5 papers
The Monge--Kantorovich problem, the Schur--Horn theorem, and the diffeomorphism group of the annulus
Anthony M. Bloch, Tudor S. Ratiu
First, we analyze the discrete Monge--Kantorovich problem, linking it with the minimization problem of linear functionals over adjoint orbits. Second, we consider its generalizatio…
Cartan Geometry and Infinite-Dimensional Kempf-Ness Theory
Tobias Diez, Akito Futaki, Tudor Ratiu
We pioneer the development of a rigorous infinite-dimensional framework for the Kempf-Ness theorem, addressing the significant challenge posed by the absence of a complexification…
The restricted Siegel disc as coadjoint orbit
François Gay-Balmaz, Tudor S. Ratiu, Alice B. Tumpach
The restricted Siegel disc is a homogeneous space related to the connected component of the Universal Teichmüller space via the period mapping. In this paper we show that…
Norm-squared of the momentum map in infinite dimensions with applications to Kähler geometry and symplectic connections
Tobias Diez, Tudor S. Ratiu
We initiate the study of the norm-squared of the momentum map as a rigorous tool in infinite dimensions. In particular, we calculate the Hessian at a critical point, show that it i…
Symmetric Discrete Optimal Control and Deep Learning
Anthony M. Bloch, Peter E. Crouch, Tudor S. Ratiu
We analyze discrete optimal control problems and their connection with back propagation and deep learning. We consider in particular the symmetric representation of the discrete ri…