Lisboa - April 2025

April 9, 2025, Department of Mathematics / CEMS.UL Universidade de Lisboa

C6 Building, room 6.2.33

Speaker: Pedro Boavida (Instituto Superior Técnico, University of Lisbon)
Title: Equivariant formality of the little disks operad
Abstract: The little disks operad is formal, which is to say, its rational homotopy type is determined by its cohomology ring. This theorem, due to Kontsevich, is a key input in many important results about the topology of manifolds (e.g. on embedding and diffeomorphism spaces and on finite-type invariants) and deformation theory (e.g. on the deformation quantization of Poisson manifolds). I will give an account of parts of this story and describe recent joint work with Joana Cirici and Geoffroy Horel showing that formality also holds when the orthogonal group action is taken into account.

Speaker: Rosa Sena-Dias (Instituto Superior Técnico, University of Lisbon)
Title: Conformally Kähler Ricci-flat toric metrics
Abstract: Biquard-Gauduchon recently classified all conformally Kähler, Ricci-flat, toric ALF metrics on the complement of smooth toric divisors in dimension 2. They deviced an ansatz for such metrics which uses an axi-symmetric harmonic function on R^3. Such harmonic functions can also be used in an ansatz for scalar-flat Kähler toric metrics on non-compact surfaces through which all such metrics, without assumptions on asymptotic growth, were classified. Our work grew out of an attempt to find a connection between these two contexts.
We will give background on toric geometry and conformally Kähler metrics. We will describe both classification results and discuss how to use the scalar-flat Kähler toric methods in the Biquard-Gauduchon setting to get a classification without assumptions on asymptotics. This is joint work with Gonçalo Oliveira.

Speaker: Helena Reis (University of Porto)
Title: Construction of algebraic vector fields out of projective structures
Abstract: Given a singular uniformizable projective structure on CP(1), we will  show how to associate it with a rational vector field with univalued  solutions.
This is a joint work with A. Elshafei and J. Rebelo.

Speaker: Tim Henke (University of Porto)
Title: The projective equivalence of the Hitchin and the Knizhnik–Zamolodchikov connections
Abstract: The Pauly Isomorphism identifies the geometric quantisation of the moduli space of parabolic bundles over a pointed Riemann surface with the space of conformal blocks associated to the same surface. This isomorphism is the mathematical formalisation of the physical CS/WZNW duality. Both spades depend crucially on the complex structure of the surface, but the physical duality is understood to be purely topological. This implies that the Pauly Isomorphism should be projectively flat with respect to variations of the complex structure. These are given by the Hitchin connection for the moduli space and the Knizhnik–Zamolodchikov connection on the sheaf of conformal blocks. For higher genera the equivalence follows from the non-parabolic case. We treat the genus 0 case that is crucial for developing topological invariants from the moduli theory.
I will explain the construction of the different spaces, define the bundles on both sides of the isomorphism and sketch the ideas that go into the proof.