35 research outputs found
Looking backward: From Euler to Riemann
We survey the main ideas in the early history of the subjects on which
Riemann worked and that led to some of his most important discoveries. The
subjects discussed include the theory of functions of a complex variable,
elliptic and Abelian integrals, the hypergeometric series, the zeta function,
topology, differential geometry, integration, and the notion of space. We shall
see that among Riemann's predecessors in all these fields, one name occupies a
prominent place, this is Leonhard Euler. The final version of this paper will
appear in the book \emph{From Riemann to differential geometry and relativity}
(L. Ji, A. Papadopoulos and S. Yamada, ed.) Berlin: Springer, 2017
New Directions in Geometric and Applied Knot Theory
The aim of this book is to present recent results in both theoretical and applied knot theory—which are at the same time stimulating for leading researchers in the field as well as accessible to non-experts. The book comprises recent research results while covering a wide range of different sub-disciplines, such as the young field of geometric knot theory, combinatorial knot theory, as well as applications in microbiology and theoretical physics
Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials
We investigate the asymptotic zero distribution of Heine-Stieltjes
polynomials - polynomial solutions of a second order differential equations
with complex polynomial coefficients. In the case when all zeros of the leading
coefficients are all real, zeros of the Heine-Stieltjes polynomials were
interpreted by Stieltjes as discrete distributions minimizing an energy
functional. In a general complex situation one deals instead with a critical
point of the energy. We introduce the notion of discrete and continuous
critical measures (saddle points of the weighted logarithmic energy on the
plane), and prove that a weak-* limit of a sequence of discrete critical
measures is a continuous critical measure. Thus, the limit zero distributions
of the Heine-Stieltjes polynomials are given by continuous critical measures.
We give a detailed description of such measures, showing their connections with
quadratic differentials. In doing that, we obtain some results on the global
structure of rational quadratic differentials on the Riemann sphere that have
an independent interest.Comment: 70 pages, 14 figures. Minor corrections, to appear in Comm. Math.
Physic