My research directions are compactly explained below with the aid of presentations I have given. For more popular descriptions visit the Royal Society Exhibition section.
We developed a method which enables one to construct isospectral objects, such as quantum graphs and drums. This method is based on some basic representation theory arguments. For an explanation of the method and some neat examples, you might want to have a look at the isospectral construction method for drums (Oxford University SIAM Student Chapter Conference, February 2011) or the same construction demonstrated on quantum graphs (satellite workshop in Edinburgh belonging to the semester on Inverse Problems at the Isaac Newton Institute, October 2011).
Quantum graphs can be extended to scattering systems when they are connected by leads to infinity. We can show that for certain extensions, the scattering matrices of isospectral graphs are conjugate to each other and their poles distributions are therefore identical. I discussed the issue of scattering from isosepctral graphs in a talk given in East Midlands Mathematical Physics Seminars, Loughborough University (December 2010). You can also watch this talk as was given in the Isaac Newton Institute Workshop – Analysis on Graphs and its Applications, Cambridge (June 2010).
You can find a basic introduction of quantum graphs in most of the presentations in this page. For some more details please visit the section Quantum Graphs Course.
Counting nodal domains can be used to resolve isospectrality. Namely, one can show that certain isospectral objects have different nodal count sequences. Thus the information contained in the nodal count is complementary to the spectral data. More about this in the resolution of isospectrality for the dihedral pair (workshop on Nodal patterns in physics and mathematics, Wittenberg, Germany, July 2006).
Wishing to investigate the content within the nodal count, we try to derive trace formulae for the nodal count sequence. The question of existence of a trace formula for nodal domains on quantum graphs and the quest for the answer is described in a talk given in a meeting awarding Cardiff Honorary Distinguished Professor to Michael Solomyak (July 2010). A more pedestrian presentation called How to count nodal domains of quantum graphs? was given in the Young Researchers Symposium of the International Congress of Mathematical Physics, Prague (August 2010).
Another approach taken in this field of research is the use of graph partitions. An energy function defined on the space of all partitions allows to beatifully characterize the number of nodal domains of a specific eigenfunction. An explanation of the partition approach to nodal domains of a quantum graph was given in Quantum graphs in Mathematics, Physics and Applications, QGRAPH Network meeting, Stockholm (December 2011).