Forthcoming Math Seminar - 26 May

[math]

MATHEMATICS SEMINARS 2009


Tuesday, 26 May at 14:00 hrs.


Professor Giorgi N. Khimshiashvili
(Georgian Academy of Sciences, Tbilisi, Georgia)

"Cyclic polygons as critical points"


Abstract:
We investigate critical points of several natural functions on the
configuration space of polygonal linkage. The main attention is given to
the signed area A and electrostatic potential E considered as functions
on the planar configuration space C(L) of a generic polygonal linkage L.
The following three settings will be discussed in some detail.

Firstly, we consider A as a function on C(L) and show that its critical
points are given by the cyclic configurations of L (as usual a
configuration is called cyclic if it can be inscribed in a circle, i.e.,
there exists a point in the reference plane equidistant from all
vertices of L). We also show that, generically, A is a Morse function
and find the Morse indices of cyclic configurations. This, in
particular, enables us to obtain useful information about C(L) by merely
examining the cyclic configurations of L, which can be effectively done
in many cases.

Next, we consider A as a function on the configuration space of an n-arm
(open polygonal n-chain) and show that its critical points are again
given by the cyclic configurations. In this case the structure of
critical points can be more complicated and we'll illustrate certain
typical phenomena by concrete examples.

Similar results hold for electrostatic potential E on the planar
configuration space of polygonal linkages satisfying some additional
conditions which guarantee boundedness of E.

The same problems become drastically more complicated in the case of
spatial linkage and we'll end by presenting a few conjectures and open
problems arising in this context.

Venue: ICTP, Leonardo Building, Seminar Room