In order for artificial neural networks to learn a task, one must solve an inverse design problem. What network will produce the desired output? We have harnessed AI approaches to design physical systems to perform functions inspired by biology, such as protein allostery. But artificial neural networks require a computer in order to learn in top-down fashion by the global process of gradient descent on a cost function. By contrast, the brain learns by local rules on its own, with each neuron adjusting itself and its synapses without knowing what all the other neurons are doing, and without the aid of an external computer. But the brain is not the only biological system that learns by local rules; I will argue that the actin cortex and the amnioserosa during the dorsal closure stage of Drosophila development can also be viewed this way.

The key role of physical and mechanical interactions in cancer emerges from a very large variety of data sources and methods - from genomics to bioimaging, from proteomics to clinical records. Thus, learning physics-driven relational information is crucial to characterize its progression at different scales.

In this talk I will discuss how mathematical and computational tools allow for learning  and better understanding of  the mechano-biology of cancer,  thanks to the integration of  patient-specific data and physics-based models. I will present a few applications developed in the last decade in which the development of  digital twins,  empowered by ad-hoc learning tools,  allows us to test new hypotheses,  to assess the model predictions against biological and clinical data, and to aid decision-making in a clinical setting.

Funding from MUR - PRIN 2020, Progetto di Eccellenza 2023-2027 and Regione Lombardia (NEWMED Grant, ID: 117599, POR FESR 2014-2020) is gratefully acknowledged.

Over the years, I've often taught a first course in asymptotics and perturbation methods, even though I don't know much about the subject. In this talk, I'll discuss a textbook example of a singularly perturbed nonlinear boundary-value problem that has revealed delightful new surprises, every time I teach it. These include a pitchfork bifurcation in the number of solutions as one varies the small parameter, and transcendentally small terms in the solutions' initial conditions that can be calculated by elementary means.

We formulate and solve a generalized inverse Navier–Stokes boundary value problem for velocity field reconstruction and simultaneous boundary segmentation of noisy Flow-MRI velocity images. We use a Bayesian framework that combines CFD, Gaussian processes, adjoint methods, and shape optimization in a unified and rigorous manner.
With this framework, we find the velocity field and flow boundaries (i.e. the digital twin) that are most likely to have produced a given noisy image. We also calculate the posterior covariances of the unknown parameters and thereby deduce the uncertainty in the reconstructed flow. First, we verify this method on synthetic noisy images of flows. Then we apply it to experimental phase contrast magnetic resonance (PC-MRI) images of an axisymmetric flow at low and high SNRs. We show that this method successfully reconstructs and segments the low SNR images, producing noiseless velocity fields that match the high SNR images, using 30 times less data.
This framework also provides additional flow information, such as the pressure field and wall shear stress, accurately and with known error bounds. We demonstrate this further on a 3-D in-vitro flow through a 3D-printed aorta and 3-D in-vivo flow through a carotid artery.

In this lecture, I will discuss the resolution of the Arthur-Barbasch-Vogan conjecture on the unitarity of special unipotent representations for any real form of a connected reductive complex Lie group, with contributions by several groups of authors (Barbasch-Ma-Sun-Zhu, Adams-Arancibia-Mezo, and Adams-Miller-van Leeuwen-Vogan). The main part of the lecture will be on the approach of the first group of authors for the case of real classical groups: counting by coherent families (combinatorial aspect), construction by theta lifting (analytic aspect), and distinguishing by invariants (algebraic-geometric aspect), resulting in a full classification, and with unitarity as a direct consequence of the construction.

I shall present a new flexible method showing that every countable model of PA admits a pointwise definable end-extension, one in which every point is definable without parameters. Also, any model of PA of size at most continuum admits an extension that is Leibnizian, meaning that any two distinct points are separated by some expressible property. Similar results hold in set theory, where one can also achieve V=L in the extension, or indeed any suitable theory holding in an inner model of the original model.