Seminários

We continue the series of talks on the paper by Gross and Zagier. A key ingredient in the paper are the so-called local symbols, which associate to each pair of coprime divisors of degree zero on a projective smooth variety a real number. This function satisfies many good properties. The first step towards understanding these symbols is to first define local heights on projective varieties, which is the topic of this talk. More concretely, I will show a method of associating to each Cartier divisor class D, in a way compatible with the group structure, a series of real-valued functions (modulo bounded functions) called the local heights associated to D, which are defined only outside the support of D. I will finish by showing how these functions assemble into a globally defined height on the variety.

Coherence theorems aim to replace complex/weak structures with (suitably) simpler/stricter equivalent ones -- a principle exemplified by MacLane’s coherence theorem for monoidal categories. In the 1970s, Max Kelly proposed a framework to study coherence in two-dimensional universal algebra, establishing the following conjecture: every pseudoalgebra for a 2-monad on the 2-category Cat should admit an equivalent (universal) strict replacement. This conjecture initiated what we call ``the two-monadic approach to coherence''. The first results in this direction were given by Power, and were subsequently extended by Blackwell–Kelly–Power and Lack. In ``On Biadjoint Triangles'' and ``On Lifting of Biadjoints and Lax Algebras'', we formulate and analyze biadjoint triangles, two-dimensional analogues of Dubuc’s adjoint triangles. Among the direct consequences of our results, we get necessary and sufficient conditions for strict replacements of pseudoalgebras, characterizing precisely when Kelly’s conjecture on strict replacements hold -- not only for pseudoalgebras but also for lax algebras. This talk will introduce examples within the setting of biadjoint triangles, emphasizing their role in the two-monadic approach to coherence.

Amongst complex manifolds, Stein manifolds are notable for the property that the topological classification of their vector bundles coincides with the holomorphic one. That is, two vector bundles over a Stein manifold are holomorphically equivalent if and only if they are topologically equivalent. For this and other reasons, Stein manifolds are fundamental to understanding the theory of complex manifolds in general. This property may also be viewed as a manifestation of Gromov's $h$-principle, for those who like using such language. The purpose of this talk is to provide a short introduction to the theory of Stein manifolds, as presented by Grauert--Remmert, with a view to presenting the essential ideas behind the proof of this property.

By Alexander's theorem, every link in the 3-sphere can be represented as the closure of a braid. Lorenz links and twisted torus links are two families that have been extensively studied and are well-described in terms of braids. In this talk, we will present a natural generalization of Lorenz links and twisted torus links that produces all links in the 3-sphere. This provides a simpler braid description for all links in the 3-sphere.