(Imagination is more important than knowledge... Albert Einstein)
My interests in Mathematics/Physics
My primary research interests focus on dynamical systems,
both conservative and dissipative, and in particular on studying
different aspects of chaos and complexity. The main areas are
- Billiards:Investigation of qualitative properties of the motion
of a point particle inside compact regions (in particular Lyapunov exponents
of such systems and their ergodicity). I am especially interested in
billiard systems in more dimensional setting
billiards between two
nonconcentric circles and generalized billiards (i.e. billiards in magnetic
fields, billiards in a finite range potential).
- Switch flow systems:
With any continuous dynamical system at hand, one can construct a
discrete dynamical system of the following type. Consider two "symmetric"
flows, v(t), w(t). By "symmetric" I mean that there exists a linear involution
G (G*G=Id) such that w(t)x=G*v(t)*Gx. For a given time T we can construct
a composition of these flows and that's our discrete system.
The question is how does the dynamics
of the composition depend on T and on a given involution.
That, of course, seems a bit artificial. There
are 2 systems, however, where the above dynamics arises naturally.
First is dynamics of a chemical reactor with a flow reversal which is the
problem I encountered while studying Chemical Engineering and it boils
down to composition of two symmetric motions (both described by PDEs).
Second is the problem of the so called "blinking vorteces" described in
the book by
J.M. Ottino: The kinematics of mixing: stretching, chaos and transport.
- Periodic orbits:
How periodic orbits determine the dynamical properties of the system.
Using cycle expansions to compute escape rates, Fredholm determinants
and Dynamical Zeta Functions and what all this has to do with semiclassical
approximation to quantum mechanics.
studying how, by feeding certain variables in chaotic
systems into a copy of themselves one can make the two systems go hand in
hand. I did some simulations and analysis for Lorenz system and for
There are too many interesting things in mathematics/physics to concentrate
only on dynamical systems. Of my side-interests, these are at the top.
- Commutativity and fractional powers
of maps: Any questions concerning commutativity and fractional
powers of maps and operators. This involves studying embeddability of
diffeomorphisms in flows, function equations (iteration groups in particular)
and some generalizations of a rotation number.
- Continued fractions: questions related to the convergence of
CF describing the curvature of a wavefront of a billiard orbit and also
topics about the classical CFs (the ones with integral partial quotients).
Their relation to Farey Trees, Mobius Transformations, Diophantine Equations
and arithmetic in finite groups (that is modulo n). As most mathematicians,
I also "admire" the Riemann Zeta function.
- Quantum Gravity: As most of scientists I have a secret admiration
for the basic questions about the nature of our universe. Being fond of
differential geometry makes me naturally curious about the
gravity and its possible
Most of physicists seem to agree that a key to
a more profound understanding of the Universe lies in the elusive relation
of General Relativity and Quantum Theory. I am secretly hoping that the
winning theory will introduce into physics a marvellous world of p-adic
numbers, since their "weird" nature would quite fit the equally "weird"
nature of quantum mechanics.
e-MATH Home Page
...the cyberquarters of AMS
The Chaos Network
...which actually is quite ordered and deterministic (see Fractals!)
...links to sites dealing with the subject (even outside math)
The Geometry Center
...a welcome page of Geometry Center in Minnesota
...a collection of unsolved problems
...another place for geometry lovers
Mathematical Quotations Server
...what people said about mathematics
Analytic Beer chronicles
...chronicles of an unorthodox mathematical seminar
All SERIOUS people are POLITELY
asked to leave Absurdistan
If you think you want to proceed, go ahead, but bear in mind
that this HOMEpage is HOMEomorphic to the Cantor set,
so mind your steps.
(a city map of Pierre de Fair)
About Math County in Absurdistan
( less seriously)
The whole south part of Weird County used to be a part of Math
County. Unfortunately, during the years when the Absurd Party came to power,
mathematicians were the only group who resisted the infiltration by its
members and the Party got angry at them. It banned mathematics from all
schools and administratively placed a big chunk of math territory into
Weird County. The consequences of this unfortunate decision are
felt even in the outside world. Just open any specialised mathematical
journal and you'll have a very weird feeling.
An example of this
administrative injustice is the city of "Codimension".
Or to be more precise the city of codimension 1. As is clear from
the name this unique city has codimension 1 and that means (for
non-mathematicians) that it is 2-dimensional.
Everything in that city, including people and their feet,
is absolutely flat.
Of course living in such a city is hell, since there is no way
how to pass under the telegraph wires or over the sewage system.
Hence, every now and then, the city officials have to disconnect
and temporarily dismantle certain parts of Codimension's
infrastructure so that the citizens could do some shopping or visit
On the other hand the townfolks pay only the flat tax, which is very low,
and that makes the living in Codimension bearable.
In case the reader
wonders how people from 3-d world could travel into 2-d Codimension,
we must remark that at the city limits there are huge hammer stations which
flatten out whoever wants to be granted access to Codimension.
It is needless to say that such access is permanent and irreversible.
And while we are in Math County, if you ever get there
don't miss the Fair in Fair - an annual
county fair in Pierre de Fair. You can get all sorts of mathematical
goodies there. Candy eigenvalues, phosphorescent compact manifolds,
Klein bottles of beer, Hollywood glamour girls with elliptic curves
and non-measurable measures,
Palace of Horror for freshman calc students with skeletons made
out of integral signs, and the biggest attraction of all
- the "Moebius Strip-tease".
While in Pierre de Fair you may also want to try
its marvellous highway system. Unfortunately the highways
are too narrow for most cars and so you will have to admire them on bicycles.
At any rate, don't take mathematics too seriously.
Just play around with it and you'll have more fun than you'd ever imagine.
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