Lifting and degeneration problems

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The principal investigator applies tools from algebraic representation theory and from number theory to study deformations of representations of profinite groups and of complexes of modules for such groups. Deformations and deformation rings have been at the center of remarkable recent progress in the theory of Galois representations and modular forms. This project has four main goals: (1) to study the connection of deformations to the non-existence of solutions to certain embedding problems; (2) to find bounds on the singularities of universal deformation rings; (3) to determine the ring structure of universal deformation rings of representations of finite groups; and (4) to develop an obstruction theory for deformations of complexes. The principal investigator also works on one other project on degenerations of modules over a finite dimensional algebra. The main goal of this second project is to use Grassmannians to study top-stable degenerations of local modules.Groups are abstract mathematical objects by which one may encode and study symmetry, for example in chemical molecules, crystals, networks, or abstract mathematical structures. Representations of groups provide a way to extract information about the internal structures of a group. Roughly speaking, representations can be thought of as "linearized snapshots" of the group which are given by explicitly described matrices. In this project, the principal investigator studies deformations of representations. The deformations of a given representation form a family of representations which are associated to this representation in a certain way. In case there is a single deformation which can be used to uniquely describe all other deformations, one talks about a universal deformation. Universal deformations provide universal constructions which can be used to solve certain problems all at once, which otherwise would have to be solved in a case-by-case fashion. This project belongs to the mathematical areas of representation theory and number theory. Research in these areas has in the past had unexpected applications to subjects such as cryptography and error correcting codes, and in this way has been a benefit to society.