BGSMath course
Introduction to Double Affine Hecke Algebras and applications
Jasper Stokman, Universiteit van Amsterdam
Marta Mazzocco, ICREA-CRM-UPC
25 January – 11 February 2027
Mondays and Thursdays, 16:00–18:00
Barcelona School of Building Construction (EPSEB), Universitat Politècnica de Catalunya (UPC)
Six two-hour sessions
Introduction to Double Affine Hecke Algebras and applications
to February 11, 2027
About the course
In the nineties Ivan Cherednik introduced the double affine Hecke algebra (DAHA) as a tool to prove the famous Macdonald constant term conjectures. By now the representation theory of the DAHA has led to important applications in areas like harmonic analysis, algebraic combinatorics, quantum topology and mathematical physics.
This course develops the DAHA theory from two complementary angles. Part I focuses on the role of DAHA in mathematical physics. It starts with recalling the well-known role of the quantum Yang–Baxter equation in quantum topology and in the theory of quantum integrable systems. Subsequently we discuss a general procedure on how representation theory of the algebra D of difference-reflection operators leads to explicit solutions of quantum Yang–Baxter-type equations. We then introduce the DAHA and show how it provides natural families of D-representations. Then we introduce the associated quantum integrable many body systems, which are spin Ruijsenaars systems, and discuss their eigenstates, which are vector-valued analogs of the celebrated Macdonald polynomials.
Part II introduces generalized DAHAs (GDAHA) attached to star-shaped affine Dynkin diagrams and shows how Fock–Goncharov higher Teichmüller theory furnishes explicit presentations of these algebras via quantum cluster X-varieties; the course closes by taking the q ↦ 1 limit and exhibiting the resulting connection to the Painlevé equations.
Part I
- The braid group and the quantum Yang–Baxter equation
- Affine symmetric group, the algebra of difference-reflection operators and related non-abelian cohomology
- The quantum Yang–Baxter equation with spectral parameter and quantum Knizhnik–Zamolodchikov equations
- DAHA and its embedding in the algebra of difference-reflection operators
- DAHA and spin Ruijsenaars systems
- Vector-valued Macdonald polynomials
Part II
- Fock–Goncharov coordinates and higher Teichmüller theory
- Generalized DAHA (GDAHA) attached to star-shaped affine diagrams
- Presenting GDAHA via quantum cluster X-varieties
- The q ↦ 1 limit and Painlevé equations
Lecturers
Jasper Stokman
Korteweg-de Vries Institute for Mathematics, Universiteit van Amsterdam
Jasper Stokman is Professor of Mathematics, holding the chair in Lie theory, at the Korteweg-de Vries Institute for Mathematics, where he is a member of the research cluster Geometry and Quantum theory. He works in Lie theory, on representations of Lie groups, quantum groups and affine and double affine Hecke algebras, and in integrable systems, from quantum many-body systems and lattice models to conformal field theory. He also studies classical, basic and elliptic hypergeometric functions associated to root systems. He is an editor of the journal SIGMA and, with Tom Koornwinder, edited Multivariable Special Functions, the second volume of the Encyclopedia of Special Functions: The Askey-Bateman Project.
Photo © Gillissen
Marta Mazzocco
ICREA-CRM-UPC
Marta Mazzocco is an ICREA Research Professor at the Universitat Politècnica de Catalunya (UPC). A mathematical physicist, she works on integrable systems, at the crossroads of analysis, geometry, mathematical physics and algebra, and her interests include isomonodromic deformations and their quantisation, (quantum) Teichmüller theory, double affine Hecke algebras and basic orthogonal polynomials. She did her doctoral work at SISSA. She previously held the Chair in Mathematics at the University of Birmingham (2018–2025) and professorships at Loughborough University and the University of Manchester, and at Cambridge she was the first woman appointed by open call to a University Lectureship in Pure Mathematics. With Eva Miranda, she co-founded the SYMCREA Excellence Cluster.
Schedule
Six two-hour sessions, on Mondays and Thursdays.
Monday 25 January
Thursday 28 January
Monday 1 February
Thursday 4 February
Monday 8 February
Thursday 11 February
List of participants
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Practical information
Venue
Barcelona School of Building Construction (EPSEB), Universitat Politècnica de Catalunya (UPC)
Avinguda de Doctor Marañón, 44-50
08028 Barcelona
Room to be confirmed.
Barcelona School of Building Construction (EPSEB), UPC
Contact
For enquiries about this event, please contact crmactivitats@crm.cat.
Policies and commitments
CRM Events Code of Conduct
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Sustainable events
We are committed to organising sustainable events that minimise environmental impact and create a positive legacy for the host community. We support organisers in designing events aligned with the UN 2030 Sustainable Development Goals, reducing negative environmental impacts and promoting responsible practices. All materials provided during our activities are responsibly sourced, including recycled pens and plastic-free badges. We work with responsible suppliers, and our catering partners use fully compostable materials while offering vegetarian and vegan options, with at least one event day being fully vegetarian.After the activity
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