Jordan’s Math Work

Jordan's Math Work
Jordan Ellenberg, New York Times Bestselling Author, at the University of Wisconsin-Madison May 12, 2021. Photo by Lauren Justice

Jordan’s Math Work on numerous mathematical topics, contributing to virtually every discipline studied during that era. His works encompassed four volumes and ranged from finite groups to topology.

Jordan has long been a strong proponent of CNAS transfer students and continues to foster academic interests through the digital agriculture fellowship he sponsors in his spare time. Furthermore, Jordan Math Games encourage parental involvement by offering detailed progress tracking and performance analytics for parent engagement purposes.

Jordan’s Math Work Finite groups

Similar to many of his French contemporaries, Jordan began his mathematical career as an engineer despite making ample time available for research. Indeed, much of Jordan’s groundbreaking work in finite groups and, later on, linear and multilinear algebra was accomplished during this phase of his professional life as an engineer.

In 1855 he entered the Ecole Polytechnique. This prestigious establishment also provided engineering training; thus allowing him to make money while pursing his mathematical interests – giving him more time for research than those who had left school to focus solely on mathematics.

He published his first work in 1857 under the name Sur le nombre des valeurs des fonctions T and then again in 1861 with his work entitled Determinants; both works allowed him to develop an important combinatorial approach towards symmetries.

Between 1860 and 1870 he pursued studies of finite groups. Although he achieved some notable results, his main aim of classifying finite abelian groups eluded him; instead he developed a classification system for primitive permutation groups which showed there are only finitely many primitive permutation groups of any given size at any one c > 1.

At this same time, he published his groundbreaking first book on group theory: the Traite des substitutions et equations algebraiques T (Traite des substitutions and Equations Algebraiques T). It quickly established itself as one of the definitive works on permutation groups and was the main textbook on group theory until Burnside’s book appeared over 100 years later.

Linear and multilinear algebra

Taylor and Francis publishes Linear and Multilinear Algebra as a journal that covers an expansive variety of linear and multilinear algebra topics, boasting an SJR score of 0.631. The publication boasts high levels of citations with its high SJR rating of over 63,000!

Jordan was widely respected for his influential contributions to linear algebra and matrix theory, including developing the determinant and matrix functions, group theory, tensors, tensors theory and his major text Traite des substitutions et de des equations algebraiques in 1870 which provided the first comprehensive examination of Galois theory – receiving him the Poncelet Prize of the Academie des Sciences for doing so.

In 1880, he married Marie-Isabelle Munet and continued his research and published many papers on elliptic curves. Additionally, he became a member of the French Academy of Science while making several significant contributions to mathematics.

An important achievement was Jordan Normal Form Theorem, an efficient method for solving matrix equations. Additionally, Jordan developed the Jordan-Gaussen Pivoting Elimination Method to detect squared errors in surveying; and also pioneered Lie algebra research with work on Tensors which helped Albert Einstein understand precession of Mercury’s Perihelion orbit in 1915.

The theory of numbers

Jordan found great enjoyment studying various academic subjects during his time at UCR, such as algebra, discrete mathematics and linear algebra. Additionally, he took up category theory – an area of mathematical study which studies mathematical structures and their relationships to one another – thereby honing his critical thinking abilities and developing resilience against challenges.

Jordan also made significant strides in the theory of numbers, defining numbers as sets with an unbounded cardinal number and developing Jordan’s theorem; both these concepts lay the groundwork for modern number theory. Furthermore, he pioneered topology research through publishing works that combined geometry, number theory, and topology.

Jordan also introduced Jordan content, a measure which determines how closely two sets are approximated – an alternative to more commonly-used distance measures between sets. Jordan content can provide a powerful way of comprehending complex mathematical problems across many fields such as engineering, computing, and science – even measuring program complexity!


Topology is a fascinating topic that appeals to both laypeople and specialists, with applications including knot theory and manifolds. If you want a basic overview, I recommend reading up on point-set topology; otherwise if your interest extends further into other topics then consider buying an expanded book on that subject matter.

Following Fuchs’ result on linear differential equations, Jordan studied finite subgroups of the general linear group of nxn matrices over complex numbers. His investigations led him to discover topological properties of certain spaces that he called Jordan normal forms; these became essential tools in solving matrix equations like Ax=b which are important in both physics and engineering.

Gispert-Chambaz suggests that Jordan used topology only as an additional feature in volume 3 in his initial edition; in subsequent editions however, analysis became the main topic. This change may have been inspired by his teaching more advanced courses on analysis during this time and encouraged him to include Jordan curve theorem at the start of volume 3, thus setting an introductory tone for future analytical books.

Jordan made a significant contribution to topology through his creation of the concept of connectedness. This distinguished it from separation, an idea first presented by Cantor. Jordan showed that any set is connected if and only if it cannot be divided apart

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