МИНИКУРС ЛАБОРАТОРИИ ЧЕБЫШЕВА
Jeanne Scott (Universidad de los Andes)
"Perturbations of isoradial triangulations and their discrete Laplace operator(s)"
Лаборатория Чебышева, ауд. 413, 14-я линия В.О., 29
Расписание лекций:
пн. 17 сентября 17:15—18:45, чт. 20 сентября 15:30—17:00
пн. 24 сентября 17:15—18:45, чт. 27 сентября 15:30—17:00
In these lectures, I'll be concerned with four discrete operators defined for an arbitrary triangulation of the plane: the Beltrami-Laplace operator, the Monge-Laplacian, the conformal Laplacian, and the Eynard-David Kähler operator. All four operators coincide with the critical Laplacian studied by Richard Kenyon when the triangulation is isoradial. In this case, an elegant formula of Kenyon's allows one to express the Green's function of the triangulation's critical Laplacian in terms of the graph's local structure; furthermore, the log-determinant of the critical Laplacian can be computed as a finite sum of local contributions if in addition one assumes the triangulation is periodic. I shall make a survey of Kenyon's results and follow this by discussing joint work with François David where we consider smooth perturbations of a periodic, isoradial triangulation and obtain a large distance expansion for the second variation of the log-determinant for the Beltrami-Laplace and Kähler operators mentioned above. This result can be interpreted as a discretisation of the second variation of the continuous Beltrami-Laplace operator known from conformal field theory. Furthermore, we can identify the role played by the central charge in this discrete setting. |