报告时间：2025年04月03日 16:30开始
报 告 人：Pierandrea VERGALLO (意大利墨西拿大学 博士后研究员）
报告地点：9-113
报告题目： Non-homogeneous Hamiltonian operators of type 1+0: classification, geometry and related systems
报告摘要：In the 80s, Dubrovin and Novikov introduced for the first time Poisson brackets with the homogeneity property in the order of derivation, revealing many applications in mathematical physics and an effective interpretation in geometry. As a first example, they showed that first-order homogeneous Hamiltonian operators play a fundamental role in the theory of integrable systems, also for their clear geometric interpretation in Riemannian geometry. Furthermore in [1], the authors extended this structure by adding a non-homogeneous term of degree zero, i.e. an ultralocal structure, which is naturally associated to non-homogeneous quasilinear systems of first-order PDEs.
In this talk, we present a complete classification of non-homogeneous 1+0 operators in n=2 and n=3 number of components, covering the case of the Hamiltonian structure for the inverted KdV equation. Starting by this leading example, we then discuss a bi-Hamiltonian formalism in which both the operators are non-homogeneous, showing a classification of compatible pairs in n=2 components. The resulting Poisson brackets reveal an interesting geometric interpretation in terms of Nijenhuis geometry and bi-pencils of compatible metrics jointly with ultralocal structures. We finally discuss non-homogeneous quasilinear systems admitting this formalism, showing necessary conditions of compatibility between the operators and the investigated systems.
This talk is based on [2,3,4] and it is a joint work with Marta dell’Atti and Alessandra Rizzo.
[1] B. A. Dubrovin, S. P. Novikov, Akad. Nauk SSSR Dokl. 279:2 pp. 294–297 (1984).
[2] M. Dell’Atti, P. Vergallo, J. Math. Phys. 64:3, 033505 (19 pp.) (2023).
[3] P. Vergallo, Boll. Unione Mat. Ital., 17:2, pp. 513–526 (2024).
[4] M. Dell’Atti, A. Rizzo, P. Vergallo, in preparation
报告人简介：Pierandrea Vergallo, 意大利墨西拿大学博士后研究员也意大利国家核物理研究所米兰分部的助理研究员。 2022 年毕业于意大利萨兰托大学，获得数学和计算机科学博士学位。他曾在英国拉夫堡大学（2018 年、2021 年、2023 年）、俄罗斯高等经济学院（2020 年）、英国诺森比亚大学（2021 年）和澳大利亚新南威尔士大学（2024 年）担任访问学者。他的科研兴趣主要涉及微分方程的哈密顿形式主义，尤其关注其与无限维可积分系统理论的联系。
