(Politecnico di Milano)
(Imperial College)
(Università degli Studi di Napoli Federico II)
(Université de Picardie J. Verne)
(Politecnico di Milano)
(Università degli Studi di Brescia)
(Université de Lille)
(Université Le Havre Normandie)
Abstract:
Neurodegenerative diseases are characterised by progressive neuronal dysfunction and loss, leading to cognitive and motor impairment. Despite major advances in neuroscience, the mechanisms governing disease onset, progression, and patient-to-patient variability remain only partially understood. In this talk, I will present a mathematical perspective discussing how mathematical modelling, scientific computing, and computational learning can help to better understand the dynamics of neurodegeneration. The proposed mathematical models integrate mechanistic descriptions of disease dynamics with advanced computational methods and clinical data across multiple spatial and temporal scales. In the second part of the lecture, I will discuss how computational learning approaches can be combined with physics-based modelling to predict individual trajectories from sparse, irregularly sampled, and multimodal clinical data. The lecture will highlight how mathematics can contribute to a better understanding of neurodegenerative diseases and support personalised care.
Abstract:
The fast dynamo conjecture, posed by Zeldovich and Sakharov and later recorded in Arnold’s book of problems, asks whether a smooth, autonomous, divergence-free flow on the three-torus can amplify magnetic fields exponentially at a rate bounded uniformly away from zero as the magnetic resistivity vanishes. In this talk, we resolve the conjecture by constructing a flow based on the classical stretch–fold–shear (SFS) mechanism. The proof uses anisotropic Banach spaces adapted to the ideal dynamics to obtain an isolated unstable eigenmode with a distributional eigenfunction. A singular spectral perturbation argument then shows that this instability persists for small positive resistivity. The construction is robust and yields an open set of smooth autonomous fast dynamos.
Abstract:
Calderón–Zygmund operators on rough Euclidean domains possess intrinsic difficulties absent from the whole-space setting. Among these, the associated operators exhibit genuinely noncancellative boundary contributions reflecting the geometry and irregularity of the boundary.
Our wavelet-resolution approach splits the operator into an almost-diagonal wavelet part and a family of cancellative and noncancellative paraproducts. The latter are handled through a Poincaré-type identity and sparse domination, leading to testing conditions expressed in terms of Triebel–Lizorkin and tree Carleson norms. Our final result is a sharp characterization of weighted Sobolev boundedness, replacing classical cancellation assumptions by substantially weaker testing conditions.
A particularly important application is to the Beltrami equation, where we extend the Astala–Iwaniec–Saksman strategy to higher Sobolev regularity, bypassing the qualitative compactness and commutator methods traditionally used to obtain higher regularity on domains. This yields quantitative estimates for the Beltrami resolvent and Caccioppoli estimates for quasiregular mappings, as well as quantitative Sobolev regularity on sufficiently regular domains.
Finally, I will describe how the resulting planar Beltrami elliptic estimates enter the analysis of the Coleman–Gurtin heat equation with rough anisotropic conductivity and memory. Partly joint work with A.W. Green and B.D. Wick.
Abstract:
We study the Liouville equation $-∆u = e^u$ on a complete, connected, non-compact, boundaryless Riemannian surface $(M,g)$ with non-negative Ricci curvature. Assuming only some asymptotic lower bound on the solution, we establish optimal classification results for both the solutions and the ambient manifold. Our results reveal a close connection between the volume growth of the manifold and the classification of both the solutions and the underlying manifold.
This is a joint work with Giulio Ciraolo and Michele Gatti.
Abstract:
We prove the existence of forces and smooth initial data for which the associated Leray-Hopf solution to the three-dimensional Navier-Stokes equations is unique and global in time, satisfies the energy equality, and exhibits infinitely many (countably many) blow-up instants within a finite time interval. We consider several classes of forces and different blow-up strengths, all of which are shown to be sharp with respect to known boundedness results in the literature. Our method also yields examples of blow-up solutions for the forced three-dimensional Euler equations. This is joint work with Paolo Galdi.
Abstract:
The innovative approach is based on a general form of the Clausius-Duhem inequality (really, an equality) where the entropy flux and the entropy production rate are given by constitutive functions. Thermodynamic restrictions and a suitable splitting of the entropy and material constitutive functions transform the Clausius-Duhem inequality into an evolutionary partial differential equation. In particular, we develop a thermodynamically-consistent model of pseudo-elasticity in a shape memory alloy under a tensile stress. The evolution properties are described by using the temperature, the martensite fraction, and the stress as independent variables. As a result, both temperature-induced and stress-induced phase transitions and their related hysteretic loops are carefully modeled by properly choosing the free energy, "dynamic functions", and the entropy production rate. Next, to also take into account spatial diffusion, a generalization is given by letting the constitutive function depend on appropriate gradients within a Lagrangian and an Eulerian formulation. Both formulations are allowed by the occurrence of the extra-entropy flux that turns out to be proportional to the pertinent rates of temperature, stress, and mass fraction.
Abstract:
We consider a damped forced nonlinear Klein-Gordon Schrödinger (KGS) system known as the parametric KGS system with Yukawa coupling.
We prove the existence of a global attractor in dimension $1$ and $2$ for this system. We point out that this system is not a dissipative perturbation of a Hamiltonian system and involves an equation that acts as a nonlinear infinite memory term for the dynamics.
We also prove that this global attractor is regular and has finite fractal dimension.
These results are joint work with Marilena Poulou (West Attica, Greece)
Abstract:
Our aim in this talk is to discuss several variants of the Cahn-Hilliard equation proposed to describe phase separation processes in binary alloys. These models are based on mechanical considerations (e.g., microforces or microconcentrations).
We focus here on the so-called strict separation property when considering the thermodynamically relevant logarithmic potentials.
The workshop will take place at:
Aula Consiglio - 7th floor, Building 14,
Politecnico di Milano, Via Bonardi 9, 20133 Milano (Italy)
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