The Rose and the Lilly- Geometrical Incompatibilities as Morphogenetic Factors

• Sharon Eran (The Hebrew University of Jerusalem)

Room: Aula Dini

Asymptotic meshes from r-variational adaptation methods for static problems

• Heiner Olbermann (UCLouvain)

Room: Aula Dini We consider the minimization of integral functionals in one dimension and their approximation by r-adaptive finite elements. Including the grid of the FEM approximation as a variable in the minimization, we are able to show that the optimal grid configurations have a well-defined limit when the number of nodes in the grid is being sent to infinity. This is done by showing that the suitably renormalized energy functionals possess a limit in the sense of Gamma-convergence. We also show some numerical results that demonstrate the convergence in practice. Joint work with Darith Hun and Nicolas Moës (both UCLouvain).

Interactive liquid crystal polymers for haptics and soft robotics

• Danqing Liu (Technische Universiteit Eindhoven)

Room: Aula Dini

Towards a two-scale model for morphogenesis

• Axel Voigt (Technische Universität Dresden)

Room: Aula Dini We propose a two-scale model to resolve essential features of developmental tissue deformations. The model couples individual cellular behavior to the mechanics at tissue scale. This is realized by a multiphase-field model addressing the motility, deformability and interaction of cells on an evolving surface. The surface evolution is due to bending elasticity, with bending properties influenced by the topology of the cellular network, which forms the surface. We discuss and motivate model assumptions, propose a numerical scheme, which essentially scales with the number of cells, and explore computationally the effect of the two-scale coupling on the global shape evolution. The approach provides a step towards more quantitative modeling of morphogenetic processes.

Gaussian morphing at constant area by sliding in metasurfaces inspired by Euglena cells

• Marino Arroyo Balaguer (UPC Universitat Politècnica de Catalunya)

Room: Aula Dini A number of strategies have been proposed to overcome the constraint imposed by Gauss Egregium theorem regarding morphing of a thin sheet. These strategies often boil down to devising a metasurface whose subunits can accommodate non-uniform in-plane stretch, e.g. pneumatic channels, swellable materials, kirigami or origami patterns. I will discuss a different approach inspired in the way Euglena cells actively change their shape to crawl in confined environments, and the realization of this idea at macroscopic scales. I will also discuss a mathematical modeling of this morphing mechanism, and how it can be used to realize a broad range of surfaces with a single metasurface. Finally, I will discuss the mechanical properties of this class of metamaterial.

"Reliable and Sustainable AI: From Mathematical Foundations to Next Generation AI Computing"

• Gitta Kutyniok (LMU München)

Room: Aula Dini "The current wave of artificial intelligence is transforming industry, society, and the sciences at an unprecedented pace. Yet, despite its remarkable progress, today’s AI still suffers from two major limitations: a lack of reliability and excessive energy consumption. This lecture will begin with an overview of this dynamic field, focusing first on reliability. We will present recent theoretical advances in the areas of generalization and explainability - core aspects of trustworthy AI that also intersect with regulatory frameworks such as the EU AI Act. From there, we will explore fundamental limitations of existing AI systems, including challenges related to computability and the energy inefficiency of current digital hardware. These challenges highlight the pressing need to rethink the foundations of AI computing. In the second part of the talk, we will turn to neuromorphic computing - a promising and rapidly evolving paradigm that emulates biological neural systems using analog hardware. We will introduce spiking neural networks, a key model in this area, and share some of our recent mathematical findings. These results point toward a new generation of AI systems that are not only provably reliable but also sustainable."

Inflatable morphing matter

• Katia Bertoldi (Harvard University)

Room: Aula Dini Inflatable morphing matter represents a frontier in programmable architecture and soft robotics, enabling dramatic shape changes driven by simple pressure inputs. In this talk, I will present a unified vision for how instabilities and geometric design can be harnessed to create inflatable systems that morph, lock, and reconfigure on demand. Starting from the fundamentals of buckling, snapping, and bistability in curved membranes and shell structures, I will show how these nonlinear phenomena can be used not as failure modes, but as functional tools to decouple input (pressure) from output (shape), and achieve multistable target states.

Weaving Gaussian baskets and complex networks

• Alison Martin

Room: Aula Dini Weaving is an orderly entanglement; it has long been a way to deal with complexity and unorganized components. Leveraging the geometry-driven nature of weaving patterns enhances physical properties such as structural efficiency and elegance of the materials in the resulting shape. The versatility of weaving patterns means that these techniques can be pushed beyond traditional craft-based limits towards novel applications across disciplines.

Index Stability for Conformally Invariant Problems

• Francesca Da Lio (ETH Zürich)

Room: Aula Dini

On the free boundary elastic flow

• Anna Dall'Acqua

Room: Aula Dini We study the length-preserving elastic flow in arbitrary codimension with free boundary on hypersurfaces. This constrained gradient flow is given by a nonlocal evolution equation with nonlinear higher-order boundary conditions. We discuss how a suitable non-flatness assumption ensures global existence and subconvergence to critical points. This is joint work with Manuel Schlierf.

Designing Structure-Preserving Deep Learning: Insights from Analysis

• Carola Bibiane Schönlieb (University of Cambridge)

Room: Aula Dini I will discuss some of our recent works on structure-preserving deep learning for the design of neural networks with specific properties - such as non-expansiveness or 1-Lipschitz regularity - and their application to imaging and to the solution of partial differential equations.

The geometry of dissipative evolution equations

• Felix Otto (Max Planck Institute)

Room: Aula Dini Arnol'd pointed out that Euler's equations for an inviscid incompressible fluid can be seen as geodesic equations on the manifold of volume-preserving maps, when endowed with the ambient $L^2$ metric. On the opposite, overdamped, end of fluid dynamics, a density driven two-phase flow in a porous medium can be interpreted as a gradient flow of the potential energy with respect to a metric that models viscous dissipation on the level of Darcy's law, but is mathematically very similar to Arnol'd's. This dissipative metric induces a distance function on densities, which following Brenier can be interpreted as an optimal transportation problem, and is well-known in statistics. While the geometry on volume-preserving transformations has mostly negative curvature, the one on densities has non-negative sectional curvature.

The history behind Gauss’s Disquisitiones circa superficies curvas

• Alberto Cogliati (Università di Padova)

Room: Aula Dini The publication of Disquisitiones circa superficies curvas (1828) is widely regarded as marking the beginning of modern differential geometry. Although important results in the geometry of curves and surfaces had already been achieved during the 18th century, Gauss’s contribution inaugurated an entirely new phase in the development of the discipline. The composition of the Disquisitiones was the result of a long process of reflection and successive revisions that occupied Gauss—albeit intermittently—for well over a decade. Despite its brevity, the work stands out for the deliberateness of the techniques employed and the meticulous care with which they are presented. My contribution aims to explore the intellectual journey that led Gauss to the final drafting of this work, with particular attention to the discovery of the Theorema Egregium and the first version of its proof, which was completed exactly 200 years ago (1825).

Morphing of thin structures: 200 years after Gauss and 100 years after Timoshenko.

• Eran Sharon
• Antonio Desimone

Room: Aula Dini

Discrete differential geometry of quad meshes with applications in computational design

• Helmut Pottmann

Room: Aula Dini Discrete differential geometry of quad meshes has so far mainly been confined to discrete counterparts of special parameterizations of surfaces. Arbitrary quad meshes received much less interest, although they are very useful for a variety of applications. In the present talk we present a discrete first fundamental form and basics of curvature theory using the diagonal meshes of a quad mesh. This approach is well suited for discrete representations of mappings between surfaces. Our focus will be on isometric maps and their usage to model developable surfaces and to solve paneling problems in architectural geometry. For the latter, we assume bendable material and work with surfaces of constant Gaussian curvature. Surprisingly, one can achieve high quality results with only a very small number of molds. Moreover, we show how easily one can handle further constraints on meshes and present results on shape morphing with mechanical metamaterials.

Stress-Free Morphing

• Luciano Teresi (Università degli Studi Roma TRE)

Room: Aula Dini We study the morphing of 3D objects within the framework of non-linear elasticity with large distortions. A distortion field induces a target metric, and the configuration which is effectively realized by a material body is the one that minimizes the distance, measured through the elastic energy, between the target metric and the actual one. Morphing through distortions might have a paramount feature: the resulting configurations might be stress-free; if this is the case, the distortions field is called compatible. We maintain that the morphing through compatible distortions is a key strategy exploited by many soft biological materials, which can exhibit very large shape-change in response to distortions controlled by stimuli such as chemicals or temperature changes, while keeping their stress state almost null.

Ricci flow with $L^p$ bounded scalar curvature.

• Miles Simon (Universität Magdeburg)

Room: Aula Dini

A Cahn-Hilliard-Willmore energy for non-oriented interfaces

• Simon Masnou (Université Claude-Bernard Lyon 1)

Room: Aula Dini The Cahn-Hilliard energy is a celebrated phase-field model for the smooth approximation of the area of domain’s boundaries. Its L2 gradient flow provides an excellent approximation, both theoretically and numerically, of the smooth mean curvature flow. In this talk, I will present a new model for approximating the area of general interfaces not associated with any interior domain, which we call non-oriented. This model was obtained by analyzing the structure of certain neural networks capable of simulating mean curvature motion for non-oriented interfaces. I will show that, instead of using neural networks, one can adopt a more classical variational approach combining a Cahn-Hilliard-type functional with an appropriate non-smooth potential and a Willmore-type stabilizing energy. I will describe some theoretical properties of this model in dimension one, and for radial functions in arbitrary dimension. A simple numerical scheme can be designed to approximate the L2 gradient flow of the model, so I will present several numerical experiments illustrating, at least formally, the connection between this new model and the mean curvature flow of interfaces of codimension 1 or 2 in space dimensions 2 and 3. It is a joint work with E. Bretin (INSA Lyon) and A. Chambolle (CNRS & Paris-Dauphine).

Shape transitions in frustrated elastic ribbons

• Maria Giovanna Mora (Università di Pavia)

Room: Aula Dini Ribbons are elastic bodies that are thin and narrow. Many ribbons in nature, from seed pods to molecular assemblies, have a non-trivial internal geometry, making them incompatible with Euclidean space. In many cases, this results in shape transitions between narrow and wide ribbons with the same internal geometry. In this talk we will show how this phenomenon can be explained mathematically in terms of the Gauss-Codazzi equations from surface theory. We will present some recent rigorous results joint with Cy Maor (Hebrew University of Jerusalem) and discuss some open questions.

Quantitative rigidity for almost isometric maps between Riemannian manifolds

• Stefan Müller (Universität Bonn)

Room: Aula Dini A classical result (‘Liouville’s theorem’) states that a sufficiently regular map u in Euclidean space whose differential Du belongs to the group SO(n) of orientation preserving isometries at every point is affine. The quantitative version of the result states that for maps of a bounded connected set U with Lipschitz boundary the L2 distance of the differential Du from a constant can be bounded in terms of the L2 distance of the differential from the set SO(n). This result is a crucial ingredient in the analysis of the rigidity and flexibility of thin elastic objects. In the talk, which is based on joint work with Sergio Conti and Georg Dolzmann, I will discuss possible generalizations of the quantitative estimate in a Riemannian setting.