Projects from Prof. Dr. Igor Burban
Algebra
TRR 358 - Affine Kac–Moody groups: algebra, analysis and arithmetic (Subprojct A05)
Affine Kac-Moody groups and related groups (like loop groups) will be studied from several perspectives. We shall investigate finiteness properties of special linear groups over Laurent polynomials over the ring of integers. We also strive to classify certain maximal Lie orders corresponding to trigonometric solutions of the classical Yang-Baxter ...
Duration: 01/2023 - 12/2026
Funded by: DFG
TRR 358 - Hereditary categories, reflection groups and non-commutative curves (Subproject C02)
There are deep connections between quiver representations and Coxeter groups involving the associated root systems, Lie algebras and quantum groups. We will study a parallel situation in which coherent sheaves on certain non-commutative curves, called exceptional curves, correspond to other types of reflection groups. Such exceptional curves ...
Duration: 01/2023 - 12/2026
Funded by: DFG
TRR 358 - Tame patterns in the representation theory of reductive groups and arithmetic geometry (Subproject C03)
One says that an associative algebra has tame representation type if a complete classification of its indecomposable representations is possible, at least in principle. For example the classification of Harish-Chandra modules for the group SL(2,R) was reduced by Gelfand to such an algebra. We shall study algebras arising from more general reductive ...
Duration: 01/2023 - 12/2026
Funded by: DFG
TRR 358 - Integral structures in geometry and representation theory
Integral structures arise in many places throughout mathematics: as lattices in Euclidean space, as integral models of reductive groups and algebraic schemes, or as integral representations of groups and associative algebras. Even questions about the most basic example of an integral structure, the ring of integers Z, very soon lead into the fields ...
Duration: 01/2022 - 12/2026
Funded by: DFG
Trigonometric solutions of the classical Yang–Baxter equation and Lie bialgebra structures on loop Lie algebras
The main goals of this project are to develop a theory of Manin triples of trigonometric solutions of the classical Yang-Baxter equation as well as to classify all bialgebra structures on the loop algebra of a simple Lie Algebra.DFG Programme Research Grants
Duration: 10/2018 - 12/2026
Funded by: DFG
TRR 191 - Symplectic structures in geometry, algebra and dynamics
Since their inception, the study of symplectic structures and the applications of symplectic techniques (as well as their odd-dimensional contact geometric counterparts) have benefited from a strong extraneous motivation. Symplectic concepts have been developed to solve problems in other fields that have resisted more traditional approaches, or ...
Duration: 01/2017 - 12/2021
Funded by: DFG
TRR 191 - Derived categfories of singular curves (Subproject A07)
In this project, we shall apply techniques of algebraic geometry and homological algebra (derived categories, Fourier-Mukai transforms, vector bundles on possibly singular Riemann surfaces) to study problems of geometric analysis. In particular, we shall investigate Bochner Laplacians and kernel functions (Bergman and Szegö kernels) attached to ...
Duration: 01/2017 - 12/2020
Funded by: DFG
TRR 191 - Symplectic methods of infinite dimensional systems (Subproject B06)
Symplectic methods lead to significant progress also for infinite-dimensional dynamical systems (mostly in the form of PDEs). This project considers two instances of such systems, and we intend to (i) construct new solutions of the Einstein equations using integrability methods (via symmetries of the spacetime); and (ii) significantly expand the ...
Duration: 01/2017 - 12/2020
Funded by: DFG
Derived–tame algebras and non–commutative nodal curves
The goal of this project is to develop a systematic theory of non-commutative nodal projective curves, similar to the theory of weighted projective lines of Geigle and Lenzing. In particular, we want to clarify when the derived category of coherent sheaves on such a curve admits a tilting object, and describe the classes of tilted algebras arising ...
Duration: 01/2015 - 12/2019
Funded by: DFG
SPP 1388: Representation Theory
Representation theory is a cross-disciplinary branch of mathematics with a wide range of applications in mathematics and in the sciences. Chemistry uses representation theory for instance to investigate symmetries of molecules, while quantum mechanics is a classical area of applications in physics. Further applications in physics and related areas ...
Duration: 01/2012 - 12/2016
Funded by: DFG