Contents
Andrei GABRIELOV : Abstract
Mystery of Point Charges
In his famous Treatise on Electricity and Magnetism (1873) J.C. Maxwell considered the following problem: How many equilibrium points a potential of n fixed point charges in $R3$ may have? Maxwell applied Morse theory (developed 50 years later) to approach this problem. He claimed that there should be at most $(n-1)2$ equilibrium points. In a footnote to the third edition (1891) of the Maxwell's book, J.J. Thomson stated that he could not find any proof of this. In the talk, recent progress in this direction will be reported, based on fewnomial theory and Voronoi diagrams.
The Wronski map, Schubert Calculus, and Pole Placement
The Wronski map associates to a $p$-tuple of polynomials of degree $m+p-1$ their Wronski determinant, a polynomial of degree $mp$. If the polynomials are linearly independent, they define a a point in the Grassmannian $G(p,m+p)$. Accordingly, the Wronski map can be considered as a map from $G(p,m+p)$ to the projective space ${\bf P}^{mp}$. The map is finite, and one can define its degree. In the complex case, this degree equals the number of standard Young tableaux for the rectangular $(m,p)$-shape. In the real case, Young tableax should be counted with the signs depending on the number of inversions. Degree of the real Wronski map is zero when $m+p$ is even, and equals the number of standard shifted Young tableaux for an appropriately defined shifted shape when $m+p$ is odd. When both $m$ and $p$ are even, the Wronski map is not surjective. These results have important applications to real Schubert Calculus and to the pole placement problem in control theory.
