STDC 2025 will feature plenary and semi-plenary talks from several notable researchers.
In this talk, I will present recent results about the poset of hyperbolic structures on Thompson's group $F$. While the global structure of this poset is as simple as one would expect, the local structure turns out to be incredibly rich, in stark contrast with the situation for the $T$ and $V$ counterparts. I will focus on the subposet of quasi-parabolic hyperbolic structures, which contains uncountably many \emph{lamplike} structures, called so as they can be described combinatorially in terms of certain hyperbolic structures on related lamplighter groups. On the other hand, there are also many non-lamplike structures, showing the vastness and complexity of this poset. Lastly, I will talk about how these actions can be extended to more general Thompson's groups $F_n$ for $n \geq 2$. This is joint work with Francesco Fournier-Facio and Matthew C.B.Zaremsky.
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The Milnor-Wood inequality, introduced in two landmark papers by John Milnor (1958) and John W. Wood (1971), is a striking result at the intersection of geometry, topology, and dynamics. It establishes sharp bounds on the Euler number of flat $\mathbb{S}^1$-bundles over surfaces, revealing deep connections between geometric curvature and topological invariants. Milnor’s original inequality highlights the boundedness of Euler invariants for flat bundles with "linear" structures, which Gromov later generalized using bounded cohomology. Wood extended Milnor's result to "non-linear" flat circle bundles, offering a perspective rooted in 1-dimensional dynamics. In the 1980s, Étienne Ghys posed the intriguing question of whether Wood’s inequality could be extended to flat-oriented $\mathbb{S}^3$-bundles. In this talk, we will also discuss the surprising ways in which inequality fails in higher-dimensional non-linear cases, showcasing the new calculations in the bounded cohomology of diffeomorphism groups.
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