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Highly symmetric partial linear spaces

Periodic Reporting for period 1 - HSPLS (Highly symmetric partial linear spaces)

Reporting period: 2017-10-02 to 2019-10-01

Symmetry is ubiquitous in nature. A group is a mathematical tool that captures this symmetry through motion. For example, a square can be rotated by 90 degrees or reflected top to bottom without its appearance being altered. By repeating this process of rotation and reflection, we obtain eight symmetry-preserving motions. The collection of these motions is an example of a group. Objects with the most symmetry are those with the biggest groups, and such objects are surprisingly rare. Amazingly, this rarity means it is often feasible to list all of these objects. By doing so, we seek to understand the very essence of symmetry.

My research is focused on objects called partial linear spaces (PLS). A PLS consists of a collection of points and a collection of lines, where each line can be thought of as a collection of points for which two conditions hold: each pair of points lies in at most one line, and each line contains at least two points. A network or graph is a PLS where every line contains exactly two points. For example, we get a graph by taking the points to be the cities in a country and connecting two cities by a line when there is a train service between them. Another example of a graph is the square, which has four points (the corners) and four lines (the edges).

My programme of research concerns PLS for which local symmetries always arise from global ones. Mathematically, a graph is homogeneous if it has this property: whenever two subgraphs (i.e. sections of the graph) are isomorphic (i.e. look the same), there is a motion taking the points of one part to the other. The graphs that are homogeneous are extremely rare and have all been identified, but what if we only care about the subgraphs with some specified structure, say those appearing in a collection X? We call this symmetry property 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 PLS, not just graphs?

I have two long-term research goals to answer these questions. The first is to understand C-homogeneous PLS where C is the collection of connected PLS (i.e. for any two points, there is a chain of lines connecting them). This leads to two research objectives. Objective 1 is to enumerate the C-homogeneous PLS. Objective 2 is to prove my conjecture that every C5-homogeneous graph containing squares but not triangles is in fact C-homogeneous, where C5 is the collection of connected graphs with at most 5 points.

My second long-term goal is to understand groups with small rank. The rank of the group H of a PLS is the number of types you get when you sort all pairs of points into collections of things that can be morphed into one another by H; these types are called orbits. For example, if the collection X contains the 3 graphs with at most 2 points, then the group of any X-homogeneous graph has rank 3, with these 3 orbits: pairs of points that are the same, pairs of points on a line, and pairs of points not on a line. The rank 3 groups are well understood, but groups of rank 4 or 5 are not. This leads to two research objectives. Objective 3 is to classify groups with rank at most 5. Objective 4 is to use this classification to investigate PLS with rank 4 groups.

Through this research, we now have a much better understanding of the symmetries of X-homogeneous PLS. I successfully proved my conjecture for Objective 2, and I am on track to complete my classification for Objective 1; surprisingly, my research indicates that there are very few families of C-homogeneous PLS. This shows that taking X to consist of connected structures is quite powerful. In contrast, for Objectives 3 and 4, I have discovered many new structures, showing that low rank is a fruitful avenue of study. For Objective 3, I am on track to complete my classification. Objective 4 turned out to be very challenging, so I completed a classification of the PLS with rank 3 groups (up to certain exceptions), an important first step towards understanding those with rank 4.
For Objective 1, I discovered that the best way to study C-homogeneous PLS is to first obtain a complete enumeration of the T-homogeneous graphs, where T is the collection of trees. A tree is a graph that looks like a tree: it is connected, and there are no cycles (i.e. closed loops). An example of a T-homogeneous graph that is not C-homogeneous is given in the project image. If we start with a C-homogeneous PLS, then there is a natural way of using this structure to construct a new T-homogeneous graph, and if we can determine all of the T-homogeneous graphs, then we can determine all of the C-homogeneous PLS. It turns out that it is enough to classify the T-homogeneous graphs that contain triangles, and I proved that such graphs have the very special property that for any two points, there is always some other point that is joined to both of them. Using the pre-existing classification of the rank 3 groups, I am now working to enumerate the T-homogeneous graphs.

For Objective 2, I proved my conjecture by first proving that in a C5-homogeneous graph G containing squares but not triangles, the number of common neighbours of points at distance 2 in G is very restricted. With this technical result, I could then use pre-existing theory to deduce that G is C-homogeneous and therefore known.

For Objective 3, I applied some deep theory that represents groups as collections of matrices, enabling me to study these groups using linear algebra, a very powerful mathematical tool. Working in collaboration with other mathematicians, we have made significant progress in using this theory to list the groups with rank at most 5. Moreover, groups with small rank have big orbits when you fix a point, and we have made some contributions to enumerating the groups where these orbits are regular, which means they are as large as possible.

For Objective 4, I used similar theory to Objective 3 to study the PLS whose groups have rank 3, focussing on those that are "primitive", as these can be thought of as the building blocks of all groups. Working with my collaborators, we were able to completely list these structures, except for some cases where the groups are quite small (for these, we suspect that there are too many examples for a classification to be possible).

All of my results will be published in high-quality research journals and made available on the open-access repository arXiv, where anyone can view them. To communicate these results, I attended 8 international conferences and workshops, and gave 5 seminar talks. I explained my research and the motivations behind it through two outreach events at Imperial College London: the Imperial Lates event "Xmaths" and the Great Exhibition Road Festival.
All of the results I have obtained are new and make a significant contribution to our understanding of the symmetries of PLS. In particular, I found lots of new examples, which will inspire future research. The results for Objectives 2 and 4 are complete, and I will obtain complete enumerations for Objectives 1 and 3, where work is still ongoing. All of my results will provide mathematicians with useful tools and resources for solving even more problems, further impacting our understanding of the nature of symmetry.
An example of a T-homogeneous graph