SPP 1388:  Representation Theory

• Subject Area: Mathematics
• Term: from 2009 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 72926371

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 include integrable lattice models, the theory of elementary particles, random matrix theory, string theory and quantum computing. Among the many branches of mathematics heavily using representation theory are algebraic geometry, topology, number theory and differential geometry.
Frobenius founded representation theory at the end of the 19th century when studying finite groups. At the beginning of the 20th century pioneers like Schur, Burnside, Cartan, Killing, Weyl, Noether and Brauer established fundamental concepts, objects and definitions. Today, representation theory is universally applicable and enjoys a variety of implementations; this makes representation theory truly interdisciplinary and turns it into a principle of order in mathematics and science.
Since the beginnings of representation theory, it has seen several crucial changes in points of view, in concepts and in approaches. This has led to new branches forming and to areas changing their directions. In recent years, the various branches of representation theory have started to move towards each other, and this process is increasingly gaining momentum. New methods and approaches are being formed, cutting across traditional boundaries. Innovative combinations of methods and new developments in techniques allow for deeper insights in fundamental problems and for stronger applications.
The Priority Programme will face and accept the challenge to support, promote and organise new collaborations and joint activities of different branches towards solving fundamental problems, developing new methods and applying these methods.

• DFG Programme: Priority Programmes
• International Connection: Switzerland, USA

Projects

• Actions of Algebraic Groups, Fans and Tilting Modules (Applicant Hille, Lutz )
• Affine Nichols algebras of diagonal type and modular tensor categories (Applicant Cuntz, Michael )
• Algebraic group techniques for finite(ly presented) groups (Applicant Plesken, Wilhelm )
• Asymptotic branching laws for finite dimensional representations of complex reductive Lie groups by geometric methods (Applicant Seppänen, Henrik )
• Block Structure, Fusion Systems and Conjectures of Brauer and Olsson (Applicant Külshammer, Burkhard )
• Branching laws for 1-parameter families of representations of Lie groups and their asymptotic behavior (Applicant Hilgert, Joachim )
• Classical Yang-Baxter equation and sheaves on degenerations of elliptic curves (Applicant Burban, Igor )
• Cluster categories and torsion theory (Applicant Holm, Thorsten )
• Cluster-categories, cluster-tilted algebras and derived equivalences (Applicant Holm, Thorsten )
• Combinatorial and geometric aspects of the representation theory of finite group schemes (Applicant Farnsteiner, Rolf )
• Complex geometry of actions and related representations (Applicant Heinzner, Peter )
• Coordinator project (Applicant Littelmann, Peter )
• Critical level representations of affine Kac-Moody algebras and the geometric Langlands program (Applicant Fiebig, Peter )
• Derived categories of sheaves over finite partially ordered sets and their homological properties (Applicant Ladkani, Sefi )
• Dualities in the representation theory and geometry of loop groups (Applicant Fiebig, Peter )
• Gabriel-Roiter measure for finite dimensional algebras (Applicant Chen, Bo )
• Geometry and representation theory in computational complexity (Applicant Bürgisser, Peter )
• Geometry and the compression of combinatorial formulas for Macdonald polynomals (Applicant Littelmann, Peter )
• Hermitian symmetric modular category O (Applicant Soergel, Wolfgang )
• Higher Schur-Weyl dualities and gradings (Applicant Stroppel, Catharina )
• Homogeneous Einstein metrics and their geometric properties (Applicant Agricola, Ilka )
• Homological Mirror Symmetry for Singularities (Applicant Ebeling, Wolfgang )
• Homological structures at the interface of abstract representation theory and algebraic Lie theory (Applicant Koenig, Steffen )
• Invariant theory of theta-representations (Applicant Yakimova, Oksana )
• Investigations into the Abelian Defect Group Conjecture (Applicant Danz, Susanne )
• Investigations on Alperin's Weight Conjecture by means of Auslander-Reiten Quivers (Applicant Naehrig, Natalie )
• Investigations on the conjectures of McKay and Alperin-McKay (Applicant Malle, Gunter )
• Koszul duality in representation theory (Applicant Soergel, Wolfgang )
• Matrix Factorizations and complete Intersection-rings (Applicant Burke, Jesse )
• Multiplicity Free Actions (Applicant Knop, Friedrich )
• Non-commutative crepant resolutions and their DT invariants (Applicant Mozgovoy, Sergey )
• p-adic group rings of finite groups (Applicant Nebe, Gabriele )
• PBW-filtration of representations, degenerate flag varieties and polytopes (Applicant Littelmann, Peter )
• Polyhedral models of representation (Applicant Bliem, Thomas )
• Polyhedral models of representations (Applicant Bliem, Thomas )
• Positivity in cluster algebras and its relations with categorifications of cluster algebras and total positivity in semisimple algebraic groups (Applicant Cerulli Irelli, Giovanni )
• Purity of stable pieces in compactifications of semisimple groups (Applicant Wedhorn, Torsten )
• Quiver moduli and quantized Donaldson-Thomas type invariants (Applicant Reineke, Markus )
• Quiver representations, singularity categories, and monoidal structures (Applicant Krause, Henning )
• Recollements and stratifications of derived module categories (Applicant Koenig, Steffen )
• Representation and category theoretic aspects of logarithmic conformal field theories (Applicant Schweigert, Christoph )
• Representation theoretic tools for equivariant and orbifold conformal field theories (Applicant Schweigert, Christoph )
• Representation Theory of the Unitary and Symmetric Group with a view towards the Quantum Marginal Problem (Applicant Christandl, Ph.D., Matthias )
• Representations of Algebraic Groups in Differential and Difference Galois Theory (Applicant Hartmann, Julia )
• Restricting Specht Modules of Finite General Linear Groups to the Unitriangular Subgroup (Applicant Dipper, Richard )
• Semibounded unitary representations of infinite dimensional Lie groups (Applicant Neeb, Karl-Hermann )
• Serre's notion of complete reducibility and geometric invariant theory (Applicant Röhrle, Gerhard )
• Shuffles and Schur positivity (Applicant Fourier, Ghislain )
• Spectral theory of Green functors and other commutative 2-rings (Applicant Dell`Ambrogio, Ivo )
• Spherical subalgebras of quantized enveloping algebras - structure theory and classification problems (Applicant Heckenberger, István )
• Structural and geometric study of representations with applications to categorification (Applicant Penkov, Ivan )
• Structure and representation of cyclotomic Hecke algebras (Applicant Malle, Gunter )
• The telescope conjecture for derived categories arising in representation theory (Applicant Krause, Henning )
• Toric structures in derived categories and representations of algebras (Applicant Perling, Markus )

• Spokesperson: Professor Dr. Peter Littelmann