# Five not-so-easy pieces:open problems about vertex rings

## TL;DR

This paper presents five open problems in the theory of vertex rings, exploring their connections to various mathematical areas and highlighting potential avenues for future research.

## Contribution

It introduces five new open problems in vertex ring theory, providing background, motivation, and partial results to stimulate further investigation.

## Key findings

- Identification of five key open problems in vertex rings
- Connections established between vertex rings and Lie theory, modular forms, and Moonshine
- Partial solutions and insights into special cases of the problems

## Abstract

We present five open problems in the theory of vertex rings. They cover a variety of different areas of research where vertex rings have been, or are threatening to be, relevant. They have also been chosen because I personally find them interesting, and because I think each of them has a chance (the title of the paper notwithstanding!) of being solved. In each case we give some explanatory background and motivation, sometimes including proofs of special cases. Beyond vertex rings per se, the topics covered include connections to real Lie theory, formal group laws, modular linear differential equations, Pierce bundles, and genus 2 Siegel modular forms and the Moonshine Module.