In this new paper (**arXiv:2007.16069**) the exact analytical solution in closed form of a modified SIR system is presented. This is, to the best of our knowledge, the first closed-form solution for a three-dimensional deterministic compartmental model of epidemics. In this dynamical system the populations S(t) and R(t) of susceptible and recovered individuals are found to be generalized logistic functions, while infective ones I(t) are given by a generalized logistic function times an exponential, all of them with the same characteristic time. The nonlinear dynamics of this modified SIR system is analyzed and the exact computation of some epidemiologically relevant quantities is performed. The main differences between this modified SIR model and original SIR one are presented and explained in terms of the zeroes of their respective conserved quantities. We recall that both models have been recently used in order to describe the essentials of the dynamics of the COVID-19 pandemic.

# Generalized noncommutative Snyder spaces and projective geometry

Given a group of kinematical symmetry generators, one can construct a compatible noncommutative spacetime and deformed phase space by means of projective geometry. This was the main idea behind the very first model of noncommutative spacetime, proposed by H.S. Snyder in 1947. In this framework, spacetime coordinates are the translation generators over a manifold that is symmetric under the required generators, while momenta are projective coordinates on such a manifold. In these proceedings (**arXiv:2007.09653**) we review the construction of Euclidean and Lorentzian noncommutative Snyder spaces and investigate the freedom left by this construction in the choice of the physical momenta, because of different available choices of projective coordinates. In particular, we derive a quasi-canonical structure for both the Euclidean and Lorentzian Snyder noncommutative models such that their phase space algebra is diagonal although no longer quadratic.

# Hamiltonian structure of compartmental epidemiological models

In this new paper (**arXiv:2006.00564**), any epidemiological compartmental model with constant population is shown to be a Hamiltonian dynamical system in which the total population plays the role of the Hamiltonian function. Moreover, some particular cases within this large class of models are shown to be bi-Hamiltonian. New interacting compartmental models among different populations, which are endowed with a Hamiltonian structure, are introduced. The Poisson structures underlying the Hamiltonian description of all these dynamical systems are explicitly presented, and their associated Casimir functions are shown to provide an efficient tool in order to find exact analytical solutions for epidemiological dynamics.

# The κ-Newtonian and κ-Carrollian algebras and their noncommutative spacetimes

In this work (**arXiv:2003.03921**) we derive the non-relativistic c→∞ and ultra-relativistic c→0 limits of the κ-deformed symmetries and corresponding spacetime in (3+1) dimensions, with and without a cosmological constant. We apply the theory of Lie bialgebra contractions to the Poisson version of the κ-(A)dS quantum algebra, and quantize the resulting contracted Poisson-Hopf algebras, thus giving rise to the κ-deformation of the Newtonian (Newton-Hooke and Galilei) and Carrollian (Para-Poincaré, Para-Euclidean and Carroll) quantum symmetries, including their deformed quadratic Casimir operators. The corresponding κ-Newtonian and κ-Carrollian noncommutative spacetimes are also obtained as the non-relativistic and ultra-relativistic limits of the κ-(A)dS noncommutative spacetime. These constructions allow us to analyze the non-trivial interplay between the quantum deformation parameter κ, the curvature parameter η and the speed of light parameter c.

# Seminar “Momentum Spaces of Arbitrary Signature for kappa-Minkowski”

Speaker:** Flavio Mercati (Università di Napoli Federico II, Italy)**

Date and time: February 5th, 12:30 h

Place: Aula 14, Facultad de Ciencias

# Lorentzian Snyder spacetimes and their Galilei and Carroll limits from projective geometry

In this new paper (**arXiv:1912.12878**) we show that the Lorentzian Snyder models, together with their non-relativistic (c→∞) and ultra-relativistic (c→0) limiting cases, can be rigorously constructed through the projective geometry description of Lorentzian, Galilean and Carrollian spaces with nonvanishing constant curvature. The projective coordinates of these spaces take the role of momenta, while translation generators over the same spaces are identified with noncommutative spacetime coordinates. In this way, one obtains a deformed phase space algebra, which fully characterizes the Snyder model and is invariant under boosts and rotations of the relevant kinematical symmetries. While the momentum space of the Lorentzian Snyder models is given by certain projective coordinates on (Anti-) de Sitter spaces, we discover that the momentum space of the Galilean (Carrollian) Snyder models is given by certain projective coordinates on curved Carroll (Newton–Hooke) spaces. This exchange between the non-relativistic and ultra-relativistic limits emerging in the transition from the geometric picture to the phase space picture is traced back to an interchange of the role of coordinates and translation operators. As a physically relevant feature, we find that in Galilean Snyder spacetimes the time coordinate does not commute with space coordinates, in contrast with previous proposals for non-relativistic Snyder models, which assume that time and space decouple in the non-relativistic limit. This remnant mixing between space and time in the non-relativistic limit is a quite general Planck-scale effect found in several quantum spacetime models.

# Seminar “Ideal tetrahedra and their duals”

Speaker:** Catherine Meusburger (FAU Erlangen-Nürnberg)**

Date and time: October 2nd, 16:00 h

Place: Aula 24, Facultad de Ciencias

# Seminar “Superintegrability and structure of higher rank quadratic algebras”

Speaker:** Ian Marquette (University of Queensland)**

Date and time: September 25th, 16:00 h

Place: Aula 24, Facultad de Ciencias

# Coreductive Lie bialgebras and dual homogeneous spaces

Quantum homogeneous spaces are noncommutative spaces with quantum group covariance. Their semiclassical counterparts are Poisson homogeneous spaces, which are quotient manifolds of Lie groups M=G/H equipped with an additional Poisson structure π which is compatible with a Poisson-Lie structure Π on G. Since the infinitesimal version of Π defines a unique Lie bialgebra structure δ on the Lie algebra 𝔤=Lie(G), in this new paper (**arXiv:1909.01000**) we exploit the idea of Lie bialgebra duality in order to introduce the notion of dual homogeneous space of a given homogeneous space M=G/H with respect to the Lie bialgebra δ. Then, by considering the natural notions of reductive and symmetric homogeneous spaces, we extend these concepts to the dual space thus showing that an even richer duality framework arises. In order to analyse the physical implications of this new duality, the case of M being a Minkowski or (Anti-) de Sitter Poisson homogeneous spacetime is fully studied, and the corresponding dual reductive and symmetric spaces are explicitly constructed in the case of the well-known κ-deformation, where the cosmological constant Λ is introduced as an explicit parameter in order to describe all Lorentzian spaces simultaneously. In particular, the fact that the dual space is reductive is shown to provide a natural condition for the representation theory of the quantum analogue of M that ensures the existence of physically meaningful uncertainty relations between the noncommutative spacetime coordinates. Finally we show that, despite the dual spaces are not endowed in general with an invariant metric, their geometry can be described by making use of K-structures.

# The κ-(A)dS noncommutative spacetime

The (3+1)-dimensional κ-(A)dS noncommutative spacetime is explicitly constructed in this new paper (**arXiv:1905.12358**) by quantizing its semiclassical counterpart, which is the κ-(A)dS Poisson homogeneous space. Under minimal physical assumptions, it is explicitly proven that this is the only possible generalization to the case of non-vanishing cosmological constant of the well-known κ-Minkowski spacetime. The κ-(A)dS noncommutative spacetime is shown to have a quadratic subalgebra of local spatial coordinates whose first-order brackets in terms of the cosmological constant parameter define a quantum sphere, while the commutators between time and space coordinates preserve the same structure of the κ-Minkowski spacetime. When expressed in ambient coordinates, the quantum κ-(A)dS spacetime is shown to be defined as a noncommutative pseudosphere.