An Algebraic Approach to Linear ODEs

The book offers a purely formal algebraic perspective on solutions of linear ODEs, carried out by working with rings of formal power series with coefficients in a Q-algebra, based on the observation that solutions of linear ODEs with complex coefficients are analytic. This approach highlights the relationships with Schubert calculus, with the theory symmetric functions, as well as representation theory. The book presents a construction of a universal basis of solutions for linear homogeneous ODEs, made of formal power series whose coefficients depend on the coefficients of the given ODE, and a procedure to derive solutions in general. The motivation for the treatment of ODEs in the book stems from Schubert calculus for Grassmannians and the theory of symmetric functions: the formalism of derivations on a Grassmann algebra is analogous to that of generalized Wronskians associated to bases of solutions of linear ODEs and their derivative, which leads to a kind of Giambelli–Jacobi–Trudi formula for Wronskians. The formalism is then extended to linear ODEs of infinite order and its relationship with the representation theory of infinite dimensional Lie algebras: the universal Jacobi–Trudi formula corresponds to the Boson–Fermion correspondence. The book includes numerous exercises and is addressed to students with a background in algebra interested in a different perspective of linear ODEs and its relationships with Schubert calculus or representation theory. It can also provide an accessible introduction to researchers.