Ivan Smalyukh

January 16, 2026

Title: Artificial matter from knots: solitons and vortices in chiral media

Abstract: Topology is key for understanding properties of many natural material systems. Moreover, topology can be used as an important design principle to create artificial materials with properties not encountered in nature. This lecture will discuss how stable solitonic and vortex knots in chiral liquid crystals, colloids and magnets can exhibit atom-like behavior, including fusion, fission as well as self-assembly into various crystals and other forms of artificial matter [1-5]. The molecular and host medium's chirality play important roles in enabling stability of the spatially localized knotted solitons, the hopfions, and vortex structures, illustrating a hierarchical interplay of chirality effects. The unusual crystals of self-assembled knots exhibit giant electrostriction, facile reconfigurability of lattice symmetries and other properties never encountered in conventional forms of matter. These experimental demonstrations and theoretical/computational findings will let us revisit and admire the beautiful history of the early model of atoms by Kelvin, Maxwell and Tait, turning this model from a blunder to a new topological metamaterial design approach. I will then show that these vortices interact with light similar to what was previously predicted for the elusive cosmic strings, with knots and crystalline arrays of vortices allowing to spatially localize non-spreading laser beams into closed loops and knots, potentially paving the way to cosmology-inspired and knot-theory-guided optical engineering.

[1] D. Hall, J.-S. B. Tai, L. Kauffman and I. I. Smalyukh. Nature Physics doi.org/10.1038/41567-025-03107-0 (2025).
[2] J.-S. B. Tai and I. I. Smalyukh. Science 365, 1449-1453 (2019).
[3] C. Meng, J.-S. Wu, and I. I. Smalyukh. Nature Materials 22, 64–72 (2023).
[3] H. Zhao, J.-S. B. Tai, J.-S. Wu, and I. I. Smalyukh. Nature Physics 19, 451–459 (2023).
[4] R. Voinescu, J.-S. B. Tai and I. I. Smalyukh. Phys Rev Lett 125, 057201 (2020).
[5] P. J. Ackerman and I. I. Smalyukh. Nature Mater 16, 426-432 (2017).

Keywords: hopfion, knot, vortex, liquid crystal, magnet, chirality, metamaterial.

Patrizio Frosini

January 30, 2026

Title: On the Role of Group Equivariant Non-Expansive Operators as a Bridge between TDA and Machine Learning

Abstract: Group Equivariant Non-Expansive Operators (GENEOs) were introduced ten years ago as a tool to reduce and modulate the invariance of persistence diagrams (originally valid for every reparameterization of the signal domain) [1]. The computation of persistence diagrams itself can be seen as an example of a GENEO. Subsequently, these operators have been independently studied and employed in various applications in data analysis and machine learning [2-7]. In this talk, we will illustrate the definitions and basic properties of the main concepts used in GENEO theory, while also highlighting their promising applications in TDA and Explainable Artificial Intelligence.

[1] Patrizio Frosini, Grzegorz Jabłoński, Combining persistent homology and invariance groups for shape comparison, Discrete & Computational Geometry, vol. 55 (2016), n. 2, pages 373-409. DOI: 10.1007/s00454-016-9761-y.
[2] Mattia G. Bergomi, Patrizio Frosini, Daniela Giorgi, Nicola Quercioli, Towards a topological-geometrical theory of group equivariant non-expansive operators for data analysis and machine learning, Nature Machine Intelligence, vol. 1, n. 9, pages 423 433 (2 September 2019). DOI: 10.1038/s42256-019-0087-3.
[3] Giovanni Bocchi, Stefano Botteghi, Martina Brasini, Patrizio Frosini, Nicola Quercioli, On the finite representation of linear group equivariant operators via permutant measures, Annals of Mathematics and Artificial Intelligence, vol. 91 (2023), n. 4, 465 487. DOI: 10.1007/s10472-022-09830-1.
[4] Giovanni Bocchi, Patrizio Frosini, Massimo Ferri, A novel approach to graph distinction through GENEOs and permutants, Scientific Reports, 15 (2025), 6259. DOI: 10.1038/s41598-025-90152-7.
[5] Giovanni Bocchi, Patrizio Frosini, Alessandra Micheletti, Alessandro Pedretti, Gianluca Palermo, Davide Gadioli, Carmen Gratteri, Filippo Lunghini, Akash Deep Biswas, Pieter F.W. Stouten, Andrea R. Beccari, Anna Fava, Carmine Talarico, GENEOnet: A breakthrough in protein binding pocket detection using group equivariant non-expansive operators, Scientific Reports, 15 (2025), 34597. DOI: 10.1038/s41598-025-18132-5.
[6] Raúl Felipe, GENEOs with respect to the projective Hilbert metric, The Journal of Geometric Analysis, vol. 35 (9) (2025), 264. DOI: 10.1007/s12220-025-02102-4.
[7] Diogo Lavado, Alessandra Micheletti, Giovanni Bocchi, Patrizio Frosini, Cláudia Soares,
SCENE-Net: Geometric induction for interpretable and low-resource 3D pole detection with Group-Equivariant Non-Expansive Operators, Computer Vision and Image Understanding, vol. 262 (2025), 104531. DOI: 10.1016/j.cviu.2025.104531.

Keywords: TDA, machine learning, GENEO.