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K-State Today

October 2, 2014

Math lecture today about abelianization, spectral networks

By Reta McDermott

Andy Neitzke, University of Texas, Austin, will present "Abelianization and Spectral Networks" as part of the Mathematics Department Colloquium 52nd William J. Spencer Lecture at 2:30 p.m. Thursday, Oct. 2, in 101 Cardwell Hall.

The abstract for the lecture is:

The question, "how do we describe a generic matrix up to conjugation?" has a well known and canonical answer: we describe it by its eigenvalues. In contrast, the question "how do we describe two generic matrices up to simultaneous conjugation?" looks much more difficult, since we cannot simultaneously diagonalize two generic matrices. Nevertheless, it turns out that there is a reasonable answer to this question, and to many generalizations thereof. The answers are purely algebraic but can be well understood in terms of certain networks of curves on surfaces, called "spectral networks." Spectral networks turn out to have many other applications as well, e.g. to the analysis of differential equations depending on a small parameter, and from there to hyperkahler geometry and the theory of Donaldson-Thomas invariants. I will survey this story. It is mainly joint work with Davide Gaiotto and Greg Moore, heavily influenced by related work of Kontsevich-Soibelman and Fock-Goncharov.