From 12 to 14 January 2026, in the rooms of U9 Building (Viale dell'Innovazione 10, Milano) Prof. Antonio Moro (Northumbria University) will give a course entitled Integrability and Complexity, followed by the seminar Integrable differential identities for the Orthogonal Ensemble.
Programme:
Course
- 12 january | 2:00 pm-3:30 pm | Room U9-11 | Solvable closure relations in mean field models on complete and fully connected graphs
- 13 january | 1:30 pm-3:00 pm | Room U9-09 | Partition functions vs tau- functions: Random matrix models, beta-integrals and Lax equations
- 14 january | 2:00 pm-3:30 pm | Room U9-05 | Unitary ensembles and Toda hierarchy: thermodynamic limit and integrable hydrodynamic-type systems
Seminar
- 14 january | 4:00 pm-5:00 pm | Room U9-05 | Integrable differential identities for the Orthogonal Ensemble
Abstract: Integrable differential identities, together with ensemble-specific initial conditions, provide an effective approach for the characterisation of relevant observables and state functions in random matrix theory. We develop this approach for the unitary and orthogonal ensembles. In particular, we focus on a reduction where the probability measure is induced by a Hamiltonian expressed as a formal series of even interaction terms. We show that the order parameters for the unitary ensemble, that is associated with the Volterra lattice, provide a solution of the modified KP equation. The analogous reduction for the orthogonal ensemble, associated with the Pfaff lattice, leads to a new integrable chain. A key step for the calculation of order parameters for the orthogonal ensemble is the evaluation of the initial condition by using a map from orthogonal to skew-orthogonal polynomials. The thermodynamic limit leads to an integrable system (a chain for the orthogonal ensemble) of hydrodynamic type. Intriguingly, we find that the solution to the initial value problem for both the discrete system and its continuum limit are given by the very same semi-discrete dynamical chain.
All interested are invited to participate