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
YouTube
Corrected slides
Abstract:
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.
