#### Milestone Year

### 1994

#### Qualitative Theory of Differential Equations

1994 Qualitative Theory of Differential Equations
and the Poincare Center-Focus Problem

In 1994, Weizmann scientists developed a new approach
to a range of problems in Qualitative Theory of Differential Equations,
concentrating around the Poincare Center-Focus problem.
This problem, which dates to the 1880's, is still widely open.
It asks for explicit conditions on a given differential equation
to have all its solutions periodic; that is, infinitely repeating themselves.
This is a central question in the study of the stability
of many important physical phenomena.
Weizmann scientists discovered deep connections
of the Poincare Center-Focus problem with the Classical Moment problem
and with the Composition algebra of polynomials.
Using this relation, major progress has been achieved on the former problem,
culminating in the proof of a close approximation of
the center configurations by the composed ones.

In 2008-2010 the study of vanishing Moments, initiated in the work
of Weizmann scientists, was used to obtain a breakthrough
in the classical Ritt theory of Compositions (dating to the 1920's).
In 2010-2012 based on these results, a second major advance
in understanding of the Poincare Center-Focus problem was obtained,
clarifying to great extent the role of the Moments in this problem.

This development led to an investigation of
the "second order iterated integrals", which generalize Moments.
During the last year, Weizmann scientists found an innovative way
to incorporate these objects into the existing study.
This new research direction bears great promise for
a breakthrough in the study of Poincare Center-Focus problem,
and it has been significantly advanced by the work of many researchers.