报告题目：Two New Developments for Noether's Two Theorems

报告地点：英国正版365官方网站长安校区文津楼3425学术报告厅

报告时间：2024年9月25日16:00 -17:00

报告简介：In the first part, I start by recalling the two well-known classes of partial differential equations that admit infinite hierarchies of higher order generalized symmetries: 1) linear and linearizable systems that admit a nontrivial point symmetry group; 2) integrable nonlinear equations such as Kortewe-de Vries, nonlinear Schrödinger, and Burgers. I will then introduce a new general class: 3) underdetermined systems of partial differential equations that admit an infinite-dimensional symmetry algebra depending on one or more arbitrary functions of the independent variables.  An important subclass of the latter are the underdetermined Euler-Lagrange equations arising from a variational principle that admits an infinite-dimensional variational symmetry algebra depending on one or more arbitrary functions of the independent variables. According to Noether's Second Theorem, the associated Euler-Lagrange equations satisfy Noether dependencies; examples include general relativity, electromagnetism, and parameter-independent variational principles. Noether's First Theorem relates strictly invariant variational problems and conservation laws of their Euler-Lagrange equations. The Noether correspondence was extended by her student Bessel-Hagen to divergence invariant variational problems. In the second part of this talk, I highlight the role of Lie algebra cohomology in the classification of the latter and conclude with some provocative remarks on the role of invariant variational problems in fundamental physics.

报告人介绍：Professor Peter J. Olver is an internationally renowned expert in the field of nonlinear partial differential equations, a fellow of the American Mathematical Society and the Society for Industrial and Applied Mathematics (SIAM), and a former dean of the School of Mathematics at the University of Minnesota. Professor Olver is mainly engaged in the research of mathematical physics, nonlinear partial differential equations and integrable systems. His series of work in symmetry group theory, fluid dynamics, variational problems, differential geometry, computer vision and image processing, and geometric numerical methods have been widely concerned and cited by his peers. He has been invited to give more than 500 academic presentations worldwide, published more than 160 academic papers, and published 5 books.