Monographic lecture course

Duality, Descent & Defects
The Higher Geometry and Category Theory of Charged Dynamics (I)

Academic year 2026/2027 — the winter semester

• Description: What, how & to what end
  
• Lecturer: Rafał R. Suszek
• Mode: hybrid
• Time & venue: Wednesdays, 10:15–12:00 (Warsaw time), room ???
• Basis for crediting: an oral exam or a report upon completion of a self-study project.
• Self-study projects:
  
  • Multiplicative forms, weak Dirac structures & cohomology..
  • The Lu-Weinstein groupoid & Poisson-Lie dualities.
  • The Poisson σ-model in the language of graded geometry.
  • Poisson defects (branes and bi-branes) in the coaugmented Poisson σ-model.
  • The Cattaneo-Felder derivation of the Kontsevitch formula.
  • The Cartan–Borel construction — from the homotopy model of the orbispace through equivariant cohomology to equivariant structures on n-gerbes.
  • Higher stacks & descent, abstractly.
  • Non-isometric groupoidal gauging in the Polyakov-Alvarez-Gawędzki σ-model.
  • Current symmetries in the WZNWG σ-model with defects and their higher-geometric lift.
  • The gerbe theory of loop dualities (e.g., the T-duality).
  • Quantum groups as symmetry models in the two-dimensional Rational Conformal Field Theory.
  • The inverse Higgs mechanism and the Hughes-Polchinsky duality.
  • The Hopf index — the fibre-bundle and cohomology perspectives.
  • ...
• Lecture notes:
  • I semester
    • Introduction (recording I)
    • Lectures I i II —
• Literature
  • R.R.S., Gaugings of groupoids, strings in shadows, and emergent Poisson σ-models, arXiv 2603.20914 [hep-th]
  • Thomas Strobl & R.R.S., Principaloid bundles, arXiv 2503.09886 [math.DG]
  • Kirill Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, Vol. 213, Cambridge University Press, 2005.
  • Ieke Moerdijk & Janez Mrčun, Introduction to Foliations and Lie Groupoids, Cambridge Studies in Advanced Mathematics, Vol. 91, Cambridge University Press, 2003.
  • Eckhard Meinrenken, Lie Groupoids and Lie Algebroids, Lecture Notes, Fall 2017, University of Toronto.
  • Marius N. Crainic & Rui L. Fernandes, Integrability of Poisson Brackets, J. Diff. Geom. 66 (2004) pp. 71-137.
  • Loring W. Tu, An Introduction to Manifolds, Universitext, Springer, 2010.
  • Rui L. Fernandes, Lectures on Differential Geometry, World Scientific, 2024.
  • Peter W. Michor, Topics in Differential Geometry, Graduate Studies in Mathematics, Vol. 93, American Mathematical Society, 2008.
  • Ivan Kolář, Peter W. Michor & Jan Slovák, Natural Operations in Differential Geometry, Springer, 1993.
  • Peter W. Michor, Gauge Theory for Fiber Bundles, Extended version of a series of lectures held at the Institute of Physics of the University of Napoli, March 28—April 1, 1988.
  • Yvonne Choquet-Bruhat, Cécile DeWitt-Morette & Margaret Dillard Bleick, Analysis, Manifolds and Physics, Part I, II, Elsevier, 1996-2000.
  • Shoshichi Kobayashi & Katsumi Nomizu, Foundations of Differential Geometry, vols. 1-2, John Wiley & Sons, 1963-69.
  • John M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, Vol. 218, Springer, 2013.
  • Борис А. Дубровин, Сергей П. Новиков, Анатолий Т. Фоменко, Modern Geometry - Methods and Applications, vol. 1,2,3, Graduate Texts in Mathematics, Vol. 93, 104, 124, Springer, 1984-90 (a translation of the original work Современная геометрия, Методы и приложения, Том 1, 2, 3, Наука, 1979).
  • Raoul Bott & Loring W. Tu, Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, Vol. 82, Springer, 1982.
  • Loring W. Tu, Introductory Lectures on Equivariant Cohomology, Annals of Mathematics Studies, Vol. 204, Princeton University Press, 2020.
  • Johan L. Dupont, Simplicial de Rham cohomology and characteristic classes of flat bundles, Topology 15 (1976) pp. 233-245
  • Marius N. Crainic, Maria A. Salazar, & Ivan Struchiner, Multiplicative forms and Spencer operators, Mathematische Zeitschrift 279 (2015) pp. 939–979
  • Nicolas Bourbaki, Éléments de mathématique. Livre II: Algèbre, chapitres 1 à 3, Springer, 2007.
  • Paul Moritz Cohn, Algebra. Volumes I-III, John Wiley & Sons, 1982-1991.
  • Serge Lang, Algebra, Springer, 2002.
  • Saunders Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics Vol. 5, Springer, 1971.
  • Tom Leinster, Basic Category Theory, Cambridge Studies in Advanced Mathematics Vol. 143, Cambridge University Press, 2014.
• Materiały uzupełniające
  • R.R.S., „Algebra. Podstawy”
  • R.R.S., „Niezbędnik różniczkowo–geometryczny”
  • R.R.S., „Niezbędnik cartanowski” — Wykłady II, III i IV z Teorii Grup II (2022/23)
  • R.R.S., „Lech, Čech i de Rham — studium kompleksów odwiecznych”
  • R.R.S., The Gauge Principle, The DGMP2026 Lectures on the Gauging of Groupoidal Symmetry, Equivariant Cohomology, Stacks, Loops, Fish, and All That

Retro Redintegratio novissima 10/10/2026 confecta est.