His first significant work, published in 1896, was on theta functions ( these are special functions of several complex variables; they are important in many areas, including the theories of abelian varieties and moduli spaces, and of quadratic forms). He proposed as a generalization of eigenvalues, the concept of the spectrum of an operator, in an 1897 paper; the concept was further extended by David Hilbert and now it forms the main object of investigation in the field of spectral theory (Wikipedia gives an interesting and quite good description for spectral theory: “spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of systems of linear equations and their generalizations.The theory is connected to that of analytic functions because the spectral properties of an operator are related to analytic functions of the spectral parameter“).
Moreover, he collaborated with Kurt Reidemeister (he is best known for his interests in combinatorial group theory, combinatorial topology, geometric group theory, and the foundations of geometry). on knot theory, showing in 1905 how to compute the knot group (Knot theory is inspired by knots which appear in daily life in shoelaces and rope, a mathematician’s knot differs in that the ends are joined together so that it cannot be undone. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space). Also, he was one of the editors of the Analysis section of Klein’s encyclopedia.
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