We first investigate the interconnection of invariants of certain group actions and time-reversibility of a class of two-dimensional polynomial systems with 1:-1 resonant singularity at the origin. 

The time-reversibility is related to the Sibirsky subvariety of the center (integrability) variety and it is known that every time-reversible system has a local analytic first integral at the origin. 

We propose a new  algorithm to obtain a generating set for the Sibirsky ideal of such polynomial  systems and investigate  some algebraic properties of this ideal.

Likelihood Geometry is the geometric study of the maximum likelihood estimation problem to an algebraic variety serving as a statistical model. In this talk, basic notions of algebraic statistics and likelihood geometry will be introduced before discussing the maximum likelihood degree of a toric variety (also known as a log-linear model in the statistics context) in more detail.

Any Poisson bracket on a differentiable manifold can be written locally as the sum of two commuting Poisson brackets: a Poisson bracket of
a symplectic form and a Poisson bracket that vanishes at a point. This is the so-called Weinstein splitting. The result is strictly local; in
general, it is impossible to globalize the two local Poisson structures.  It turns out that for holomorphic Poisson structures on compact Kähler
manifolds admitting a simply connected compact symplectic leaf, the local Weinstein splitting globalizes and produces a global splitting of

The tunneling effect predicted by B.Josephson (Nobel Prize, 1973) concerns the Josephson junction: two superconductors

separated by a narrow dielectric. It states existence of a supercurrent through it and equations governing it. The overdamped Josephson junction

is modeled by a family of differential equations on 2-torus depending on 3 parameters. In this talk we are going to study these parameters.

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 and have presence in applications in science. 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 with such properties. 

This is joint work with António Salgueiro.

We are interested in equivariant embeddings of homogeneous spaces. Given the standard action
of C^2 on itself, we look for complex surfaces M acted upon by C^2 so as M should include an
invariant copy of C^2 where the restriction of the (extended) action recovers C^2
acting on itself. In particular, for a given equivariant C^2-embedding,  we shall provide a local classification of the action  by means of a description of the corresponding local transformation group.

This is joint work with Julio Rebelo, Helena Reis and Ana Cristina Ferreira.

 Sasaki manifolds are the odd dimensional analogues of Kähler manifolds. In this talk, I will discuss the equivariant index of the horizontal Dolbeault complex on toric Sasaki manifolds. We show that the index localises to certain closed Reeb orbits and can be expressed as a sum over lattice points of the moment cone. This kind of index problems have appeared recently in the evaluation of partition functions of certain supersymmetric gauge theories. 

On a symplectic manifold $M$ with a hamiltonian action by a group $K$, it is natural to perform geometric quantization with respect to $K$-invariant polarizations. We first consider the case of the cotangent bundle $T^\ast K$ (based on arXiv:2301.10853), where parallel transport along the geodesics yields a realization of the Peter--Weyl Theorem.

In the second part, we dicuss the universal role of construction for the cotangent bundle plays in the parametrization of invariant Mabuchi geodesic rays for general hamiltonian $K$-manifolds.

After reviewing the formalism of Simpson's shapes for the non-abelian Hodge theory associated to a smooth projective curve X, we will describe the construction of a certain class of functors between the derived categories of the Dolbeault, De Rahm and Betti shapes of X into the derived categories of the corresponding moduli stacks. We shall describe as well some of their main properties, in particular their interaction with the so-called Wilson operators.

This is a report on the thesis of Robert Hanson.