MATH COUNTY

(Imagination is more important than knowledge... Albert Einstein)

My interests in Mathematics/Physics (seriously)

Primary

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.
Synchronization: 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 low-dimensional maps.

Secondary

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 quantum interpretation. 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.

Math Links

e-MATH Home Page

...the cyberquarters of AMS

The Chaos Network

...which actually is quite ordered and deterministic (see Fractals!)

Complex systems

...links to sites dealing with the subject (even outside math)

The Geometry Center

...a welcome page of Geometry Center in Minnesota

Unsolved Problems

...a collection of unsolved problems

Geometry Forum

...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 NOW!
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 each other. 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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