My research interests (* and
possible PhD projects*) fall into the following categories:

1. **Partial
differential equations (PDEs)**

I am interested in various questions concerning the theory of linear and

nonlinear *evolution* partial differential equations. These include

*Smoothing estimates
for evolution PDEs*

Many evolution PDEs have so-called smoothing properties. It roughly means that,

when averaged with respect to time, their solutions become more regular. This topic has

many very interesting relations to the harmonic analysis and spectral theory, as well

as to the well-posedness questions for nonlinear PDEs. Examples of equations are

hyperbolic equations (such as the wave equation, Klein-Gordon equation, dissipative

wave equations, as well as coupled equations and systems), Schrodinger, KdV, and

many others.

My
research in this direction is mostly joint with M.
Sugimoto, and the results in this

direction include the critical case of the Agmon-Hormander limiting absorption principle,

the critical cases of Ben-Artzi-Klainerman and Kato-Yajima
smoothing estimates for the

Schrodinger
and other equations, and estimates for norms for the critical cases of the

trace theorems. As an application, new
global in time well-posedness results for the

derivative nonlinear Schrodinger equations
have been obtained.

Moreover,
extending the works by Ben-Artzi and Devinatz,
Constantin and Saut,

Kenig,

we established
a number of new smoothing estimates, and, more importantly, by

introducing
new methods of canonical transforms and comparison principles for

evolution
equations, we showed that essentially all known smoothing estimates for a

variety of
a-priori unrelated evolution equations, are equivalent to each other and can

be obtained one from another (for
example, the gain of half derivative for the Schrodinger

equation
can be obtained from the gain of one derivative for the KdV
equation, etc.).

My current
work in this direction includes the extension of ideas of canonical transforms

and
comparison principles to evolution equations with potentials and to equations
with

variable
coefficients (in particular, the interesting case is equations with potentials,

also
allowing resonances at zero, which is a joint research with Matania
Ben-Artzi and

Mitsuru
Sugimoto).

*Regularity, dispersive
and Strichartz estimates for hyperbolic PDEs*

Again, these are one of the main tools for the analysis of nonlinear PDEs. The analysis

here relies on the asymptotic properties and decay rates of solutions to linear equations.

The methods involve various estimates for so-called Fourier integral operators, which

is a class of operators of great interest by themselves. Especially interesting and challenging

are higher order equations or systems of equations, with multiple characterisitics.

I am also interested in applications of such analysis (e.g. to Kirchhoff equations),

as well as in their nonlinear perturbations and scattering
theory.

A
comprehensive analysis of dispersive estimates for hyperbolic equations of
higher

orders
with constant coefficients and for hyperbolic systems is carried out in the
memoir

Ruzhansky M., Smith J., *Dispersive and Strichartz estimates for hyperbolic equations with *

*constant coefficients*, MSJ Memoirs, 22, *Mathematical Society of Japan, * 2010. (157pp)

The main
difficulties here is the appearance of multiple characteristics of general form

and the
results include the time decay estimates for different frequency zones (large
frequencies,

small
frequencies, frequencies around multiplicities, etc.) A certain geometric index
is

associated
to each equation, and the dispersive and Strichartz
estimates are obtained

depending
on this index, with the corresponding global in time well-posedness
results

for
hyperbolic equations. The results are applied to Grad systems of gas dynamics

(which are
hyperbolic systems of large size depending on the number of moments),

and to the
infinite dimensional systems appearing in the analysis of the Fokker-Planck

equations.
In a series of papers with T. Matsuyama and J. Wirth the results are

extended
to equations and systems with time-dependent coefficients, with the main

application
to the Kirchhoff equations. The current research deals with equations with

variable
coefficients and the corresponding geometric analysis. In particular, in this

direction
can be mentioned the joint work with I. Kamotski
where Sobolev space

estimates
were established for a general class of hyperbolic systems with variable

multiplicities
and the Weyl spectral asymptotics
for the corresponding elliptic systems

(also with
variable multiplicities) have been obtained.

**2. Microlocal and
harmonic analysis**

Microlocal analysis can be viewed as an extension of the Fourier analysis for the study

of partial differential operators. It allows to describe various properties of operators and

functions, such as the propagation and location of singularities, questions of regularity,

smoothness, etc. I am mainly interested in questions related to the so-called

*Fourier integral
operators* which is a fascinating class of operators that has relations to

many other areas of mathematics such as symplectic geometry, harmonic analysis,

spectral theory, PDEs, etc. Harmonic analysis aspects of interest to me are their

properties in Sobolev, modulation,
Wiener amalgam, and other function spaces.

My results
in this direction include the geometric analysis of the smooth factorization

condition for the L^{p}^{
}boundedness of Fourier integral operators
introduces by Seeger, Sogge,

and Stein.
Their L^{p}^{ }results for Fourier
integral operators with real-valued phases as well as

the L^{2
}results for Fourier integral operators with complex-valued phases of Melin and

Sjostrand,
and Hormander, have been extended to the L^{p}^{ }estimates
for Fourier integral operators

with
complex-valued phases in the monograph

Ruzhansky, M. *Regularity theory of Fourier integral
operators with complex phases and *

*singularities** of
affine **fibrations**,* CWI Tracts ,
vol. 131,

Recently I
came back to this topic in the global setting (partly motivated by the method
of

the canonical
transforms for the smoothing estimates), and the results include the global

(and weighted) L^{2 }and Sobolev
spaces results on the boundedness of Fourier integral

operators
and the global calculus for such operators under minimal decay assumptions

(jointly with M. Sugimoto), global L^{p}^{
}results and the determination of weights

for the global L^{p}^{
}estimates for Fourier integral operators (jointly with S. Coriasco), as

well as
results on the Beurling-Helsons theorems for
modulation and Wiener amalgam

spaces and
their relation to Fourier integral operators (joint with M. Sugimoto, N.
Tomita,

and J.
Toft).

Another
result in this direction, motivated by applications to dispersive estimates, in
the

multi-dimensional
van der Corput lemma for oscillatory integrals
(allowing complex

phases and
the dependence of parameters), obtaining the multi-dimensional decay rate for

degenerate
stationary points, thus bridging the gap between the one-dimensional decay rate

in the
classical van der Corput lemma and the multi-dimensional
decay rate for non-degenerate

stationary
poins provided by the stationary phase method.

^{ }

**3.
Pseudo-differential operators on Lie groups and manifolds**

I am 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.

The simplest case of a compact commutative Lie group is a torus (even a circle) and the

Fourier analysis of operators there is already very interesting. Here a global theory of

operators on such manifolds is of particular importance since it allows to capture many

geometric and other underlying algebraic properties of the manifold (a Lie group), and

to relate them to various properties of operators.

Some of my
recent work in this area can be found in the monograph

M.
Ruzhansky, V. Turunen, *Pseudo-differential
operators and symmetries, *Birkhauser, 2010. (724pp)

This work
is also closely related to the area of Time-Frequency Analysis

The work
in this direction are joint with V. Turunen, and the main result here is the
development

of the
global quantisation of operators on compact Lie groups and homogeneous spaces

yielding
globally defined matrix-valued full symbols and their calculus. Many properties
of

operators
can be expressed in terms of these symbols, and the knowledge of the full
symbols

allows one to formulate manifold
counterparts of well-known results on R^{n }that require the knowledge

of the full symbols (such as the
sharp Garding inequality, etc., which require the
knowledge of

the full
symbol, while the standard theory of pseudo-differential operators via
localisations

yields
essentially only the principal symbols with an invariant global
interpretation). The obtained

characterisation of the standard Hormander classes of pseudo-differential operators on
manifolds

also
allowed us to construct liftings of
pseudo-differential operators from the homogeneous space to

a Lie
group acting on it, thus extending the corresponding constructions well-known
for invariant

differential
operators, to the full pseudo-differential setting. The corresponding
matrix-valued

quantisation
of operators on S^{3}=~SU(2) has been also linked to the explicit
representation theory

of SU(2) and several related physical quantities (such as quantum numbers, Pauli matrices, etc.)

For my papers in these and other topics see my list of publications