ICTP/SISSA Joint Colloquium in Mathematics on 23 January

[Math]

ICTP/SISSA Joint Colloquium in Mathematics


Wednesday, 23 January, at 15.00 hrs.


Professor Vladimir I. Arnold
Steklov Mathematical Institute, Moscow, Russia


"Straight lines of a plane and real algebraic geometry"

Abstract

Which values may attain the number $M$ of the connected components
of the complement to $n$ distinct straight lines in the real plane?

For $n=4$ the possible numbers of components are (5, 8, 9, 10, 11),
but the general problem for $n$ straight lines is unsolved.

Namely, between the minimal number of components, $n+1$, and
the maximal one, $(n^2+n+2)/2$, there are gaps formed by the
unattainable numbers.

The first gap is $n+1<M<2n$: these values are unattainable.
For the $k$-th gap the stable boundaries  $a_k(n)<M< b_k(n)$ are known,
providing the answer for the stable case (where the number of straight
lines is sufficiently high, $n>C(k)$).  For the unstable values of $n$,
the $k$-th gap may be smaller than the stable answer. Its exact boundary
is unknown even for the 3-rd gap (where the stability starts from $n=14$
straight lines).

In real algebraic geometry even the simplest problems are difficult
(and even the contribution of Hilbert to his 16-th problem,  
discussing these
questions, was wrong).



Venue:  ICTP Main Lecture Hall, Main Building, entrance level