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Getting even closer!

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We found all the pieces and everything is assembled! Now it is time to remove the paper.

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New Paper on Graph Gonality

The UK Math Lab has submitted a new paper to the arXiv! The paper, entitled "On the Semigroup of Graph Gonality Sequences", can be found here. https://arxiv.org/abs/2306.11118 This paper concerns a certain game played on graphs. (We'll call our players Brill and Noether.) Brill is given d poker chips to place on the vertices of a graph. In the example below, d is equal to 2, and Brill may choose to place the chips like this: Noether then "steals" r chips, introducing a debt of r chips. In the example, r is equal to 1, and Noether might choose to steal a chip like this: Brill's goal is then to eliminate the debt via a sequence of chip-firing moves. The way this works is that Brill selects a vertex to "fire", which results in that vertex giving 1 chip to each of its neighbors. If Brill fires the top left vertex, the result is this: This eliminates the debt at the center vertex, but introduces a new debt at the top left. ...

The special linear group quilt!

The new quilt is done! This quilt is the multiplication table for \(Sl(2,F_3)\), the group of 2 by 2 matrices whose entries are in the field with 3 elements and whose determinant is 1. It looks both very similar to the \(S_4\) quilt below and very different! These are both multiplication tables for groups of order 24 and the colors are arranged in the same orders on the top and left sides. (Except we didn't use exactly the same colors!) Then things start to get very different. If you stand far away from the new quilt you can image that the 8 by 8 blocks of yellow/orange, red/purple, and blue/green are each a single block. So instead of thinking of the 64 squares below as distinct think of them as one big yellow/orange block. This shows the order 8 subgroup of \(Sl(2,F_3)\) is normal and grouping these squares together is the quotient group. From this perspective, the quilt is a multiplication table for \(\mathbb{Z}/3\)....

Matroids

Matroids are a broad class of objects that underlie everything from concrete problems such as job assignments to finite affine geometry such as the Fano plane. One place matroids appear is in algebraic geometry. Intersection theory is a branch of algebraic geometry which studies how curves (and more) intersect inside a space. One classical result from this field is Bezout's theorem . Intersection theory is one of the most studied branches of algebraic geometry today with much still unknown about the general case. Tools and examples are needed to test conjectures and matroids give rise to a rich class of examples with concrete descriptions. Associated with each matroid is an algebraic variety such that the combinatorics of the matroid perfectly encode the intersection theory on that variety. By understanding the chains of flats of a matroid, we can prove statements about the intersection theory of the corresponding variety. Current work by the Matlab group at the ...

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