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.

The Hopfield Neural Network has played, ever since its introduction in 1982 by John Hopfield, a fundamental role in the inter-disciplinary study of storage and retrieval capabilities of neural networks, further highlighted by the recent 2024 Physics Nobel Prize. From its strong link with biological pattern retrieval mechanisms to its high-capacity Dense Associative Memory variants and connections to generative models, the Hopfield Neural Network has found relevance both in Neuroscience, as well as the most modern of AI systems. Much of our theoretical knowledge of these systems however, comes from a surprising and powerful link with Statistical Mechanics, first established and explored in seminal works of Amit, Gutfreund and Sompolinsky in the second half of the 1980s: the interpretation of associative memories as spin-glass systems. In this talk, we will present this duality, as well as the mathematical techniques from spin-glass systems that allow us to accurately and rigorously predict the behavior of different types of associative memories, capable of undertaking various different tasks.

The Becker-Döring equation is among the wide class of coagulation-fragmentation equations and is one of the earliest descriptions of particle growth in the theory of nucleation from supersaturated vapour. This model describes the growth and decay of clusters, consisting of identical monomers, only by the addition and removal of monomers.

In this talk, motivated by enzymatic reactions in biology, we will introduce and analyse a variant of the Becker-Döring equations; this model incorporates irreversible fragmentation and monomer injection. After some recollection of coagulation-fragmentation equations, we will first present theoretical results on our model; we will establish global existence and uniqueness under some suitable conditions, and then we will focus on the long-time behaviour of our solution. Finally, we will present an efficient scheme that preserves the asymptotic and allows fast computation by sub-sampling the clusters.