The Galois theory of maps on surfaces and delta-matrodis

Goran Malic

Frank Adams 1,

This being my final pure postgrad talk, I thought it'd be appropriate that for once I talk about what I actually do. A map M on a surface X is a bipartite graph embedded into X in a certain way. This embedding gives X the structure of a Riemann surface and an algebraic curve defined over the algebraic numbers. Therefore, the absolute Galois group (the group of all field automorphisms of the algebraic numbers fixing the rationals point-wise) acts on X and the result of this action is a Riemann surface X' with a map M' on it.
 
At the same time, the maximal subsets of edges of M whose removal does not disconnect X correspond to the bases of a certain Lagrangian subspace of the orthogonal vector space $\mathbb Q^n\oplus\mathbb Q^n$, where $n$ is the number of edges of M. The combinatorics of the bases are encoded in what is called a delta-matroid.
 
The aim of my thesis was to look at what happens to the delta-matroid after the Galois group acts on the surface which induces it. I won't spoil that for you, so you'll just have come to the talk to see what I've found.
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