The aim of this project is to develop and strengthen collaboration among participants, to share complementary knowledge, and to pursue international collaborations in the transversal and competitive field of the project. Four of the six participants have got a new position in 2006 and one in 2005: the participants would like to expand this dynamic through this ANR project. The six participants are conducting their research on the deformation theory of algebraic or geometric structures appearing in representation theory or in mathematical physics. This is a vast area, touching upon many different subjects besides representation theory, such as topology, algebraic geometry, combinatorics, conformal field theory,... Recent years have seen spectacular results emerge from interaction between these fields, such as Witten's approach to the geometric Langlands program via four dimensional gauge theories, or Khovanov's construction of new quantum knot invariants using a mixture of topological quantum field theory and singularity theory. The participants all specialize in one or several aspects of the theory, and the knowledge the participants possess in the field will be illustrated by the detailed description of the different topics in which the participants intend to do research. The aim of this ANR project is to strengthen the collaboration that has emerged and is emerging among the participants; to give them the opportunity to share their different perspectives and to use these different points of views to obtain significantly new results; to organize international conferences of common interest; and to increase the international visibility of the French activity of young researchers in this field. All participants have worked or are working with internationally recognized experts of the field. The participants have already met several times to discuss this ANR project. A first research meeting (small CIRM research group) is already planned in Luminy in spring 2008.