Abstract TBA
The talks are all happening at University of Southampton (in person), on Highfield Campus (SO17 1BJ). For the building and the room, see the individual postings. Email the organiser ( a (dot) guidolin (at) soton.ac.uk ) if you have questions.
More Pure Maths talks at Southampton: we run Pure Maths Colloquia, Topology Seminar, and Pure Postgraduate Seminar (link soon).
Abstract TBA
The eigenvalues and eigenvectors of the graph Laplacian has a long history in computational harmonic analysis, primarily utilised in creating local and global features of graphs. However, the eigenvectors are known to be unstable under perturbations to the graph, and subject to a choice of sign or basis in the eigenspace. Here, we show that these two issues can be solved by considering the projection-valued spectral measure of the Laplacian. By construction, it combines both eigenvalues and eigenvectors into one coherent object, and is independent of the choice of eigenbasis. Furthermore, we prove that vertex features derived from the projection-valued spectral measure is Lipschitz stable with respect to perturbations to the Laplacian, which overcomes the instability of using eigenvectors. We show that these vertex features empirically improve the performance of graph neural networks on graph regression tasks, and can be used to detect approximate symmetries in point clouds and graphs.
In 1971, Wilson introduced the structure lattice of a just-infinite group and provided a classification of just-infinite groups based on it. In the late 1990s, when Grigorchuk introduced the class of branch groups, Wilson showed that the structure graph, a subgraph of the structure lattice, encodes all the possible actions of a group as a branch group. This opened the door to the study of branch actions and, in particular it started the problem of classifying all the possible branch actions of a given branch group.
In this talk, we generalize the notion of structure graph from branch groups to weakly branch groups and show its applications to the study of the rigidity of weakly branch actions, i.e. when the action of a group as a weakly branch group on a given tree is unique. We further discuss applications to the first-order theory of weakly branch groups generalizing analogous results of Wilson for branch groups.
In 1966 Jane Matthews claimed that the conjugacy problem is solvable in the standard restricted wreath product A \wr B if and only if (i) the conjugacy problem is solvable in A and B and (ii) B has a solvable power problem. We will discuss this result, showing that an additional condition is required - that either A is abelian or B has a solvable order problem. We will also seek to extend this result to the permutational restricted wreath product A \wr_S B where, rather than acting on itself, B acts on a set S.
For an infinite graph, a periodic colouring of the vertices is one that is invariant under a subset of the symmetries of the graph. As such, a natural consideration are those locally finite quasi-transitive graphs with finite quotient. Recent work from Abrishami and collaborators investigates whether all such graphs permit proper periodic colourings, with conclusions dependent on the number of ends. An observation of Hopf tells us any infinite locally finite, quasi-transitive graph has 1, 2, or ∞-many ends. One case was left uncertain; do all locally finite quasi-transitive planar graphs with 1-end permit a periodic colouring? In this talk I will review the results of Abrishami and collaborators before describing how, through a theorem of Babai providing embeddings of 3-connected graphs into the Euclidean or hyperbolic planes, we can leverage properties of isometry groups to complete this investigation.