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I will give a gentle introduction to the moduli space of graphs and its fine moduli space cousin known as Outer Space. This moduli space of graphs has many applications to various branches of mathematical physics, algebraic geometry, and geometric group theory. It is a natural object to consider while studying Feynman amplitudes in parametric space, and it can be seen as the configuration space of one-dimensional quantum gravity. I will explain how this moduli space of graphs recently became the largest provider of information on the homology of the moduli space of curves of genus g and how associated graph complexes can be used to shed light on the 'dark-matter problems' of these moduli space's cohomology.

I will give a gentle introduction to the moduli space of graphs and its fine moduli space cousin known as Outer Space. This moduli space of graphs has many applications to various branches of mathematical physics, algebraic geometry, and geometric group theory. It is a natural object to consider while studying Feynman amplitudes in parametric space, and it can be seen as the configuration space of one-dimensional quantum gravity. I will explain how this moduli space of graphs recently became the largest provider of information on the homology of the moduli space of curves of genus g and how associated graph complexes can be used to shed light on the 'dark-matter problems' of these moduli space's cohomology.

This talk is based on my recent joint works arXiv:2405.18625, arXiv:2307.03831 with Joaquin Liniado and Florian Girelli.
Based on Lie 2-groups, I will introduce a 3d topological-holomorphic integrable field theory W, which can be understood as a higher-dimensional version of the Wess-Zumino-Witten model. By studying its higher currents and holonomies, it is revealed that W is related to both the raviolo VOAs of Garner- Williams --- a type of derived higher quantum algebra --- and the lasagna modules of Manolescu-Walker-Wedrich --- a type of 4d higher-skein invariant. I will then analyze the Noether charges of W, and prove that its symmetries are encoded by a derived version of the Kac-Moody algebra. If time allows, I will discuss how W enjoys a certain notion of "higher Lax integrability".