University of Kentucky Math Lab

Search This Blog

Progress!

August 17, 2026

Get link
Facebook
X
Pinterest
Email
Other Apps

All that is left is the binding and then this quilt is done!

2d Visualization

Get link
Facebook
X
Pinterest
Email
Other Apps

Comments

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 \(\mathbb{Z}/3\)....

Transfer Systems - David Mehrle

Topology is the branch of mathematics that deals with questions of shape and space without measuring. It studies properties of shapes that are preserved under continuous deformations, such as stretching, twisting, or bending. Modern topology takes this to the extreme and studies shapes that are often infinite-dimensional and very hard to visualize! Nevertheless, topologists studying the sub-field called equivariant stable homotopy theory have distilled some questions into the study of transfer systems. Transfer systems are concrete combinatorial objects that are easy to draw! Things that we learn about transfer systems often have important consequences for those infinite-dimensional shapes in equivariant stable homotopy theory. Our math lab group has been studying duality for a certain class of transfer systems called flat transfer systems. Just knowing how many transfer systems there are can be a tough question. But if we can understand how transfer systems pair up with thi...

Powered by Blogger

Progress!

Comments

Post a Comment