ~at~ uwaterloo ~.~ ca if you need the password]

The classical Tutte polynomial is a polynomial-valued invariant of graphs and matroids, and is arguably the most important and best studied graph polynomial. It’s important not only because it encodes a large amount of combinatorial information about a graph, but also because of its applications to areas such as statistical physics (as the Ising and Potts models) and knot theory (as the Jones and homfly polynomials). Unsurprisingly, given its role in combinatorics, the Tutte polynomial has been extended to many different settings for many different purposes. 

In this talk I’ll discuss an extension of the Tutte polynomial to delta-matroids. (Delta-matroids are a generalisation of matroids introduced by Bouchet and others in the 1980s.) I’ll describe some of the advantages of the delta-matroid setting over the classical matroid setting, as well as some of its limitations; some of the properties of the classical polynomial that extend to the new setting, and some that don’t seem to. Through this, I hope to convey where the delta-matroid polynomial fits in the world of graph polynomials, and why I think it is a particularly interesting generalisation of the Tutte polynomial.  

I’ll not assume any familiarity with the Tutte polynomial or with delta-matroids. 

Online talk: Rudi Pendavingh (plus an announcement)

I’m happy to announce another seminar series many of our readers may be interested in, titled “Algebraic Matroids and Rigidity Theory”. It is at 10am EST on Thursdays and is organized by Daniel Bernstein. Please email him at [dibernst ~at ~ mit ~.~ edu] or [bernstein.daniel ~at~ gmail ~.~ com] to get on the mailing list and for the password.

We do not intend on having a talk on May 18 for Victoria Day :). Here’s the info for Rudi’s talk next week.

Mon, May 11 3pm EST (8pm BST, 7am Tue NZST)
Rudi Pendavingh, Eindhoven University of Technology
Counting valuated matroid types
Corrected slides

If $M$ is a matroid with bases $\mathcal{B}$, then a valuation of $M$ is a function $\nu:\mathcal{B}\rightarrow \mathbb{R}$ satisfying the following symmetric exchange axiom:

  • If $B, B’\in \mathcal{B}$ and $e\in B\setminus B’$, then there is an $f\in B’\setminus B$ so that $$\nu(B)+\nu(B’)\leq \nu(B-e+f)+\nu(B’+e-f)$$
The combinatorial type of a given valuation essentially comprises the information for which $B,B’,e,f$ equality is attained in this definition. 
A matroid is rigid if all its valuations are of the same combinatorial type. By a theorem of Lafforge, a rigid matroid has a discrete set of linear representations over each field. By work of Bollen, Draisma, and myself, a rigid matroid which is algebraic in characteristic $p$ is also linear in characteristic $p$. More generally, if a matroid is algebraic in characteristic $p$, then the matroid has some valuation which satisfies a certain condition on its combinatorial type. Testing this condition involved enumerating the combinatorial types.
In this talk, we present bounds on the number of combinatorial types of valuations. The method of proof suggests ways to enumerate the combinatorial types of valuations of a given matroid more efficiently.
This is joint work with Simon Soto Ochoa.