One day workshop in Geometry and Topology

One day workshop dedicated to recent advances in Differential Geometry and Topology, with a broad spectrum of topics.

Schedule (with schedule changes)

09:30  -  Reception / coffee break

10:00  -  António Salgueiro (CMUC, Uni. Coimbra): On the unknottability and splittability of 2-string tangles

11:00  -  Francesco Cattafi (Uni. Würzburg): From Poisson geometry to pseudogroups and geometric structures 

12:00  -  Lunch

14:00  -   Marko Stošić (CAMGSD, IST): Invariants of knots and 3-manifolds, and knots-quivers correspondence

15:00  -  Ivan Beschastnyi (CIDMA, Uni. Aveiro): Stochastic developments on sub-Riemannian manifolds

16:00  -  Coffee Break

Registration

The event is free and open to everyone. But please register to help us prepare the logistics. 

Location

Room 5.5, Department of Mathematics, University of Coimbra (How to get here).

Abstracts

A. Salgueiro: On the unknottability and splittability of 2-string tangles

A tangle T is a pair formed by a ball and collection of properly embedded disjoint arcs in B. Tangles are useful for decomposing links; one such example is the decomposition of a knot into prime factors. In this talk we will discuss the case where the unknot and split links have decompositions with a given tangle T as factor. As an application we obtain a complete classification for tangles up to seven crossings. This is joint work with João Nogueira (University of Coimbra). 

F. Cattafi: From Poisson geometry to pseudogroups and geometric structures

The space of (local) symmetries of a given geometric structure has the natural structure of a Lie (pseudo)group. Conversely, geometric structures admitting a local model can be described via the pseudogroup of symmetries of such local model.

This philosophy can be made precise at various levels of generality (depending on the definition of "geometric structure") and using different tools/methods. In this talk I will present a new framework, which include previous formalisms (e.g. that of G-structures) and allows us to prove integrability theorems. The novelty of this point of view consists however in the fact that it uncovers the (beautiful!) hidden structures behind Lie pseudogroups and geometric structures. Indeed, the relevant objects which make this approach work are Lie groupoids endowed with a multiplicative "PDE-structure", their principal actions and the related Morita theory. Poisson geometry provides the guiding principle to understand those objects, which are directly inspired from, respectively, symplectic groupoids, principal Hamiltonian bundles and symplectic Morita equivalence.

If time permits, I will also discuss applications to Cartan geometries.

This is based on joint works with Luca Accornero, Marius Crainic and María Amelia Salazar.

M. Stošić: Invariants of knots and 3-manifolds, and knots-quivers correspondence

In this talk I will give an overview of  knots-quivers correspondence, including the first formulation as well as recent developments.

The correspondence started with the quantum knot invariants (namely colored HOMFLY-PT polynomials), and it has been extended significantly to include  other knot invariants, like Gukov-Manolescu \hat{Z}-invariants of knot complements.

I will present some basic facts about these correspondences, as well as some of their surprising applications.

I. Beschastnyi: Stochastic developments on sub-Riemannian manifolds  

Organizers

• João Nogueira

• João Nuno Mestre