Asymptotic and computational aspects of complex differential equations
Partially supported by GNFM-INdAM and SISSA, Trieste
13 February 2017 - 17 February 2017Aims
Differential equations appear naturally in many areas of mathematics and mathematical physics. Special functions which solve some differential equations have a long and remarkable history and they attract a lot of attention nowadays. For instance, the Painleve’ equations are the core of modern special function theory and find applications in mathematical physics, integrable system, enumerative geometry, Frobenius manifolds, evolutionary PDEs, Random Matrix theory.
The objective of the workshop is to bring together the leading experts in differential equations in the complex domain, special functions, summability theory, asymptotic analysis, isomonodromy deformations and Riemann-Hilbert problems, in order to discuss recent progresses and open questions. Some lectures will be general overviews for young researchers. Moreover, some talks will be devoted to applications, particularly the analytic theory of Frobenius manifolds and Quantum Cohomology. The participants will learn new methods in the related areas and identify new topics for future research. It is expected that the workshop will be an excellent opportunity to promote existing and start new collaborations between the participants.
Research Directions
There are a lot of open problems in the area related to the workshop program, which we plan to explore during the workshop. In particular, we are interested in concrete computational aspects of the theory of solving complex differential equations and linear systems. These problems and many others are important to the development of the theory of differential equations and will have a substantial impact. The proposed workshop is, therefore, interesting, timely and necessary for the overall progress in differential equations and special functions.
The topic will include:
- 1. Multisummability. The theory of summability is being widely developed from the initial ideas on asymptotic expansions due to Poincare. In the 1970th, the work on k- summability of formal power series by J.P. Ramis (France) and the results provided by J. Ecalle (France), and B. Braaksma (the Netherlands) among others, put forward the foundations of multisummability, and the ideas that are nowadays of great growth. The methods of multisummability have become powerful tools for solving ordinary differential equations. Borel transform is one of the important tools to study summability. The theory of summability of formal solutions to partial differential equations and difference equations has just started to develop. R. Schaefke, D. Sauzin, S. Michalik
- 2. Asymptotic methods O. Costin, R. Costin
- 3. Computational methods of asymptotic analysis M. Barkatou
- 4. Painleve’ equations, Riemann-Hilbert problems. F. Zullo, D. Guzzetti. O. Costin, M. Bertola
- 5. Applications of linear differential equations and their isomonodromy deformation equations to the analytic theory of Frobenius manifolds. In the workshop, special attention will be devoted to the explicit computational methods of Stokes matrices and connection matrices related to quantum cohomology of Fano varieties. B. Dubrovin, G. Cotti, D. Guzzetti