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My research deals with PDEs and, more generally, analytical methods related to geometry and topology as well as methods coming from mathematical physics. Particular areas of specialisation include: Spectral geometry; extremal eigenvalue problems. I have been working on isoperimetric inequalities and multiplicity bounds for various eigenvalue problems. I am also intersted in extremal eigenvalue problems. One of the past projects is concerned with the development of direct methods for such extremal problems on Riemannian surfaces and understanding of possible singularities of extremal metrics. Harmonic maps and their generalisations. Another interest of mine is general equations of the type of harmonic maps, related analytic phenomena, and their applications to geometry. One of my contributions here is concerned with exploring the relationship between bubble convergence and the curvature concentration. In another paper I developed a number of applications of the socalled pseudoharmonic maps in conformal/complex geometry. Moduli spaces of solutions to elliptic PDEs. In my PhD thesis and shortly after I have been studying moduli spaces formed by solutions of PDEs on mappings between manifolds together with Sergei Kuksin. My favourite application of these results is to the topology of the evaluation map; it establishes a relationship between the symplectic version of the Gottlieb vanishing phenomenon and the occurrence of rational curves. 
Preprints and publications:

Slides of some talks:
