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Hitchin systems, spectral networks and noncommutative clusters

Hot Topics: Cluster algebras and wall-crossing March 28, 2016 - April 01, 2016

March 29, 2016 (04:00 PM PDT - 05:00 PM PDT)
Speaker(s): Andrew Neitzke (Yale University)
Location: MSRI:
Tags/Keywords
  • algebraic combinatorics

  • cluster algebras

  • Hitchin representation

  • Hitchin system

  • local systems

  • moduli spaces

  • hyperkahler

  • scattering diagram

  • parallel transport

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

14483

Abstract

Hitchin's integrable system is one of the fundamental examples of mirror symmetry in the sense of Strominger-Yau-Zaslow.

Many of the structures described in this workshop can be seen very concretely in this example. This example has moreover some extra structure, coming from the fact that it is actually a hyperkahler space.

I will review an approach to these moduli spaces which arose in my joint work with Davide Gaiotto and Greg Moore. The key player in the story is a set of cluster-type coordinate systems, very closely related to those appearing in the work of Fock-Goncharov. We will also see scattering diagrams very similar to those in the work of Gross-Hacking-Keel.

In the end I will describe a new point: when one tries to extend the cluster description over certain singular loci of the moduli space, the most natural description seems to involve not an ordinary cluster algebra, but rather a noncommutative version thereof. One special case of this story appears closely related to the "noncommutative marked surfaces" recently introduced by Berenstein-Retakh; there is also related work of Goncharov-Kontsevich.

Supplements
25691?type=thumb Neitzke Notes 273 KB application/pdf Download
Video/Audio Files

14483

H.264 Video 14483.mp4 387 MB video/mp4 rtsp://videos.msri.org/data/000/025/626/original/14483.mp4 Download
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