Tropical geometry and logarithmic compactifications of reductive algebraic groups

In this talk I will present two approaches towards the tropicalization of a reductive algebraic group G, one via Mumford’s toroidal compactification, the other via de Concini and Procesi’s wonderful compacitification. The Bruhat-Tits building of G and its root system will play a crucial role in both approaches. Using these insights I will propose two corresponding logarithmic compactifications of G. The first approach will provide us with a new logarithmic perspective on toric (and more generally parabolic) vector bundles, the other will allow us to study the geometry of the free group character variety at infinity, thereby providing evidence for the geometric P=W conjecture. Depending on the preferences of the audience I might also engage in some wild speculations concerning a yet-to-be-discovered logarithmic incarnation of Simpson’s non-abelian Hodge correspondence. Parts of this talk are based on ongoing joint work with Lorenzo Fantini and Alex Kuronya.

The synchronous discussion for Martin Ulirsch’s talk is taking place not in zoom-chat, but at https://tinyurl.com/2021-06-04-mu  (and will be deleted after ~3-7 days).