Seminários e cursos curtos

We present our solutions to two long standing open problems, one from probability theory formulated by Malyshev in 1970 and another one from a crossroad of geometry and dynamics, going back to Darboux in 1879. The Malyshev problem is of finding effective, explicit necessary and sufficient conditions in the closed form to characterize all random walks in the quarter plane with a finite group of the random walk of order 2n, for all n ≥ 2. Previously known results covered the cases n = 2, 3, and 4. We also describe all n-periodic Darboux transformations for four-bar link problems for all n ≥ 2, thus completely solving the Darboux problem, that he solved for n = 2, and which was recently extended to n = 3. The talk is based on a joint work with Milena Radnovic (arXiv:2512.21976).

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.

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).

Since the groundbreaking work of M. Gromov in the 1980s many tools have been developed for distinguishing open symplectic domains. However, until recently, similar questions in the contact geometric setup were largely open. For instance, it was not known whether there are open domains in the standard contact vector space of dimension $>3$ which are diffeomorphic but not contactomorphic to it (in dimension $3$ it is known that all of them are). In my lecture I will discuss Floer theoretic tools for answering this type of questions. As one application I will construct a continuous family of pairwise non-contactomorphic open balls in the standard contact ${\mathbb R}^5$. The lecture is based on a joint work in progress with K. Ajij, Mahan Mj, Dishant Pancholi and L. Polterovich.

Bartnik introduced a notion of quasi-local mass based on a minimization procedure over asymptotically flat extensions. After reviewing the fundamental ideas behind Bartnik's approach, I will introduce a spacetime version of a Bartnik-type quasi-local mass and establish its positivity and monotonicity in time for two classes of domains associated with black holes. This is joint work with L. Andersson, M. Khuri and W. Simon.