Speaker: Folkert Müller-Hoissen (Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany)
Date and time: 18.01.2023, 17:00 Moscow time (GMT +03:00)
Title: Higher Bruhat and Tamari orders, simplex and polygon equations
Abstract: We first present an introduction to Bruhat and higher Bruhat (partial) orders, highlighting the very simple underlying ideas. Higher Bruhat orders have been introduced by Manin and Schechtman in 1986, who also revealed them as the crucial structure behind the hierarchy of simplex equations, of which the famous Yang-Baxter equation and the tetrahedron (Zamolodchikov) equation are first members.
Via a certain decomposition of the higher Bruhat orders we arrive at "higher Tamari orders", as defined in a joint work with Aristophanes Dimakis (Tamari Memorial Festschrift, Progress in Mathematics, vol. 299, 2012, pp. 391-423), where this structure arose from an exploration of a class of soliton solutions of the Kadomtsev-Petviashvili (KP) equation, which form rooted binary trees at fixed time. We had conjectured that these higher Tamari orders are equivalent to what was known as higher Stasheff-Tamari orders (defined in terms of triangulations of cyclic polytopes), and this has recently been proved by Nicholas Williams (arXiv:2012.10371).
Following my work with Dimakis, SIGMA 11 (2015) 042, we explain how in the same way as the higher Bruhat orders determine the hierarchy of simplex equations, the higher Tamari orders determine a hierarchy of "polygon equations", of which the famous pentagon equation is a member.
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