CMUC Differential Geometry
PhD seminar
Wednesday, June 11th 2025, Departamento de Matemática da Universidade de Coimbra
Wednesday, June 11th 2025, Departamento de Matemática da Universidade de Coimbra
This seminar consists of talks of doctoral students of CMUC and IST Lisboa about their PhD projects. Each talk has a maximum duration of 50 minutes.
10:30–11:10 Gonçalo Oliveira
Poissonization of Jacobi manifolds and explicit bi-realization
Symplectic and Poisson manifolds provide the setting for conservative classical mechanics (via the study of Hamiltonian systems). Related to them, contact manifolds provide a setting for the study of dissipative systems, among other applications. Jacobi manifolds generalize contact structures, allowing for degeneracies, in a similar way to how Poisson manifolds generalize symplectic manifolds. Hamiltonian systems on Jacobi manifolds are useful to study time-dependent or non-conservative Hamiltonian systems.
11:10–12:00 Žan Grad
Yang–Mills theory for multiplicative Ehresmann connections
Classical Yang–Mills theory is a theoretical cornerstone of elementary particle physics. Roughly speaking, it is an application of calculus of variations to the theory of principal bundles, seeking to minimize the norm of the curvature of principal bundle connections. In this talk, we present a broad generalization of this framework to Lie groupoids and Lie algebroids, where principal bundle connections are replaced with (infinitesimal) multiplicative Ehresmann connections. This extends the classical theory to the non-integrable and non-transitive setting, capturing the dynamics of gauge fields in both longitudinal and transverse directions relative to the (singular) orbit foliation. We will present several examples, highlight the novel phenomena in the theory, and discuss connections to other known generalizations.
12:00–13:30 Lunch break
13:30–14:20 Pedro Pessoa
G-structures
A G-structure is meant to encompass several interesting geometric structures on manifolds. I will go through two definitions of G-structure, one more classical and the other using the soldering form, and show that they are equivalent. Then, I will define morphisms of G-structures and show that this definition is adequate by seeing that it reproduces well known notions of morphism for specific choices of G.
14:20–15:10 Lennart Obster
Linearising Lie algebroids
Linearisation of a particular geometric structure comes down to finding “adapted” tubular neighbourhoods. One can try to construct such a tubular neighbourhood using the flow of a smartly chosen vector field. It can be complicated to find such vector fields, but often they arise for a reason: rigidity. Proving rigidity amounts to detecting isolated points of the moduli space, whose points represent isomorphism classes. Assuming the “tangent space” at a point in this space is zero, one can use a type of inverse function theorem to prove rigidity. We will focus on making this last procedure more concrete and explain the progress that we made in the (ongoing) work of proving a rigidity theorem for Lie algebroids.
15:10–15:40 Coffee break
15:40–16:30 Sebastian Daza
Equivariant deformation problems and homotopy operators
When studying an algebraic or geometric structure (e.g. a Lie algebra/algebroid), it is fruitful to understand the space of all possible structures (e.g. the space of Lie algebra(oid) structures). One way to make sense of this statement is by identifying an ambient space of almost structures and an equation on this space that characterizes the actual structures. In this talk, the space of structures is given as the zeros of a section of a vector bundle E over M. Accordingly, the space of structures inherits the topology of M (the space of almost structures). We are interested in the following question: When does this topology admits locally a smooth structure? We will see how to associate to this problem an algebraic structure, namely an L_infty algebra, and use homotopy operators to provide answers to this question.