## Overview

Professor Ruzhansky''s research includes partial differential equations of hyperbolic type, Schrödinger equations, more general dispersive and evolution equations, as well as elliptic and hyperbolic systems of equations. His main interest is in various (local and global) properties of solutions to these equations, their geometric analysis, and further applications of these properties to semilinear and nonlinear equations.

Examples of the properties include microlocal analysis, properties of Fourier integral operators, estimates in Lp, Sobolev and other function spaces, singularities of the Hamiltonian flows, time asymptotics, dispersive, smoothing, and Strichartz estimates, etc.

He is also interested in the theory of pseudo-differential operators on spaces with symmetries, such as Lie groups, symmetric, and homogeneous spaces, as well as more general manifolds.

## Funding

- EPSRC Pathways to Impact Award (£24,250), 2011
- EPSRC Leadership Fellowship EP/G007233/1 “Phase space analysis of evolution equations”, 2008-2013
- EPSRC Grant EP/E062873/01 "Asymptotic properties of solutions to hyperbolic equations", 2007-2010
- Grants from IMU, LMS, EPSRC, OxPDE, OCCAM, BICS, and IC Fund, for the funding of the 7
^{th}ISAAC Congress, Imperial College London, 2009 - JSPS Invitational Research Fellowship for the project "Smoothing for dispersive equations and applications to nonlinear problems", 2007-2008
- Leverhulme Research Fellowship for the project "Smoothing properties of Schrodinger equations", 2006-2007
- London Mathematical Society conference grant for the organisation of the conference "Fourier analysis and hyperbolic PDEs", held at Imperial College London, organised by Michael Ruzhansky and Ingo Witt, 2005
- Royal Society grant for the collaboration between scientists from UK and Japan, jointly with Yuri Safarov, 2003-2004
- EPSRC Grant GR/R67583/01 "Regularity of solutions and spectral asymptotics for systems of partial differential equations", 2002-2004

## Research Student Supervision

ANDREWS,G, A closed class of SG Fourier integral operators with applications

CONNOLLY,D, current

ROTTENSTEINER,D, current

SMITH,J, Global time estimates for solutions to higher order strictly hyperbolic PDEs