An Algebraic Approach to Linear ODEs

Erscheinungsjahr 2026
ISBN: 978-3-032-36200-1
Verlag: Springer Nature Switzerland AG

Buch, Englisch, 106 Seiten, Format (B × H): 155 mm x 235 mm

Reihe: SpringerBriefs in Mathematics

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.

Zielgruppe

Graduate

Autoren/Hrsg.

Marelli, Giovanni

Mugochi, Martin

Amukugo, Sigfried

Lazarus, Elise

Lichela, Jesse

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Mathematik | Informatik Mathematik Algebra

Inhaltsverzeichnis

Part I.       .- Chapter 1. Algebraic Preliminaries.- Chapter 2. Formal power series Over Q-algebras.- Chapter 3. Universal Solutions to Linear ODEs.- Chapter 4. Generalized Wronskians.- Chapter 5. Exponential of Matrices.- Part II       .- Chapter 6. Derivations on an Exterior Algebra.- Chapter 7. The Boson-Fermion Correspondence.

Über Autor(innen)

Born in Italy, Giovanni Marelli received a degree in physics from University of Milano and a PhD in mathematics from SISSA. After that, he held positions at the University of Salamanca, Kyoto University, the Korea Institute of Advanced Study, the University of Antioquia (Colombia), and the University of Namibia. His research interests are in algebraic geometry, including non-Archimedean geometry, homotopical algebra, and mirror symmetry.

Martin Mugochi  was born in Zimbabwe, where he received B.Sc. Honours and M.Phil. degrees in mathematics from the University of Zimbabwe (UZ), including a study visit at the University of Cambridge. He further received a PhD in mathematics from UNISA in South Africa. He started his academic career at the UZ and is now based at the University of Namibia. His research area is mainly in categorical topology, particulary working with frames/ locales and (sometimes) toposes.

Born in Namibia,  Sigfried Amukugo received a degree in mathematics from the University 'Marta Abreu' of Las Villas (in Cuba) and a master's degree in mathematics from the University of Botswana (in Botswana). After that, he held positions as a tutor and lecturer at the University of Namibia. He is also currently a PhD student under the supervision of Giovanni Marelli and Letterio Gatto. His research interests are in algebraic geometry and combinatorics.

Elise Lazarus , born in Namibia, holds an honors degree and a Master of Science in applied mathematics from the University of Namibia as well as a Master of Science in mathematics from the University of KwaZulu-Natal, South Africa. She is a lecturer and a PhD student in the Department of Computing, Mathematical, and Statistical Sciences at the University of Namibia, Windhoek. Her research interests include algebraic geometry, differential geometry, and topology.

Jesse Lichela was born in the Zambezi Region of Namibia. He received a Bachelor of Science degree in Mathematics and Computer Science and an Honours degree in Aeronautical Science from the University of Namibia, before earning a Master of Science degree in Pure Mathematics from the University of Zambia. He has extensive experience teaching mathematics and mentoring students at tertiary levels. His research interests lie in functional analysis, particularly ordered Banach algebras, and spectral theory, with a growing interest in Mathematics for Computer Science.