Kirill's research interests are in a variety of areas within the wide subject area of multiscale analysis, where the predominant approach is the asymptotics of solutions of differential equations. During the 23 years of his research career (since the time he was an undergraduate student at St.Peterburg State University, Russia), he has made progress in understanding the emergence of higher-order effects in the overall behaviour of composites, discovered and described a number of phenomena in wave propagation, introduced a number of new techniques in the mathematical analysis of nonlinear composites, and most recently, obtained a series of results on operator-norm convergence of solutions to problems of periodic homogenisation.
Yulia is a specialist in the spectral analysis of quantum graphs, which occupy an intermediate position between one-dimensional (ODE) and multi-dimensional (PDE) differential operators. During her PhD studies at the Institute of Mathematics of NAS (Kyiv), Yulia solved the problem of the reconstruction of the topology of a finite graph from its spectrum and, in particular, described what graphs are isospectral.
Reynaldo is a UUKi Rutherford Fellows from the Technical University of Oaxaca, Mexico. His research has been focussed on the use of variational techniques for the study of solutions to the n-body problem. In his research project at Bath, Reynaldo develops a new approach to nonlinear problems in homogenisation using recent ideas in the analysis of operators describing time-dispersive media.
Serena, who is in the second year of her PhD studies at Bath, works on the analysis of differential operators on periodic singular and thin structures, which can be represented, for example, by appropriate periodic measures. Serena has proved operator-norm resolvent convergence for scalar version of such operators in the long-wave regime ("homogenisation"), and she is currently working on the development of her technique to the case of Maxwell's equations of electromagnetism and the system of linearised (acoustic) elasticity.
Aaron's research is focused on new approaches to the description of defects in liquid crystals, using asymptotic analysis of PDE in domains with singular boundaries. Aaron is in the first year of his PhD studies, and so far he has studied the behaviour of solutions to the Laplace equation in a punctured square domain with piecewise-constant boundary conditions on the boundary of the square. This models the orientation of molecules in a nematic liquid crystal. Aaron is now working on the vector version of his approach.
Will joined the group as a summer project student in 2016, and since September 2017 have has been developing his research project as part of the SAMBa Centre for Doctoral Training. Will has carried out analysis of the interface conditions between a dielectric and a Lorentz medium in the context of Maxwell's equations of electromagnetism (a version of the so-called Leontovich conditions) and has studied the waves propagating along the interface and decaying exponentially away from it. He has also studied numerically the related dispersion relations.
Milos is working in a summer internship project, supported by the London Mathematical Society and the University of Bath. Milos' project concerns the derivation of explicit expressions for solutions to a one-dimensional scattering problem for scalar and vector ordinary differential equations with piecewise-constant coefficients. He is planning to investigate the obtained solution numerically, which in combination with the analytical approach will provide new understanding of scattering phenomena in elastic and electromagnetic composite media.
Forthcoming highlight: Uzy Smilansky, Professor Emeritus at Weizmann Institute of Science, will visit the group for 6 months in 2020-21. Professor Smilansky is a recognised leader in several research areas on the physics-mathematics interface. He is known for raising important questions about the behaviour of complex system that have initiated substantial strands of research, as well as for solving open problems that have provided new insights. Professor Smilansky will be supported by David Parkin Visiting Professorship awarded to him by the University of Bath.