My research interests can be broadly described as modelling with, and the analysis of, partial differential equations (PDEs). I have been mostly working on problems originating in the mechanics of solids and electromagnetism, although this is by no means restrictive: it is well known that a great number of phenomena of different physical nature can be treated along similar lines. More often than not, the analogy proceeds from the shared structure of the mathematical objects invoked in the theoretical study. For example, the diffusion equation can be used to model the distribution of dislocations in plastically deformed solids, and for analysing the spread of populations in ecology. And it is this unifying role of the research in applied mathematics that fascinates me, and motivates my work.

My interest in the mathematical study of solids and waves has stemmed from the prevailing interests of the research groups I have been part of during my academic career to date. Thus, as an undergraduate student I was thoroughly exposed to the activities of the St.Petersburg school of diffraction and mathematical methods in geophysics, and since coming to the UK I have been working on the mathematical study of composite materials as part of the related UK community. The mathematical tools I have used in my research are seemingly unrelated: asymptotic analysis, homogenisation theory, spectral theory of differential equations, calculus of variations, regularity theory for PDEs. However, they are all united by the common theme of modelling across length-scales. For instance, minimising sequences to some variational integrals often represent fine microstructures observed in real materials, but the choice of a sequence may depend on what we find reasonable to assume about the regularity of certain physical quantities. This in turn affects the choice of a numerical algorithm used to calculate such microstructures, and is crucial indeed in whether we are able to give an adequate description of the physical reality! Bridging length-scale gaps thus presents a powerful paradigm for the use of mathematics in understanding a wealth of phenomena in physics, materials science, engineering and life sciences. This, in brief is the main subject of my academic preoccupation.

As with any scientific work, my current research draws to some extent from what I have done in the past. As an undergraduate at St.Petersburg State University I derived uniform large-index asymptotic expansions for Legendre functions, and more generally, for solutions to the so-called Fuchsian ODEs. These expansions have since been used in the study of some diffraction/scattering problems. My PhD project at the University of Bath focused on the derivation of size effects in the overall behaviour of heterogeneous media. Such effects have been observed experimentally, and my results helped provide their quantitative description, as well as carry out an efficient numerical study. The development of this work, which followed during my time as a Junior Research Fellow in Oxford, involved the study of nonlinear periodic media and of the so-called ``double-porosity'' composites. In the latter, size effects are intimately linked to the ``band-gap'' phenomenon for the spectrum, which is highly relevant to recent advances in experimental optics and elasticity. At the same time, my continued work in the subject of wave propagation resulted in a series of papers on waves localised in the vicinity of curved boundaries of elastic bodies, such as Rayleigh or edge waves.

Currently I am working on the following research projects:

1) Modelling of size effects in the overall behaviour of large dislocation ensembles in elasto-plastic solids via homogenisation, and comparing the results to the available experimental evidence (with V. Deshpande and J. Willis);

2) Providing a description of the high-frequency boundary-layer spectrum of the Maxwell operator using such concepts as Bloch-wave decomposition and Bloch measures, two-scale convergence and the Hausdorff convergence of spectra of PDEs (with S. Guenneau);

3) Analysing and performing a numerical study of the effect of micro-resonance in periodic and random media of the double-porosity type via the use of an advanced version of the two-scale convergence method;

4) Combining the methods of the mathematical theory of homogenisation and surface wave propagating in the study of distributed trapped modes in periodic composites, including the case of high contrast between the mechanical characteristics of the constituent media;

5) Studying the relation between the quasi-steady states in dynamics for non-linear models with phase transitions and convergence results for sequences of the associated Fokker-Planck equations (with P. Padilla).

The above list lays out approximate boundaries of my research at the moment, and can thus be thought of as a grainy snapshot of a more detailed research programme I pursue. Indeed the topics of my current and past study must be looked at in conjunction, with the unifying aim of the multi-scale analysis constantly born in mind. For example, my results on size effects in periodic composites are are in close relationship with my present study of spectral properties of high-contrast media. At the same time, however, these results are highly relevant to the strain-gradient plasticity theories, whose relation, in turn, to the known ``lower-order'' theories, such as the discrete dislocation plasticity, is still to be substantiated. Another important guide for my work is its relevance to the practical issues to be addressed, from the design of more efficient materials and structures to better insight into seismic waves and tumour growth; the length-scale interactions is what underpins these seemingly different contexts, and gives mathematics the edge in their progress.