SPP 1489:  Algorithmic and Experimental Methods in Algebra, Geometry and Number Theory

The field of computer algebra allows one to compute in and with a multitude of mathematical structures. It is interdisciplinary in nature, with links to quite a number of areas in mathematics, with applications in mathematics and other branches of science, and with constantly new and often surprising developments.
Particular fruitful interactions unfold between computer algebra and algebraic geometry, number theory and group theory. Algebraic algorithms open up new ways of accessing subareas of these key disciplines of mathematics, and they are fundamental to practical applications of the disciplines. Conversely, challenges arising in algebraic geometry, number theory and group theory quite often lead to algorithmic breakthroughs which, in turn, open the door for new theoretical and practical applications of computer algebra.
The goal of the Priority Programme is to considerably further the algorithmic and experimental methods in the afore mentioned disciplines, to combine the different methods where needed, and to apply them to central questions in theory and praxis. Moreover, the programme is meant to support the further development of free computer algebra systems, which are (co-)based in Germany and which in the framework of different projects may require crosslinking on different levels.
Of particular interest are interactions with application areas inside and outside of mathematics such as system and control theory, coding theory, cryptography, CAD, algebraic combinatorics and algebraic statistics as well as hybrid methods, which combine numerical and symbolic approaches.
The software packages and data bases developed will be freely available via the webserver of the Priority Programme, which will serve as a general platform for algorithmic and experimental mathematics. A comprehensive scheme of schools and workshops will contribute to the training of young researchers.

DFG Programme Priority Programmes

International Connection Australia, Denmark, France, Italy, Luxembourg, Romania, Turkey, United Kingdom, USA

• Algorithmic and Experimental Arithmetic Geometry (Applicant Stoll, Michael )
• Algorithmic and experimental aspects of modular Galois representations over finite fields and modulo prime powers (Applicant Wiese, Gabor )
• Algorithmic aspects of branch groups (Applicant Bartholdi, Laurent )
• Algorithmic methods for arithmetic surfaces and regular, minimal models (Applicant Frühbis-Krüger, Anne )
• Algorithmic methods in the modular representation theory of diagram algebras (Applicant Shalile, Armin )
• Algorithmic tropical intersection theory on moduli spaces (Applicants Gathmann, Andreas ;  Markwig, Hannah )
• Algorithms for rational cones and toric geometry (Applicant Bruns, Winfried )
• Algorithms for the Computation of Canonical Forms and Groups of Automorphisms of Linear Codes over Finite Rings and Related Objects (Applicant Wassermann, Alfred )
• Applications of cohomology in group theory and number theory (Applicant Eick, Bettina )
• Arithmetic methods for finitely generated matrix groups (Applicant Nebe, Gabriele )
• Arrangements of complex reflection groups: Geometry and combinatorics (Applicants Cuntz, Michael ;  Röhrle, Gerhard )
• Asymptotics of wildly ramified Galois extensions of local or global function fields (Applicant Klüners, Jürgen )
• Bifurcations and Singularities of Algebraic Differential Equations (Applicants Seiler, Werner M. ;  Weber, Andreas )
• Class group computation in large fields (Applicant Fieker, Claus )
• Class groups and unramifed extensions of number fields (Applicant Malle, Gunter )
• Classification of nilpotent associative algebras and coclass theory (Applicant Eick, Bettina )
• Combinatorial and geometric structures for reflection groups and groupoids (Applicants Cuntz, Michael ;  Stump, Christian )
• Complex Multiplication: Class invariants and cryptographic applications (Applicant Heß, Florian )
• Computational aspects of block theory of finite groups (Applicant Müller, Jürgen )
• Computational aspects of modular forms and p-adic Galois representations (Applicant Böckle, Gebhard )
• Computational aspects of motivic Hadamard products (Applicant Dettweiler, Michael )
• Computational aspects of the Cohomology of Coxeter arrangements: On Conjectures of Lehrer-Solomon and Felder-Veselov (Applicant Röhrle, Gerhard )
• Computational Cohomology for basic algebras (Applicant Green, David J. )
• Computational Galois Theory for Local Fields (Applicant Klüners, Jürgen )
• Computational methods for abelian varieties over number fields with complex multiplication (Applicants Stein, Andreas ;  Weng, Annegret )
• Computer algebra for geometric evolution equations (Applicant Schnürer, Oliver )
• Computing with Coxeter groups and Hecke algebras (CHEVIE/PyCox) (Applicant Geck, Meinolf )
• Computing with Hecke algebras (Applicant Külshammer, Burkhard )
• Constructive derived equivalences and equivariant vector bundles (Applicant Barakat, Mohamed )
• Coordinator Project (Applicant Decker, Wolfram )
• Degree Bounds for Gröbner Bases of Important Classes of Polynomial Ideals and Efficient Algorithms (GBiC PolyA) (Applicant Mayr, Ernst W. )
• Design, analysis, and implementation of efficient and reliable algorithms for complex geometric objects (Applicant Sagraloff, Michael )
• Development, implementation and applications of fundamental algorithms, relying on Gröbner bases in free associative algebras (Applicants Kreuzer, Martin ;  Levandovskyy, Viktor )
• Effective methods for spectrahedra in real and convex algebraic geometry (Applicant Theobald, Thorsten )
• Elimination and Counting via Thomas Decomposition (Applicant Plesken, Wilhelm )
• Experiments with cellular structures (Applicant Koenig, Steffen )
• Explicit Chabauty-Kim theory for the thrice punctured line (Applicant Dan-Cohen, Ishai )
• Exploiting torus actions in algebraic geometry (Applicant Altmann, Klaus )
• Fundamental Algorithms in Singular (Applicants Decker, Wolfram ;  Pfister, Gerhard ;  Schulze, Mathias )
• Generic Character Tables in GAP (Applicant Lübeck, Frank )
• Geometric Aspects of Differential Equations (Applicant Dettweiler, Michael )
• Geometry of Deligne-Lusztig varieties (Applicant Görtz, Ulrich )
• Improving and Combining Gröbner bases and SAT solving techniques for algebraic cryptanalysis (Applicants Buchmann, Johannes ;  Greuel, Gert-Martin )
• L-functions and other arithmetic invariants of curves of genus greater than or equal to 3 (Applicants Bouw, Irene Ingeborg ;  Wewers, Stefan )
• Monodromy Algorithms in Singular (Applicant van Straten, Duco )
• Mori dream spaces: Theory, algorithms and implementation (Applicant Hausen, Jürgen )
• Polyhedral Fan Structures in Toric and Tropical Geometry (Applicant Joswig, Michael )
• Semistable resolutions of local models (Applicant Görtz, Ulrich )
• Standard basis methods for path algebra quotients (Applicant King, Simon Alexander )
• Symmetries of singular del Pezzo surfaces in algebraic and arithmetic geometry (Applicants Derenthal, Ulrich ;  Hausen, Jürgen )
• Syzygies, experiments in algebraic geometry and unirationality questions for moduli spaces (Applicant Schreyer, Frank-Olaf )
• Syzygies, Hurwitz spaces and Ulrich sheaves (Applicant Farkas, Gavril )
• The Equivariant Tamagawa Number Conjecture for the base change of an abelian variety (Applicant Bley, Werner )
• The Generalized Fermat Equation with exponents 2, 3, n (Applicant Stoll, Michael )
• The study of the birational geometry of various moduli spaces of curves with the help of the computer algebra system Macaulay (Applicant Farkas, Gavril )
• Toroidal methods for computing zeta functions of groups and rings (Applicant Voll, Christopher )
• Units in integral group rings (Applicant Kimmerle, Wolfgang )

Spokesperson Professor Dr. Wolfram Decker