500

Date | Jun 03, 2024 |
---|---|

Speaker | Eugen Rogozinnikov |

Dept. | Max Planck Institute for Mathematics in the Sciences |

Room | 129-301 |

Time | 17:00-18:15 |

※ 일시/장소

- 6월 3일(월), 17:00~18:15, 129동 301호

- 6월 5일(수), 17:00~18:15, 129동 309호

- 6월 11일(화), 17:00~18:15, 129동 301호

- 6월 12일(수), 17:00~18:15, 129동 301호

Abstract: Spectral networks were introduced in a seminal article by Davide Gaiotto, Gregory W. Moore and Andrew Neitzke published in 2013. These are networks of trajectories on surfaces that naturally arise in the study of various four-dimensional quantum field theories. From a purely geometric point of view, they yield a new map between flat connections over a Riemann surface and flat abelian connections on a spectral covering of the surface. At the same time, these networks of trajectories provide local coordinate systems on the moduli space of flat connections that are valuable in the study of higher Teichmüller spaces.

In the first part of this mini-course, I will review key concepts from geometric group theory, including hyperbolic groups and boundaries at infinity of hyperbolic groups and spaces. Following this, I will discuss the theory of vector bundles and the Riemann-Hilbert correspondence. In the second part, I will define spectral networks explicitly for surfaces with punctures. I will also present and discuss their most prominent applications in geometry: non-abelianization and abelianization, which connect higher Teichmüller spaces of a base surface to abelian character varieties of its ramified cover, the spectral curve.

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