Integrable systems and geometry

Integrable Systems and Geometry have long been fundamental components of mathematical physics, dating back to the pioneering contributions of renowned figures such as J.-L. Lagrange, C.F. Gauss, N.H. Abel, C.G.J. Jacobi, J. Liouville, G.F.B. Riemann, S. Kowalevskaya, and P. Painlevé (to name a few).

Since then, this area has played a pivotal role in modern mathematics and its applications across various scientific disciplines, as the study of dynamical systems with special symmetries and geometric structures offers insights into more general dynamical systems. The convergence of Geometry and Integrable Systems continues to provide a fertile ground for mathematical inquiry, fostering advancements in both fields through interdisciplinary collaboration.

Some keywords that describe our research are:

  • Isomonodromic deformations and the Riemann-Hilbert correspondence
  • ODE/IM correspondence
  • Random matrix theory
  • Asymptotic analysis and the complex WKB method
  • Enumerative geometry (Gromov-Witten invariants, Frobenius manifolds)
  • Hierarchies of integrable PDEs (such as the KdV or KP hierarchies)
  • Special functions (Painlevé equations, Heun equation, theta functions)

Members involved in this area: Jonathan Bradley-Thrush, Julián Barragán Amado, Jean Douçot, Giordano Cotti, Davide Masoero, Giulio Ruzza, and the PhD student Gabriele Degano

Some recent postdocs: Xavier Blot, Riccardo Conti

We currently have three members with FCT CEEC individual grants, and in the last few years we have hosted three FCT research projects, including the project “(Ir)regular singularities & Quantum Field Theory“, you can also visit our eponymous YouTube channel as an example of our outreach and dissemination activities. Several of our members also belong to the COST Action Calista. See below for more details, including a list of some recent and upcoming conferences we have organized or co-organized.

List of selected recent publications of our team members:

ODE/IM Correspondence

Enumerative geometry

  • G. Cotti, Borel (α,β)-multitransforms and Quantum Leray-Hirsch: integral representations of solutions of quantum differential equations for ℙ1-bundles, J. Math. Pures Appl. 183 (2024), 102-136
  • T. Bridgeland, D. Masoero, On the monodromy of the deformed cubic oscillator, Math. Ann. 385 (2023), 193-258
  • G. Cotti, Cyclic stratum of Frobenius manifolds, Borel-Laplace (α,β)-multitransforms, and integral representations of solutions of Quantum Differential Equations, Mem. Eur. Math. Soc. Vol 2 (2022), 134pp.

Random matrix theory and Integrable PDEs

  • M. Cafasso, G. Ruzza, Integrable equations associated with the finite‐temperature deformation of the discrete Bessel point process, J. London Math. Soc. 108 (2023), 273-308
  • C. Charlier, T. Claeys, G. Ruzza, Uniform tail asymptotics for Airy kernel determinant solutions to KdV and for the narrow wedge solution to KPZ, J. Funct. Anal. 283 (2022), 109608
  • M. Cafasso, T. Claeys, G. Ruzza, Airy Kernel Determinant Solutions to the KdV Equation and Integro-Differential Painlevé Equations, Comm. Math. Phys. 386 (2021), 1107-1153

Special functions

  • J. Barragán Amado, K. Kwon, B. Gwak, Absorption cross section in gravity’s rainbow from confluent Heun equation, Class. Quantum Grav. 41 (2024), 035005
  • G. Degano, D. Guzzetti, The sixth Painlevé equation as isomonodromy deformation of an irregular system: monodromy data, coalescing eigenvalues, locally holomorphic transcendents and Frobenius manifolds, Nonlinearity 36 (2023), 4110-4168
  • J. G. Bradley-Thrush, Properties of the Appell–Lerch function (I), Ramanujan J. 57 (2022), 291-367
  • D. Masoero, P. Roffelsen, Roots of generalised Hermite polynomials when both parameters are large, Nonlinearity 34 (2021), 1663-1732

Isomonodromic deformations and Riemann-Hilbert correspondence

List of recent FCT research projects:

  • FCT Exploratory Project, “Geometria de deformações isomonodrômicas não genéricas: análise exploratória e aplicações” 2022.03702.PTDC (2023-2025, P.I. Giordano Cotti)
  • FCT Project, “Irregular connections on algebraic curves and Quantum Field Theory” PTDC/MAT-PUR/30234/2017 (2018-2022, P.I. Davide Masoero)
  • FCT Exploratory Project, “A Mathematical Framework for the ODE/IM Correspondence” (2017-2021, P.I. Davide Masoero)

List of individual FCT CEEC grants:

Selected conferences organized or co-organized by our group: