The question addressed in this talk is that of finding the truth value of a given conditional statement. The speaker finds the answers found in Classical Logic, the strict conditional and Stalnaker’s Theory of Conditionals to be unsatisfactory and proposes a new approach to the problem, which allows for the assessment of the truth value of both indicative and counterfactual conditionals and is based on the provision of an argument that links the consequent and antecedent.

Nonequilibrium stationary measures are generally not known explicitly and depend on the specific microscopic dynamics of the system. For some interacting particle systems, however, they admit an exact representation in terms of the Matrix Product Ansatz (MPA), an algebraic construction based on ordered products of matrices.

In this talk, we provide a probabilistic characterization of the class of probability measures that can be represented by the MPA. We introduce a constructive procedure, based on a suitable enlargement of the state space, showing that a probability measure admits a representation in terms of non-negative matrices via the MPA if and only if it can be expressed as a mixture of inhomogeneous product measures, where the mixing law is given by a Markov bridge.

We illustrate this construction through several examples of interacting particle systems. Finally, we discuss how the resulting probabilistic structure can be exploited to obtain large deviation principles for this class of measures.

In the classical theory of fluid mechanics, Newtonian fluids are characterized by a linear relationship between the stress tensor and the symmetric part of the velocity gradient, leading to the standard Navier-Stokes model. However, many complex materials, such as polymers, gels, and certain biological fluids, exhibit nonlinear rheological behavior better described by power-law models, where the viscosity depends on the magnitude of the shear rate. In this framework, the case $p<2$ corresponds to shear-thinning fluids, whose effective viscosity decreases as the shear rate increases. These nonlinear models naturally lead to evolutionary systems involving the $p$-Laplacian operator or its variants, and introduce analytical challenges not present in the Newtonian setting.

In [1,2], we study the well-posedness of a parabolic $p$-Laplacian system with a convective term, derived from the power-law system in the subquadratic case ($p<2$), by replacing the symmetric gradient with the full gradient and eliminating the pressure term. It should be noted that these modifications make the results less relevant from a Fluid Dynamics perspective, since the corresponding constitutive law does not comply with the principle of material invariance. Nevertheless, they are useful to better delimit the expectations for possible results in the fluid dynamics context.

We establish existence and a maximum principle for regular solutions (for $p \in \left(\frac{3}{2}, 2\right)$) and weak solutions (for $p \in \left(1, 2\right)$) for an initial datum $v_\circ (x) \in L^\infty (\Omega)$; for regular solutions we analyze the property of extinction in a finite time under suitable smallness assumptions on the initial datum. Moreover, for $v_\circ (x) \in L^\infty (\Omega)\cap W^{1,2}_0(\Omega),$ we are able to prove the uniqueness of regular solutions for $p\in \left(\frac{5}{3}, 2\right)$.

The talk is based on two joint works with Francesca Crispo and Michael M. Růžička.

[1] F. Crispo, A.P. Di Feola, On a parabolic p-Laplacian system with a convective term, Annali di Matematica Pura ed Applicata (1923 -), 204, (2025), no.3, 1119–1146.

[2] A.P. Di Feola, M. Růžička, Existence of global weak solutions to a parabolic $p$-Laplacian problem with convective term, arXiv:2510.05847, (2025).

In this talk we examine some aspects of the classical-quantum correspondence induced by symplectic maps and metaplectic operators. We first recall the notion of the metaplectic group, the double covering of the symplectic group.

We then extend this construction to the complex setting and define the metaplectic semigroup associated with the semigroup of positive complex symplectic linear maps. In this context, we review the various definitions appearing in the literature, notably those due to M. Brunet and P. Kramer, L. Hörmander, and R. Howe.

We finally establish several properties of the metaplectic semigroup, with particular emphasis on applications to time-frequency analysis and to evolution equations with complex quadratic Hamiltonians.

This talk is based on joint work with G. Giacchi, M. Malagutti, A. Parmeggiani and L. Rodino.

Since a landmark paper by Jordan, Otto, and Kinderlehrer (98'), it is now well known that some evolution PDEs, such as diffusion and advection PDEs, can be interpreted as gradient flows with respect to the Wasserstein distance. Since then, there have been ongoing efforts to integrate various evolutionary processes into this framework. In this talk, I will introduce the Moran process and its high popuation limit the Kimura equation, and explain how they are related to Wasserstein gradient flows. Indeed, the degeneracy of the diffusion at the boundaries leads to the study of a conditioned version of the dynamics, that can be seen as a Wasserstein gradient flow, with degenerate underlying geometry, involving the Shahshahani metric. Finally, we will see that the dissipation induced by the hidden gradient flow in the continuous setting is a good approximation of the dissipation in the discrete setting in its high population limit.