In nature, symmetry is ubiquitous. Mathematicians use a tool called a group to capture this symmetry through motion. To illustrate, a square can be rotated by 90 degrees or reflected top to bottom without altering its appearance. By repeating this process, eight symmetry-preserving motions can be obtained. This collection of motions is a group. “Objects with the most symmetry are those with the biggest groups – and such objects are surprisingly rare,” says Joanna Fawcett, a Marie Skłodowska-Curie Individual Fellow in the Department of Mathematics at Imperial College London. “Because this rarity means we can list all of these objects, we are better able to understand the very essence of symmetry.” Fawcett’s research focuses primarily on group theory, combinatorics, discrete geometry, and representation theory. Through the support of the EU-funded HSPLS project, she recently conducted in-depth research into objects called partial linear spaces (PLSs), or a collection of points and lines where each line can be thought of as a collection of points. The goal was to understand the PLS for which local symmetries always arise from global ones.
Three goals, many questions
When lines all have exactly two points, PLSs are called graphs or networks. Mathematically, a graph is homogeneous whenever two subgraphs look the same. These homogenous graphs are extremely rare and have all been identified. What Fawcett is interested in are the subgraphs with some specified structure, such as those appearing in a collection X (a symmetry property called X-homogeneity). “As we vary the possibilities for X, can we still enumerate the X-homogeneous graphs, and can we also do this for all PLSs, not just graphs?” asks Fawcett. “Moreover, the group of a homogeneous PLS is special because it has a small rank, and our understanding of the groups with rank 4 or 5 has remained incomplete for over 30 years – can we remedy this?” To answer these questions, Fawcett’s research focused on achieving three main goals. First, to enumerate the C-homogeneous PLS, where C consists of all connected PLS; then prove her conjecture that any C5-homogeneous graph containing squares but not triangles is C-homogeneous, where C5 consists of the connected graphs with at most 5 points. Lastly, complete the classification of the groups with rank 4 or 5.
A useful tool for understanding symmetry
According to Fawcett, once a few loose ends are tied up, she will have successfully completed her three goals: “Combining the results achieved in this project with my earlier work means our understanding of the C7-homogeneous graphs is nearly complete.” Fawcett notes that her work on the third goal will provide group theorists with a useful tool for understanding symmetry. “It feels like we are at the tip of the iceberg for understanding the effects of the choices for the collection X when studying X-homogeneous PLSs,” adds Fawcett. “Although studying X-homogeneity in PLSs revealed that these objects have even more symmetry than we first believed, if we’re going to get to the bottom of this mystery, we still need to consider lots of other possibilities for X in a more systematic way.”
HSPLS, partial linear spaces, Marie Skłodowska-Curie, mathematicians, symmetry, mathematics, group theory