Charlotte Scott Centre for Algebra
School of Engineering and Physical Sciences, University of Lincoln
Algebra Research Seminar: Talk by Dr Dillon Mayhew (Leeds)
Dr Dillon Mayhew (University of Leeds) visited the Charlotte Scott Research Centre for Algebra to discuss maths with Lincoln algebraist Dr Simon Smith.
On 5 March 2025 Dr Mayhew spoke in our Algebra Seminar about the current state of the art around definability of matroids. Title of the talk: Monadic definability of classes of matroidsAbstract:
Matroids are axiomatic abstractions of notions of independence that arise in linear algebra and geometry. There are connections between matroid theory and combinatorial optimisation as well as algebraic geometry.
Monadic second-order logic is a formal logical language that can naturally be applied to finite discrete structures such as graphs and matroids. If a class of graphs or matroids can be characterised by a sentence in the monadic second-order language, then that class becomes much more approachable from the perspective of complexity theory and computability theory. Therefore we are well motivated to explore the boundary between classes that are monadically definable and those that are not.
For every group H, there is a corresponding class of matroids, known as H-gain-graphic matroids. Work in progress by myself and Funk shows that when H is finite, the class of H-gain-graphic matroids is monadically definable. But when H is infinite, the situation is less clear. Some infinite groups lead to definable classes, and other groups give rise to non-definable classes. At the moment, it is not yet clear where the boundary between these behaviours will lie.
This seminar is based on discussions with Sapir Ben-Shahar, Matt Conder, Daryl Funk, Angus Matthews, Mike Newman, and Gabriel Verret. I will not assume any knowledge of matroids, and will include an introduction to the subject.
About Yuri Santos
Charlotte Scott Centre for Algebra 