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Lecture Notes: Zagreb 2026

Topological Textures: theory and mathematical analysis, simulation and experiment · Training School, 11–14 May 2026

a sheet with a knot, a vortex lattice and a braid
Lecture Notes — Training School — Zagreb 2026

Topological Textures: Theory and Mathematical Analysis, Simulation and Experiment

The second POLYTOPO Training School was held at the Ruđer Bošković Institute in Zagreb on 11–14 May 2026. Its lectures treated topological textures across a wide range of physical systems, from quantum vortices and anyons to solitons and knots in soft matter, with an emphasis on theoretical, mathematical and computational methods, informed by experiment. The lecturers have kindly shared their notes, which you can download below.

11–14 May 2026 Zagreb, Croatia 4 lecturers 11 sets of notes
Lecturers · notes
Michael Sigal

University of Toronto, Canada

Quantum vortices and vortex lattices

Four lectures, in three sets of notes.

PDFLectures 1–2 Abelian gauge theories and equivariant states444 kB PDFLecture 3 The Ginzburg–Landau equations457 kB PDFLecture 4 The Kohn–Sham equation (density functional theory)3.1 MB
Ivan Smalyukh

University of Colorado Boulder, USA, and WPI-SKCM², Hiroshima University, Japan

Topological solitons and knotted vortices in soft matter and magnetic solids

Four lectures, each with its own set of notes.

PDFLecture 1 Topological solitons and knotted vortices in soft matter and magnetic solids13 MB PDFLecture 2 Order parameter spaces, homotopy theory and classification of defects in liquid crystals17 MB PDFLecture 3 Entanglement and dynamics of liquid-crystal vortices23 MB PDFLecture 4 Chirality-enabled knot solitons and knotted vortices as topological metaatoms45 MB
Gareth Alexander

University of Warwick, U.K.

Geometric topology and textures in soft matter

Three lectures, each with its own set of notes.

PDFLecture 1 Textures in the plane14 MB PDFLecture 2 Escape from the plane9.3 MB PDFLecture 3 Hopfions and chiral topology14 MB
Douglas Lundholm

Uppsala University, Sweden

Mathematics of the 2D anyon gas

Four lectures in one set of notes: quantum statistics and transmutation; local exclusion and stability for ideal anyons; the almost-bosonic, extended, interacting anyon gas; non-abelian anyons and topological quantum computing.

PDFAll lectures Mathematics of the 2D anyon gas5.9 MB
Previous school ← Budapest 2025. Physics and Mathematics of Topological Textures. Notes by Markus Garst and Radu Ignat, and the source code of Philipp Rybakov. Overview All lecture notes of the POLYTOPO Training Schools
The school

The programme, the venue and the organisers are on the page of the Training School on Topological Textures: theory and mathematical analysis, simulation and experiment, Zagreb, 11–14 May 2026.

The notes

The notes are the work of the lecturers, who kindly made them available. Please credit the author when you use them.

Lecture Notes