Lisboa - February 2024

February 9, 2024, Mathematics Department / CAMGSD of Instituto Superior Técnico, Universidade de Lisboa

Speaker: André Oliveira (Universidade do Porto, Universidade de Trás-os-Montes e Alto Douro)
Title: Lie algebroid connections and Higgs bundles
Abstract: Out of a given holomorphic Lie algebroid L on a compact Riemann surface X, one can consider a corresponding lambda-connection on a vector bundle over X. This naturally degenerates onto a (twisted) Higgs bundle on X. Via a generalization of the classical construction by Simpson of lambda-connections, such degeneration induces an associated one at the level of moduli spaces, using the so-called L-Hodge moduli space. We use this to study geometric and topological properties of the moduli spaces of L-connections on X, as simple as their dimension or more complicated like their motivic class. This joint work with David Alfaya.

Speaker: Davide Masoero (Universidade de Lisboa)
Title: Affine Opers and Bethe Equations
Abstract: The ODE/IM correspondence is a conjectural and surprising link between nonlocal observables of integrable quantum field theories and monodromy data of linear analytic ODEs. Let g be a simple Lie algebra over the complex field, (g,1) the corresponding untwisted Kac-Moody algebra and Lan(g,1) the Langlands dual of (g,1). In 2011, extending previous results in the physics literature, B. Feigin and E. Frenkel conjectured that observables of the Quantum g-KdV model can be expressed in terms of monodromy data of a class of Lan(g,1)-affine opers known as Feigin-Frenkel-Hernandez opers. In this talk, we introduce the relevant objects (such as affine opers and Bethe equations) and describe the state-of-.the-art of the Feigin-Frenkel conjecture.

Speaker: Ana Cristina Castro Ferreira (Universidade do Minho)
Title: Geodesic completeness of pseudo-Riemannian Lie groups
Abstract: A striking difference between Riemannian and pseudo-Riemannian metrics is that pseudo-Riemannian ones often fail to be geodesically complete even in the compact case. We will present some developments in the classification of Lie groups with all their left-invariant pseudo-Riemannian metrics complete. More concretely, we will discuss the specifics of geodesic completeness when the manifold in question is a Lie group and recall the Euler-Arnold theorem as well as the seminal work of Marsden for the compact (homogeneous) case. We will see how an interpretation in Riemannian terms of his techniques provided us with tools for characterising completeness even for general manifolds. As for Lie groups, we will show how a certain notion of "linear growth'' allowed us to establish large classes of Lie groups whose left-invariant metrics are all complete. Time permitting, we will also discuss the generalisation of the Euler-Arnold formalism to the holomorphic-Riemann setting and discuss the classification of geodesic completeness for 3-dimensional (non-unimodular) Lie groups. This is a series of joint works with S. Chaib, A. Elshafei, H. Reis, M. Sánchez and A. Zeghib.

Speaker: João Nuno Mestre (Universidade de Coimbra)
Title: Deformations of complex Lie groupoids
Abstract: Lie groupoids can encode geometric objects such as smooth actions, and foliations; deformations of Lie groupoids also relate to deformations of these objects. In this talk we’ll first see the deformation cohomology of a (real) Lie groupoid and mention relations to deformations the mentioned examples. We will then see the cohomology controlling deformations of a complex Lie groupoid: it combines deformation cohomology of the groupoid structure and Kodaira-Spencer cohomology of the underlying complex manifold, via a double complex. The talk is based on ongoing work with Luca Vitagliano.