Geometric structures for differential equations and field theory

My recent work develops a coherent programme around k-contact geometry, jet bundles, integrable systems, Hamilton–Jacobi theory and geometric reduction. The aim is to identify intrinsic geometric structures that simultaneously encode PDEs, non-conservative dynamics, symmetries, transformations and reduction procedures, rather than treating these topics through separate formalisms.

The programme is now moving from foundational structures to recognition results, transformations, reductions and applications to field theories and integrable systems.

• Jet Bundles as Higher-Order Polarised k-Contact Manifolds, arXiv:2606.09263.
• Foundations on k-contact geometry, with X. Rivas and T. Sobczak, arXiv:2409.11001.
• Marsden–Meyer–Weinstein reduction for k-contact field theories, with X. Rivas, S. Vilariño and B.M. Zawora, arXiv:2505.05462.
• J. Lange, J. de Lucas and M. Krych, A Guide to Applications of k-Contact Geometry in Dissipative Field Equations, Journal of Physics A: Mathematical and Theoretical (2026), DOI: 10.1088/1751-8121/aeaf24; arXiv:2605.13313.
• J. de Lucas, J. Lange, X. Rivas and C. Sardón, Hamilton–Jacobi theory for non-conservative field theories in the k-contact framework, Journal of Physics A: Mathematical and Theoretical 59 (2026), 355202.

The complete publication list is available on the full information page.

This selection complements the recent-work section. It highlights two monographs, selected papers in major journals, and several publications that rank among my most cited works. The complete publication list and the current citation record are linked below.

Monographs

• J.F. Cariñena and J. de Lucas, Lie systems: theory, generalisations, and applications, Dissertationes Mathematicae 479 (2011), 1–162.
• J. de Lucas and C. Sardón, A Guide to Lie Systems with Compatible Geometric Structures, World Scientific (2020), 408 pp.

Selected journal articles

• J. de Lucas and A.M. Grundland, A cohomological approach to immersed submanifolds via integrable systems, Selecta Mathematica (N.S.) 24 (2018), 4749–4780.
• J. de Lucas and S. Vilariño, k-Symplectic Lie systems: theory and applications, Journal of Differential Equations 258 (2015), 2221–2255.
• A. Ballesteros, A. Blasco, F.J. Herranz, J. de Lucas and C. Sardón, Lie–Hamilton systems on the plane: properties, classification and applications, Journal of Differential Equations 258 (2015), 2873–2907.
• A.M. Grundland and J. de Lucas, A Lie systems approach to the Riccati hierarchy and partial differential equations, Journal of Differential Equations 263 (2017), 299–337.
• X. Gràcia, J. de Lucas, X. Rivas and N. Román-Roy, On Darboux theorems for geometric structures induced by closed forms, RACSAM 118 (2024), article 131.

Further high-citation landmarks

• J.F. Cariñena, J. de Lucas and M.F. Rañada, Recent Applications of the Theory of Lie Systems in Ermakov Systems, SIGMA 4 (2008), 031.
• J.F. Cariñena, J. de Lucas and C. Sardón, Lie–Hamilton systems: theory and applications, International Journal of Geometric Methods in Modern Physics 10 (2013), 1350047.
• A. Ballesteros, J.F. Cariñena, F.J. Herranz, J. de Lucas and C. Sardón, From constants of motion to superposition rules for Lie–Hamilton systems, Journal of Physics A 46 (2013), 285203.

Google Scholar profile and current citation record · Complete publication list

The Gamma Research Group is an internationally oriented research environment at the University of Warsaw, focused on geometric methods in mathematics and mathematical physics. It supports research on jets, symmetries, PDEs, field theories, geometric mechanics, integrable systems and modern reduction methods.

• Weekly online Gamma seminar, with recorded talks and lecture series on the Gamma Seminar YouTube channel.
• Hybrid Reading Groupoid sessions on Lie groupoids and Lie algebroids.
• Monographic courses on Geometric Mechanics I and II- Mechanics and Field Theory and student research projects.
• Strong collaborations with the Universitat Rovira i Virgili in Tarragona, UPC Barcelona, CRM Montréal and other institutions.

Selected conference organisation and network activity

• 2024 — XVIII International Young Researchers Workshop in Geometry, Dynamics and Field Theory, University of Warsaw, Warsaw.
• 2025 — Geometric Science of Information 2025, Saint-Malo: session on geometric approaches to differential equations.
• 2026 — 2nd International Workshop on Geometric Field Theory, University of Warsaw, Warsaw, 6–8 July 2026.
• 2026 — International Workshop on k-Contact Geometry and Applications, University of Łódź, 17–18 December 2026.
• 2027 — Geometric Science of Information, Saint-Malo: session on geometric approaches to differential equations in preparation.

I have supervised 18 defended bachelor's theses and 9 defended master's theses, with 2 further master's theses in progress. My doctoral supervision and co-supervision includes 3 defended theses, 1 thesis deposited and under evaluation, and 2 other ongoing doctorates. Julia Lange's doctoral examination is expected in December 2026 and her thesis defence in January 2027. Several of my students have received research prizes and academic distinctions, and some former students now hold permanent faculty positions. Recent student topics include k-contact geometry, supersymplectic geometry, Cartan bundles, orbifold reduction, Lie group symmetries, vector bundle invariants, Clifford algebras, Lie algebroids, energy-momentum methods and energy-Casimir methods.

Open the full students section

Current visible group and collaboration network includes B.M. Zawora, J. Lange, A. Maskalaniec, T. Sobczak, X. Rivas, S. Vilariño, A. Lopez-Gordon and other collaborators working on k-contact geometry, reductions, field theories, integrable systems and geometric mechanics.

• 2026 — Invited participant, Workshop on Recent Developments in Symplectic Dynamics, School of Mathematics, Institute for Advanced Study, Princeton.
• 2026 — Invited talk, Department of Mathematics, University of Illinois Urbana-Champaign.
• 2025–2026 — Expert evaluator for the Marie Skłodowska-Curie Actions Postdoctoral Fellowships (MSCA-PF), European Research Executive Agency.
• 2023 — Simons–CRM Professorship, Centre de Recherches Mathématiques, University of Montréal.
• 2021 — Dean Prize for the best researcher under 40, Faculty of Physics, University of Warsaw.
• 2011 — Special Award for Doctoral Theses, awarded in a university-wide competition across the natural and exact sciences at the University of Zaragoza.
• 2013, 2017, 2020 — University-wide didactic awards at the University of Warsaw; further student-council nominations in 2022 and 2024.

Complete publication, supervision, teaching and activity records are available on the detailed information page and in the detailed curriculum.

• Full publications
• Full student and thesis list
• Gamma Research Group
• Talks and slides
• Teaching materials
• Useful links
• Detailed curriculum