Factorization and Integrable Systems

An Overview of Matrix Factorization Theory and Operator Applications.- 0 Introduction.- 1 General theorems.- 2 Factorization in decomposing Banach algebras.- 3 Generalized Lp factorization.- 4 State space method.- References.- Matrix Riemann-Hilbert Problems Related to Branched Coverings of ??1.- 1 Introduction.- 2 Riemann-Hilbert problem with quasi-permutation monodromies and algebraic curves.- 3 Riemann surfaces. Rauch variational formulas.- 4 Solution of Riemann-Hilbert problems with quasi-permutation monodromies and Szegö kernel.- 5 Isomonodromic tau-function and Cauchy-Riemann determinants.- Acknowledgements.- References.- Quantum Groups and Integrable Models.- 1 Introduction.- 2 Algebraic Bethe Ansatz and QISM.- 3 Yang-Baxter equation.- 4 Thermodynamic limits.- 5 Conclusion.- Acknowledgements.- References.- Integrable Systems and Factorization Problems.- 1 Introduction.- 2 A few preliminaries: Poisson brackets, coadjoint orbits, etc..- 3 Classical r-matrices and Lax equations.- 4 Classical Yang-Baxter identity.- 5 A finite-dimensional example.- 6 Loop algebras and the Riemann problem.- 7 More examples.- 8 Zero curvature equations.- 9 Difference equations and Poisson-Lie groups.- Acknowledgements.- References.- Programme of the Summer School on Factorization and Integrable Systems.