CAS Seminar

Anton Kutsenko

(Jacobs University)

"Finite PDEs and finite ODEs are isomorphic"

Date: Tue, October 9, 2018
Time: 13:15
Place: Research I Seminar Room

Abstract: Assuming finite derivatives with arbitrary small rational steps of discretization, we show that non-linear stochastic ODEs, linear ODEs, and linear PDEs (all of them with bounded continuous or discontinuous coefficients) generate the same \(C^*\)-algebra, namely the universal UHF-algebra. Thus, topologically and algebraically, they are equivalent.

More precisely, the standard view is that PDEs are much more complex than ODEs, but, as will be shown below, for finite derivatives this is not true. We consider the \(C^*\)-algebras \({\mathscr H}_{N,M}\) consisting of \(N\)-dimensional finite differential operators with \(M\times M\)-matrix-valued bounded periodic coefficients. We show that any \({\mathscr H}_{N,M}\) is \(*\)-isomorphic to the universal uniformly hyperfinite algebra (UHF algebra) \( \bigotimes_{n=1}^{\infty}\mathbb{C}^{n\times n}. \) This is a complete characterization of the differential algebras. In particular, for different \(N,M\in\mathbb{N}\) the algebras \({\mathscr H}_{N,M}\) are topologically and algebraically isomorphic to each other. In this sense, there is no difference between multidimensional matrix valued PDEs \({\mathscr H}_{N,M}\) and one-dimensional scalar ODEs \({\mathscr H}_{1,1}\). So, the multidimensional world can be emulated by the one-dimensional one.

The effects of non-linear and stochastic terms, and extended algebras of integro-differential operators are also discussed. Some of the results are published in

[1] Anton A. Kutsenko (2017) Mixed multidimensional integral operators with piecewise constant kernels and their representations, Linear and Multilinear Algebra, DOI: 10.1080/03081087.2017.1415294