A carregar...
Publicação
Morse Theory and the Maxwell conjecture on point charges
| Nome: | Descrição: | Tamanho: | Formato: | |
|---|---|---|---|---|
| 2.71 MB | Adobe PDF |
Autores
Orientador(es)
Resumo(s)
In 1873, in his famous “Treatise on Electricity and Magnetism”, J. C. Maxwell conjectured that the number of equilibrium points of a potential generated by a finite number n of fixed point charges is at most (−1)2. Later, at the beginning of the twentieth century, Marston Morse developed his theory, and in 1969, together with Stewart Cairns, in their book “Critical Point Theory in Global Analysis and Differential Topology” proved that in most cases it is at least −1 using Morse Theory. The proof for the upper bound is still an open problem, but recently new tools have been used to find new upper bounds in specific cases, namely the use of Voronoi diagrams and a connection between the dimension of an effective Voronoi cell and the Morse Index of a critical point of the Potential. This is done in the article “Mystery of Point Charges” by Gabrielov, Novikov and Shapiro, in 2004, though the upper bounds found here grow much faster than the one from Maxwell. The conjecture has been proved only in very specific cases, for example, if we have 3 points with positive charges, then the electrostatic potential generated by these charges has at most 4 equilibrium points. (The upper bound is attained if the charges are placed in the vertices of an equilateral triangle.) In this dissertation, we will introduce Maxwell’s conjecture. Then, in some detail, we will introduce Morse Theory and go through the proof of the lower bound. Finally, we will explore what has been done more recently in the pursuit of the upper bound, namely the use of Voronoi diagrams in the “Mystery of Point Charges” article. All the concepts and results needed from Algebraic Topology and Voronoi diagrams will be mentioned in the appendices.
Descrição
Tese de Mestrado, Matemática, 2025, Universidade de Lisboa, Faculdade de Ciências
Palavras-chave
Morse Theory; Potential Theory Critical Points/Points of Equilibrium Voronoi Diagrams Algebraic Topology