Research team

Fundamental Mathematics

Expertise

Application related to non commutative space e.g. in economy, process evaluation etc.

Equivariant Brauer groups and Galois deformations 2. 01/01/2011 - 31/12/2014

Abstract

We study the Brauer group of some concrete monoidal categories, with emphasis on the equivariant Brauer group of a triangular pointed Hopf algebra, the Brauer group of differential graded algebras, and the Brauer group of a coquasitriangular dual pointed Hopf algebra. We will study the groupoid of biGalois objects over a Hopf algebra. We will introduce a categorical version of the Clifford functor and the Brauer-Wall group.

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Deformations and cohomology in non-commutative derived geometry. 01/10/2008 - 31/05/2011

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My research project is at the crossroads of non-commutative geometry (in the sense of Kontsevich, Van den Bergh, . . . ) and homotopical derived geometry (in the sense of Toën, . . . ). An important inspiration is the fact [6] that a smooth proper scheme is equivalent in the derived sense to a differential graded (dg) algebra [28], and smoothness and properness boil down to properties of this dg algebra. Hence, dg algebras become models of "noncommutative schemes" [37], [60]. This approach has proven useful in topics ranging from deformation quantization to homological mirror symmetry. In this spirit, we study dg algebras [28], their twins, A1-algebras [27], stacks, and in particular deformations and Hochschild cohomology of these objects.

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Properties of crystalline graded rings with as base ring (degree 0) a Dedekind Domain. 01/10/2007 - 30/06/2009

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FWO-Visiting Postdoctoral Fellowship. (Florin PANAITE, Romania) 01/04/2007 - 31/03/2008

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Modular forms in non-commutatieve geometry. 01/01/2007 - 31/10/2007

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During the previous years the candidate has studied Clifford algebra valued modular forms on arithmetic subgroups of the orthogonal group O(1,n) and that are annihilated by Dirac type operators. The aim of this project is to apply and to extend these techniques to generalizations of Clifford algebras. This shall give further insight in the study of discrete quantum groups and Hopf algebras in the framework of non-commutative geometry.

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Sheaves on a non-commutative topology: further development of the theory and its applications in algebra, geometry and logic. 01/10/2006 - 30/09/2009

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Projective Representations - Generalized Clifford Algebra - Dirac Formalism. 01/10/2005 - 30/09/2007

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New techniques in Hopf algebras and graded ring theory. 01/01/2005 - 31/12/2006

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LIEGRITS - Flags, Quivers and Invariant Theory in Lie Representation Theory. 01/02/2004 - 31/01/2008

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01/10/2003 - 30/09/2005

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Construction and applications of non-communatieve geometry: from algebra to physics. 01/01/2001 - 31/12/2004

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Hopf algebras in algebra, topology, geometry and physics. 11/12/2000 - 11/12/2003

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01/10/2000 - 30/06/2001

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    European Priority Programme : 'Noncommutative Geometry'. 01/03/2000 - 31/12/2004

    Abstract

    Recent interactions between physics and noncommutative algebra gave rise to the creation of a new area in mathematics : 'Noncommutative Geometry'. The European Science Foundation selected this for a European Priority Programme that was funded by 12 member countries. The NOG-programme is involved in the organization of congresses, workshops and summerschools and also provides fellowships and travel grants for research cooperation. See web-page win-www.uia.ac.be/u/nog2000 for more information.

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    20/12/1999 - 20/12/2002

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      Non commutative geometry and cohomology. 01/10/1998 - 31/12/1999

      Abstract

      Non commutative geometry, originally a part of abstract algebra nowadays is popular because of its applications in physics (quantum groups, Witten gauge algebras, ...), where different branches of the geometry are combined : differential geometry and C-algebras, quantum cohomology theories etc....

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        Invariants and representation theory. 01/10/1997 - 30/09/2000

        Abstract

        Representations of Algebraic and Quantum Groups with Particular attention towards Lie algebras and Quantied Envelop Algebras. The team contains the reading Research Groups in this field.

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          Classificiation of simple modules over Weyl algebras. 01/10/1997 - 30/06/1998

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          The project aims at the complete classification of Simple Modules over a class of recently very popular algebras. For higher WEYL- algebras and Rings of Differential Operators on Surfaces we aim to classify Holonomic Simple Modules.

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            Mathematical methods of technology; motion planning for mobile robots. 01/10/1997 - 31/03/1998

            Abstract

            Motion planning for mobile robots is the part of the programme where a simulation-desktop software is developed and used for programming and calibrating mobile (welding-) robots.

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              Construction of a new part of the curriculum : mathematical methods of technology. 01/06/1997 - 31/12/1998

              Abstract

              Modern Pure Mathematics recently found new exciting applications in technology, particularly in connection with robotics (Motion Planning and Vision) or telecommunication(Coding). The Mathematical Theories necessary have been developed in a European Programme. Now these are furthered on an Educational level.

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                Noncommunative algebra and geometry with focus on representation theory. 01/04/1997 - 30/11/1998

                Abstract

                The representation theory of quantum groups and quantum spaces has several connections to non-communative geometry, e.g. determination of prime spectra, irrudicible representations, holonomic modules etc...

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                  Fundamental Mathematics. 01/01/1997 - 31/12/1997

                  Abstract

                  The library is, in all its aspects, the research lakoratory in mathematica. This is certainly the case for fundamental mathematica. With the promised money books will be purchased. As such the research in algebra end analysis/stochastics will be enhanced.

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                    Creation of a new curriculum in mathematical methods of technology. 01/01/1997 - 31/12/1997

                    Abstract

                    Pure mathematical techniques nowadays find new application in robotics (mobile robots, motion planning, vision) and Telecommunications (coding theory, encryptography...). New courses are being developed and tested.

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                      Hopf algebras and co-galois theory. 20/12/1996 - 19/06/2000

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                      Study of actions and co-actions of Hopf Algebras, in particular Quantum Groups. Galois and co-Galois extensions with respect to Hopf Algebras are one of the main topics.

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                        Scientific collaboration with the University of Bucarest on analytical chemistry, algebra and Romanian language. 01/01/1996 - 31/12/1997

                        Abstract

                        To start collaboration through the exchange of postdoctoral researchers within the specialisations mentioned in the title.

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                          01/01/1996 - 30/06/1996

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                            Brauer invariants in the representation theory of finite groups. 01/10/1995 - 31/12/1996

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                            Invariants connected to the Brauer group, usually of cohomological nature, appear in the representation theory of finite groups via projective representatitions and Schur multipliers.

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                              Hopf algebra and quantum group actions and coactions. 01/10/1995 - 30/06/1996

                              Abstract

                              Group actions and group gradings extend to Hopf algebra as well as quantum group actions and coactions. The structural properties of these objects may be studied in terms of invariants and semi-invariants.

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                                AMS-Benelux-meeting : Mathematics 2000. 01/01/1995 - 31/12/1995

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                                Treats topics at the forefront of actual scientific developments, redefining the position of mathematics in the world today. New applications of otherwise pure mathematics are investigated.

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                                  Invariants and representation theory of algebras and groups. 01/11/1994 - 31/10/1998

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                                  Invariant theory, orbit methods and crystalization theory in connection with quantized envelopping algebras and quantum groups. Via Hall algebras a connection with the representation theory of finite dimensional algebras is created and studied.

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                                    Quantized algebras: representations and weight modules. 01/10/1994 - 31/12/1996

                                    Abstract

                                    We study certain classes of quantized algebras and their representations in connection with their non-communative geometry. Specific classes of modules are treated in detail. We try to apply "braided techniques" to this theory.

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                                      Quantum sections and gauge algebras for rings of differential operators over algebraic varieties. 01/10/1994 - 30/09/1996

                                      Abstract

                                      Iterated gauge algebras associated to rings of differential operators on projective varieties are schematic algebras, therefore it is possible to study their non-communative geometry by looking at their quantum sections.

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                                        Analytical and algebraic methods for quantum fluctuations in the hydrodynamical limit. 01/10/1994 - 31/08/1996

                                        Abstract

                                        The methods of quantum-deformations applied to C-algebras have applications in physics. This project aims to build a bridge between quantumfluctuations appearing in the hydrodynamic limit and Hopf-algebraic methods in quantum group theory.

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                                          01/01/1994 - 31/12/1994

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                                            Enveloping algebras, quantum groups and representative theory. 01/10/1993 - 30/09/1996

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                                            The noncommutative qeometry of quantum algebras including quantum enveloping algebras of Lie algebras or super color Lie algebras, gauge algebras and general schematic algebras, is being studied from a representation theoretic point of view.

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                                              Quantum sections of Micro-structure sheaves and Gauge-algebras for enveloping algebras of Lie Algebras 01/10/1993 - 30/09/1995

                                              Abstract

                                              The graded Rees rings of filtrations having a quantum-space for the associated graded ring may be viewed as objects over a non-commutative projective space. We aim to study the arithmetical properties of these objects

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                                                Quantum sections and gauge algebras for rings of differential operators over algebraic varieties. 01/10/1992 - 30/09/1994

                                                Abstract

                                                Iterated gauge algebras associated to rings of differential operators on projective varieties are schematic algebras, therefore it is possible to study their non-communative geometry by looking at their quantum sections.

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                                                  Analytical and algebraic methods for quantum fluctuations in the hydrodynamical limit. 01/10/1992 - 30/09/1994

                                                  Abstract

                                                  The methods of quantum-deformations applied to C-algebras have applications in physics. This project aims to build a bridge between quantumfluctuations appearing in the hydrodynamic limit and Hopf-algebraic methods in quantum group theory.

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                                                    K-theoretic approach to the Brauer group and quadratic forms. 01/10/1992 - 28/02/1993

                                                    Abstract

                                                    Mercuriev-Suslin 's theorem relates the K-group K2 of a field to the cohomology H2, hence to the Brauer group. The 2-torsion part is connected to certain quadratic forms over the ground field.

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                                                      Structure of finite dimensional algebras and their representations. 01/01/1992 - 31/12/1994

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                                                      The structure of finite dimensional associative algebras and Lie algebras is being investigated.

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                                                        Quantum sections of Micro-structure sheaves and Gauge-algebras for enveloping algebras of Lie Algebras 01/10/1991 - 30/09/1993

                                                        Abstract

                                                        The graded Rees rings of filtrations having a quantum-space for the associated graded ring may be viewed as objects over a non-commutative projective space. We aim to study the arithmetical properties of these objects

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                                                          01/10/1991 - 30/09/1992

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                                                            01/01/1991 - 31/12/1991

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                                                              01/10/1990 - 30/09/1992

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                                                                01/10/1990 - 30/09/1991

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                                                                  Hopf algebra actions and the ring of invariants or semi-invariants related to quantum spaces and quantum groups. 01/10/1989 - 30/09/1992

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                                                                  Hopf algebra actions on algebra extensions may be viewed as an extension of classical Galois theory. Induction and coinduction from invariants is being studied.

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                                                                    01/01/1989 - 31/12/1990

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                                                                      01/10/1988 - 30/09/1991

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                                                                        Geometry of matrixinvariants and arithmetic geometry. 01/10/1986 - 30/09/1997

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                                                                        Rationality problem for quotients of PFLn-varieties. Connection between ringtheoretical properties of Sklyanin algebras and arithmetic of elliptic curves.

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