Robert P. Langlands

Editorial comments: This text was prepared as a complement to a lecture at the Conference on Beauty held at the Notre Dame Institute for Advanced Study in January, 2010.

Author's comments (2021-06-22): The paper as presented here is not the published paper. That was unfortunately modified, namely slightly abridged, by the editors without consulting the author and without his approval. The present paper, the original paper, is the preferred form.

Author's comments: This text is provisional from a mathematical point of view, but it may be some time before the obstacles described in the concluding sections are overcome. Serious progress has been made by Ali Altuğ.

Author's comments: There are two papers on the Bethe Ansatz, but the work is far from complete. I have always wanted to return not only to the algebraic geometrical arguments initiated in the second paper, which seem to me of considerable intrinsic mathematical interest as algebraic geometry, but also to the notion of Wellenkomplex and the Puiseux expansions introduced briefly at the end of the first paper. So far, I have not found the time.