Integrable Systems and Factorization Problems

The present lectures were prepared for the Faro International Summer School on Factorization and Integrable Systems in September 2000. They were intended for participants with the background in Analysis and Operator Theory but without special knowledge of Geometry and Lie Groups. In order to make the main ideas reasonably clear, I tried to use only matrix algebras such as $\frak{gl}(n)$ and its natural subalgebras; Lie groups used are either GL(n) and its subgroups, or loop groups consisting of matrix-valued functions on the circle (possibly admitting an extension to parts of the Riemann sphere). I hope this makes the environment sufficiently easy to live in for an analyst. The main goal is to explain how the factorization problems (typically, the matrix Riemann problem) generate the entire small world of Integrable Systems along with the geometry of the phase space, Hamiltonian structure, Lax representations, integrals of motion and explicit solutions. The key tool will be the \emph{% classical r-matrix} (an object whose other guise is the well-known Hilbert transform). I do not give technical details, unless they may be exposed in a few lines; on the other hand, all motivations are given in full scale whenever possible.Comment: LaTeX 2.09, 69 pages. Introductory lectures on Integrable systems, Classical r-matrices and Factorization problem

The pathological role of Wnt5a in psoriasis and psoriatic arthritis.

Psoriasis (PsO) is a chronic inflammatory skin disease with both local and systemic components. PsO-associated arthritis, known as psoriatic arthritis (PsA), develops in approximately 13%-25% of PsO patients. Various factors associated with both PsO and PsA indicate that these conditions are part of a single disease. Identification of novel targets for the development of drugs to treat both PsO and PsA is desirable to provide more patient-friendly treatment regimens. Such targets will likely represent 'common checkpoints' of inflammation, for example key components or transduction cascades of the signalling pathways involved. Emerging evidence supports involvement of the non-canonical Wnt signalling pathways in the development of both PsO and PsA, especially the Wnt5a-activated signalling cascades. These, together with interlinked factors, are crucial in the interactions among keratinocytes, immune cells and inflammatory factors in PsO, as well as among chondrocytes, osteoblasts and osteoclasts that trigger both subchondral bone remodelling and cartilage catabolism in PsA. This review focuses on the pathological role of Wnt5a signalling and its interaction with other interlinked pathways in both PsO and PsA, and also on the main challenges for future research, particularly with respect to molecules targeting Wnt signalling pathways for the treatment of PsO and PsA