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Exploiting torus actions in algebraic geometry
Project Leader: Klaus Altmann (FU Berlin)
In many situations of algebraic geometry there exist actions of algebraic tori on objects, morphisms, or families. Algebraically, this is reflected by an (initially maybe invisible) multigrading of rank being the dimension of the torus. In the extension of this rank, this allows one to translate complicated and expensive (in terms of computing effort) algebraic geometry into algorithmically easier combinatorics and discrete/convex geometry. For many years this has been done for full torus actions ("toric varieties"). More recently, this method has also been developed and used for tori of smaller dimension.
To make possible a usage of these theories in praxis, one needs the creation and implementation (in Singular) of algorithmic tools to allow free movement in a combination of algebraic and convex geometry. However, any implementation requires preparation, i.e. a further development of the computer algebra systems in question. Splendid packages exist for both convex and algebraic geometry. But none of these allow one to work with objects of both areas simultaneously.
The Equivariant Tamagawa Number Conjecture for the base change of an abelian variety
Project Leader: Werner Bley (Kassel)
This project is a complement to the algorithmic part of BL 395/3-1 where the Equivariant Tamagawa Number Conjecture (ETNC) of Burns and Flach is studied in the case of Tate motives. In this new project we want to consider ETNC for the base change of an abelian variety A which is defined over Q. Here ETNC is an equivariant refinement of the famous Birch and Swinnerton-Dyer conjecture. More explicitly, ETNC describes the leading terms of twisted Hasse-Weil-L-functions in terms of cohomological data associated to the motive which is attached to A.
The aim of the project is to derive explicit formulations of these equivariant conjectures which makes them amenable to numerical verifications and to develop and implement algorithms in order to provide numerical evidence.
Algorithms for quotients of Bruhat-Tits buildings and automorphic forms
Project Leader: Gebhard Böckle (Heidelberg)
The theory of automorphic forms is one of the core topics of algebraic number theory. It has a long history and is at the same time, with such spectacular successes as the proof of Fermat's last theorem, the Sato-Tate conjecture or the Serre conjecture, one of the most flourishing topics. Presently many new directions are explored. Basic computer algorithms axe an invaluable tool to yield experimental data which further advance the theory.
The present proposal provides one approach to obtain and implement such algorithms. It also explains a number of interesting experiments which we hope to perform. The approach applies to number as well as function fields. Its relation to classical automorphic forms is explained by the Langlands program - and in often conjectural. The intention is to compute Galois representations associated to automorphic via their Hecke eigensystems.
The key tool is the algorithmic computation of quotients of Bruhat-Tits buildings by cocompact arithmetic subgroups. The result is a finite simplicial set and thus a finite combinatorial object. Automorphic forms are sections of local systems on these finite quotients. Their systems of Hecke eigenvalues can be computed combinatorially.
Algebraic-statistic attacks: Algorithms and tools
Project Leaders: Johannes Buchmann (TU Darmstadt), Gert-Martin Greuel (TU Kaiserslautern)
In the era of ubiquitous use of the Internet the questions of privacy and confidentiality play a very important role. Therefore, evaluating cryptographic primitives that provide the above properties has always been crucial for applications like online banking, eCommerce, e-mail communications etc. Block ciphers are well-established building blocks for constructing cryptographic protocols. In this project we address the cryptanalysis of block ciphers. In recent years algebraic cryptanalysis of block ciphers became a rapidly developing area of research.
On the other side, some limitations of the method became apparent over the time. Our goal is, therefore, to pursue a quite recent trend of combining new algebraic and conventional statistical attacks. We will pay a special attention to ciphers that employ modular arithmetic to provide non-linearity: algebraic aspects of such an analysis are novel. Within this project we will develop cryptanalytic methods of combined attacks as well as tools from computer algebra that are absolutely necessary to provide efficient attacks. Fast contradiction finding and system solving in the specific context of algebraic-statistic cryptanalysis is a challenge we address in this project. We believe that united competence and experience of the two applying groups will give a good chance to successfully fulfill the goals of the project.
Fundamental Algorithms in Singular
Project Leaders: Wolfram Decker (TU Kaiserslautern), Thomas Markwig (Kaiserslautern), Gerhard Pfister (TU Kaiserslautern)
The overall goal of the DFG priority programme SPP1489 is to push forward and combine computer algebra methods from different areas of mathematics, and to apply the resulting algorithms to central problems of theoretical and practical interest. To achieve this goal, the programme aims at creating a free and open source platform for linking computer algebra systems specializing in the areas covered by the programme. First and foremost, however, the individual systems have to be extended in view of the needs of the participating researchers. Our proposal aims at doing this for the system SINGULAR which, as we believe, will be a key player in this context. Our goal is to improve the performance of the most fundamental algorithms for polynomial computations, but also to redesign some of the algorithms for applications in areas such as arithmetic geometry and non-commutative algebra. At the same time, we have a number of advanced computational tools in mind. These include the combination of computational methods in algebraic and convex geometry, with potential applications in toric and tropical geometry, and the computation of cohomology, with potential applications to Deligne–Lusztig varieties and monodromy. Another goal is to link symbolic and numerical algorithms, with particular emphasis on new algorithms for primary decomposition.
Computational aspects of motivic Hadamard products
Project Leader: Michael Dettweiler (Bayreuth)
The usual Hadamard product f * g of two power series f and g is well known to be related to the convolution on the multiplicative group of the complex numbers. If f and g are motivic power series in the sense that they describe the variation of periods of cohomology groups of two family of varieties, then also f * g is motivic by twisting the original families over the square of the multiplicative group. In this way one obtains families of Hadamard product motives which are important for many applications in mathematics and physics.
The following motives occur as Hadamard products and illustrate the importance of this class: Motives of generalized hypergeometric differential equations, playing a role in the two-body problem in quantum mechanics, and motives of families of Calabi-Yau-varieties occurring in the mirror symmetry conjecture of string theory. The latter motives also play an important role in the recent proof of the Sato-Tate conjecture.
The aim of this proposal is the development and the implementation of algorithms for the computation of the monodromy, Hodge Type and Galois representations for Hadamard product motives. The resulting programs should be applied among others on modularity questions of (rigid) Calabi-Yau varieties and the study of polylogarithms and Feynman integrals.
Applications of cohomology in group theory and number theory
Project Leader: Bettina Eick (TU Braunschweig)
Constructions and classifications of group extensions play a central role in many applications of group theory. Computational group theory provides effective methods to construct up to strong isomorphism extensions with an elementary abelian module. This restriction on the module limits the possible applications significantly.
Our aim is to use cohomology theory in the development of a new effective method to construct up to strong isomorphism extensions with an arbitrary finite group as module. Then we will investigate variations of this method. For example, we want to develop an effective method to construct up to isomorphism all extensions in which the module embeds as the Fitting subgroup or as a term of the derived or lower central series.
Further, we plan to apply our new algorithms in group and number theory. First, we want to extend the Small Groups Library to all groups of the orders at most 10.000 with few exceptions on the orders. Then, we want to investigate the construction up to isomorphism of the finite metabelian groups with a given derived subgroup and derived quotient. These groups play a role in number theory, as they are related to the Galois groups of unramified extensions of number fields.
The study of the birational geometry of various moduli spaces of curves with the help of the computer algebra system Macaulay
Project Leader: Gavril Farkas (HU Berlin)
The moduli space of curves M_g is the universal parameter space for algebraic curves (Riemann surfaces) of given genus, in the sense that its points correspond to isomorphism classes of curves of genus g. The study of the geometry and topology of M_g is a central problem in algebraic geometry. A well-known principle, due to Mumford, asserts that all moduli spaces parameterizing curves of genus g >= 2 (with or without marked points or level structures), are varieties of general type, with a finite number of exceptions that occur in relatively small genus, when these varieties tend to be uniruled, or even unirational. A variety X is said to be uniruled, when through a general point x in X there passes a rational curve f : P^1 -> X, whereas X is said to be of general type, when the canonical bundle K_X has the maximum number of sections. From the point of view of classification theory, uniruledness is the opposite of being of general type.
The aim of this project is to compute the Kodaira dimension of various covers of M_g. These include universal Picard varieties and moduli spaces classifying pairs consisting of a curve together with a point of order / in the Jacobian variety respectively. The proofs are expected to rely on syzygy calculations assisted by the Macaulay system.
Algorithmic methods for arithmetic surfaces and regular, minimal models
Project Leaders: Anne Frühbis-Krüger (Hannover), Florian Heß (Oldenburg)
Regular and minimal models of algebraic curves over number fields are arithmetic surfaces that play an important role in arithmetic geometry. This research project aims at developing algorithms for such arithmetic surfaces and for the computation of regular and minimal models. The main topics are a desingularisation procedure following Lipman, functionality for the intersection pairing, exceptional divisors, blow ups and blow downs. On the basis of these algorithms applications to the Birch and Swinnerton-Dyer conjecture and other related areas are finally investigated.
Algorithmic Methods in Tropical Geometry
Project Leaders: Andreas Gathmann (Kaiserslautern), Hannah Markwig (Saarbrücken), Anders Jensen (Göttingen), Thomas Markwig (Kaiserslautern)
The new research field of tropical geometry has recently led to considerable progress in algebraic geometry and particularly enumerative geometry. With the help of tropical geometry, many algebraic or geometric questions can be reduced to combinatorial problems concerning graphs or polyhedral complexes. However, the structure of these combinatorial problems is involved, which makes the theoretical as well as practical use of these results complicated. Therefore, we would like to develop resp. implement algorithms for several of these problems which are of particular interest. Thus, we do not only want to gain new results in tropical and hence algebraic geometry, but we also want to simplify the application of tropical geometry.
Geometry of Deligne-Lusztig varieties
Project Leader: Ulrich Görtz (Duisburg-Essen)
Representation theory, in other words the study of symmetries, is an important area of mathematics. For a certain class of "representations" Pierre Deligne and George Lusztig exhibited a close connection between the algebraic notion of representation, and certain geometric objects, now called Deligne-Lusztig varieties. Since then, Deligne-Lusztig varieties have been thoroughly investigated from a representation theoretic point of view. Much less is known about their geometric properties, though. The goal of the proposed project is to improve our understanding of the geometry of Deligne-Lusztig varieties using explicit and algorithmic methods in order to study concrete examples, and hopefully eventually to reach new general results.
Deligne-Lusztig varieties are smooth quasi-projective varieties over finite fields. An important specific question is whether there exist Deligne-Lusztig varieties which are not affine varieties.
Computational Cohomology for basic algebras
Project Leader: David J. Green (Jena)
Experience shows that advances in the field of computational homological algebra lead to new theoretical insights, and that theoretical advances in turn result in better computational methods.
By combining the work of several authors – including the applicant's previous DFG project on the cohomology of p-groups – we will develop methods and software to compute cohomology rings and Ext-algebras for a variety of groups and basic algebras, using noncommutative Gr¨obner bases to construct minimal resolutions.
This will allow us to compute a series of interesting test cases: both in group cohomology and for conjectures about finite dimensional algebras such as the Strong No Loops conjecture. The software will make use of the systems Gap and Singular, and will be made available as a Sage package.
In addition to the design and implementation of suitable algorithms and the evaluation of the computational results obtained, we will also seek to generalise known degree bounds for cohomology rings to the case of Ext-algebras.
Mori dream spaces: Theory, algorithms and implementation
Project Leader: Jürgen Hausen (Tübingen)
Mori dream spaces are varieties with a finitely generated Cox ring. Examples are the toric varieties and the Fano varieties. Based on the fact that Mori dream spaces are quotients of affne varieties, one obtains a concrete description of them by means of algebraic-combinatorial data. This generalizes the description of toric varieties in terms of lattice fans and makes Mori dream spaces accessible for computations. We intend to study the geometry of Mori dream spaces in terms of their describing data and to develop and implement algorithms for concrete computations.
Experiments with cellular structures
Project Leader: Steffen König (Stuttgart)
The objects of this proposal are five classes of algebras of interest in representation theory, knot theory and mathematical physics:
  • Classical Schur algebras (associated with the algebraic group GLn),
  • group algebras of symmetric groups,
  • Brauer algebras,
  • partition algebras,
  • the recently discovered Schur algebras of Brauer algebras.
We will focus on a crucial and computationally accessible part of their cellular structure, the multiplication inside cellular subquotients, and on the information about simple modules, semisimplicity and other fundamental properties to be derived from this multiplication.
The project consists of three main parts: First, we will develop algorithms and computer programs to calculate the relevant structure constants, even for large examples. Secondly, we will run experiments, especially by varying some of the discrete and of the continuous parameters of the algebras. Thirdly, we will use the output of the experiments to formulate precise conjectures about structure and independence of parameters and to get starting points for proofs of general results.
Triangulations and other decompositions of lattice polytopes in toric and tropical geometry
Project Leader: Michael Joswig (TU Berlin)
Lattice polytopes are objects at a junction between combinatorics and algebraic geometry. The study of their triangulations, coarsest subdivisions, mixed subdivisions, and other decompositions is motivated by the mutual interaction between these fields as well as by applications in number theory, optimization, statistics, mathematical physics, and algorithmic biology.
Attacking fundamental open problems in this area requires to combine theoretical insight with algorithmic ingenuity and computer experiments. Specific topics addressed in this proposal include the following: unimodular triangulations of lattice polytopes (in particular, matroid polytopes), the relationship between smoothness and normality of a toric variety, combinatorial and geometric interpretations of h^*-polynomials, and symmetric lattice polytopes. Interaction with other researchers in the Priority Program 1489 provides a unique opportunity which makes it likely to make substantial progress.
Asymptotics of wildly ramified Galois extensions of local or global function fields
Project Leader: Jürgen Klüners (Paderborn)
The discipline of counting Galois extensions of global fields has been very active in the past years. Gunter Malle conjectured a precise asymptotic behavior of the cardinality of extensions with given Galois group for large discriminants. A recent counterexample draws attention to the case in positive characteristic, in which the group order is divisible by the characteristic. These cases were mostly ignored so far and regard function fields over finite fields of characteristic p and their wildly ramified extensions.
The analysis of the counterexamples shows that a corresponding question on local function fields require investigation as well. By Hasse's Einseinheitensatz we get infinitely many extensions with given Abelian Galois group of order divisible by p and again we may ask for their distribution for large discriminants.
In the proposed project we wish to understand the distribution of wildly ramified extensions of local or global function fields and derive a new conjecture on their asymptotic behavior to close the gap in Malle's conjecture.
Beside theoretical aspects this involves extensive computer algebraic experiments, which should initiate and endorse the new conjecture as well as provide a new database on local function fields.
Development, implementation and applications of fundamental algorithms, relying on Gröbner bases in free associative algebras
Project Leaders: Martin Kreuzer (Passau), Viktor Levandovskyy (Aachen)
This project is part of mathematical Computer Algebra. It has applications in Ring Theory, Representation Theory, Computer Science and other disciplines.
In order to perform Gröbner basis-like computations in a free associative algebra, one can use the recently developed letterplace correspondence between ideals and perform the computations in a large commutative polynomial ring. Such rings have been intensively studied before, in particular from a computer algebraic point of view. As a result, very effective data structures are known and fundamental algorithms have been optimized and implemented in computer algebra systems. The corresponding situation for non-commutative rings is less developed. The systematic use of the letterplace correspondence provides new insights and new levels of efficiency into the challenging realm of computations in free algebras and their factor rings.
We aim at the creation of an extension LETTERPLACE of the well-known computer algebra system SINGULAR. This will for the first time provide efficient implementations of Gröbner bases and all the basic algorithms involving Gröbner bases in these algebras, for instance syzygy modules, elimination, kernels of ring and module homomorphisms. A preliminary implementation of the Letterplace Gröbner basis algorithm is available in the kernel of SINGULAR and it has already demonstrated very good performance.
The next step will be to use these implementations to tackle hitherto unaccessible problems. For instance, we intend compute the K-dimension, explicit K-bases and (truncated) Hilbert series for non-commutative K-algebras. Another area of applications are computations in monoid and group rings where we plan to adress questions such as finiteness of a finitely presented group, the generalized word problem, the conjugator search problem, freeness tests for groups and the structure of the group of torsion elements of a group algebra. To guide these applications, we intend to collaborate with several research groups in Germany and across the world.
Computing with Hecke algebras
Project Leaders: Burkhard Külshammer (Jena), Jürgen Müller (Jena)
Hecke algebras are a variety of structures playing a central role in algebra, in particular in the theory of finite groups and their representations. Here, we are interested in two classes of Hecke algebras: Iwahori-Hecke algebras and cyclotomic Hecke algebras, both of which are intimately connected to the representation theory of finite groups of Lie type.
Motivated by theoretical questions concerning the structure and representation theory of these algebras, our aim is to develop new computational methods to handle Hecke algebras and their representations. To this end we will make combined use of techniques coming from various branches of computational mathematics, in particular from group theory and commutative algebra. These tools will then be applied to examine substantial interesting, but otherwise inaccessible examples, in order to collect data, to possibly detect previously unknown patterns, and thus to gain structural insights.
Generic Character Tables in GAP
Project Leader: Frank Lübeck (Aachen)
A character table encodes concisely and efficiently a lot of information about a particular finite group. In this project we consider generic character tables which describe and parameterize the character tables of certain infinite series of groups.
The series of groups we consider are finite groups of Lie type. The classification of finite simple groups shows that most simple groups are closely related to finite groups of Lie type.
The main aim of the proposed project is to develop a new software package, called GenCharLib, containing generic character tables as well as programs to compute with them. This will be implemented in the computer algebra system GAP.
GenCharLib will serve the needs of specialists in the area by facilitating further extensions and applications. It will also make generic character tables more accessible to mathematicians who are not specialists in this field. Finally, it will enhance existing GAP-functions automatically, by providing missing methods for some computations.
The package GenCharLib will build upon experience with a previous package that was based on the computer algebra system Maple.
This is a subproject of the CHEVIE project which provides computational access to the representation theory of finite groups of Lie type and related structures.
The proposed project was already mentioned under point (PB5) of the proposal for the Priority Programme 1489.
Class groups and unramified extensions of number fields
Project Leader: Gunter Malle (Kaiserslautern)
The proposed research concerns the distribution of unramified extensions of number fields. For abelian extensions this is just the distribution of class groups. The applicant has recently found evidence that the Cohen–Lenstra heuristic breaks down for the part of the class group of order not prime to the order of the roots of unity in the base field. One major aim of this project is to obtain concrete numerical predictions from the global function field case. This involves deriving explicit formulas for the number of elements in finite symplectic groups over residue rings with given eigenspaces.
In the non-abelian case we propose to study higher class groups of small derived length from a group theoretical point of view via the corresponding extension problems, and also to obtain first insights on unramified extensions with non-abelian simple Galois group by experimental methods.
Historically, class field theory, the Cohen–Lenstra heuristic and the conjectures by the applicant on distributions of Galois groups and class groups were developed on the basis of examples and tables of statistics for large sets of fields. Correspondingly, besides theoretical parts our project has substantial computational and experimental aspects.
Degree Bounds for Gröbner Bases of Important Classes of Polynomial Ideals and Efficient Algorithms (GBiC PolyA)
Project Leader: Ernst W. Mayr (Garching)
Since their introduction by Buchberger [1], Gröbner bases play a vital role in algorithmic algebra. However, often their calculation is infeasible for large examples, even though algorithms and computers improved. Thus, it is important to identify classes of ideals for which the calculation of Gröbner bases can be done efficiently, and to further develop known algorithms.
The crucial parameter of Gröbner bases is the maximal degree of their polynomials as can be seen from all known proofs of complexity bounds. Hence, this parameter will be treated for different classes of ideal often occuring in practice (e.g. radical ideals, prime ideals, and toric ideals). The resulting insights shall be applied in order to enhance algorithms computing Gröbner bases and alike.
Elimination and Counting via Thomas Decomposition
Project Leaders: Wilhelm Plesken (Aachen), Daniel Robertz (Aachen)
The Thomas decomposition based on a formal triangulation algorithm by J. M. Thomas in the 1930s, now implemented in its full generality for the first time, decomposes the set of solutions of a system of equations and inequations into disjoint subsets corresponding to so called simple systems. The system might be either polynomial or polynomial differential. In the polynomial case one obtains a counting polynomial for the solutions, which depends slightly on the chosen coordinate system. Here the challenge is to study this polynomial for generic and nongeneric coordinates (where methods from representation theory of algebraic groups are to be used), to extract more refined combinatorial information from the Thomas decomposition, and to improve the implementation by getting structural insight into the decomposition. In the differential case the aim is to define counting polynomials for free Taylor coefficients via a common generalization of the algebraic case and Janet's approach to linear PDEs. We also expect applications to the algebraic analysis of numerical methods for solving nonlinear PDEs.
Computational aspects of the Cohomology of Coxeter arrangements: On Conjectures of Lehrer-Solomon and Felder-Veselov
Project Leader: Gerhard Röhrle (Bochum)
Coxeter groups are groups of symmetries which form part of many physical theories about the world we live in. Moreover, these groups are frequently found at the heart of deep mathematical theories, most notably Lie theory. This project is concerned with two seemingly unrelated algebras, the Orlik-Solomon algebra and the Descent algebra, both of which describe certain geometric and combinatorial aspects of the underlying Coxeter group. Experimental evidence suggests the existence of surprising connections between these two algebras. These have been formulated as two precise conjectures about decompositions of certain natural representations of these algebras. A proof, even of special cases of these conjectures, will greatly enhance our understanding of Coxeter groups and the algebras associated to them and potentially impact on other theories involving these groups of symmetries.
Design, analysis, and implementation of efficient and reliable algorithms for complex geometric objects
Project Leader: Michael Sagraloff (Saarbrücken)
The proposed research concentrates on the design and development of efficient algorithms to handle complex geometric objects with quality guarantees. Algorithms of this kind constitute an important basis for applications in Computer Aided Design, robotics or computer vision.
Our overall philosophy requires that our solutions cope with any input and that the output matches the mathematically exact result. Moreover, we request two-fold efficiency: While we aim at proving low complexity of our algorithms, we also want them to compete with existing non-reliable software on inputs that can be handled by these implementations. That is, the runtime should adaptively depend on the difficulty of the input.
It is a challenge to achieve reliability and efficiency simultaneously. A canonical way to tackle degenerate situations is by means of computer algebra methods (Gröbner bases, resultants, etc.) based on exact symbolic computations. Although constituting powerful tools, their efficiency suffers from several drawbacks such as coefficient blowups during computation, non-adaptiveness and difficulties in parallelizing the computation.
By combining fast approximate with exact symbolic methods we expect adaptiveness as well as a significant speed up of the overall approach. We want to achieve this by the development of adaptive root separation and perturbation bounds for univariate polynomials and polynomial systems based on additional information gained from the approximate computation. Furthermore, the number of costly symbolic computation steps over integers should be reduced or replaced by modular computations.
Computer algebra for geometric evolution equations
Project Leader: Oliver Schnürer (Konstanz)
Many challenging problems in geometry concern flow equations like e. g. mean curvature flow or Ricci flow. The behavior of solutions to such flow equations is often controlled as follows: For a geometrically significant quantity, the evolution equation is computed. It is proved that this quantity is monotone, i. e. a Lyapunov function. This allows to control solutions of the flow equation.
The computation of evolution equations for prospective Lyapunov functions is purely algebraic but usually quite tedious. We propose to develop a program that does these algebraic computations in many different situations. We also wish to use algebraic and experimental methods to select prospective Lyapunov functions and to check, whether the resulting evolution equations allow to deduce monotonicities.
Based on these Lyapunov functions we wish to provide a tool to systematically prove new theorems for geometric evolution equations. We want to focus on the behavior of solutions for large times or near singularities.
Syzygies, experiments in algebraic geometry and unirationality questions for moduli spaces
Project Leader: Frank-Olaf Schreyer (Saarbrücken)
An algebraic variety M is unirational, if there exists a dominant P^n -> M. On the other extreme, M is of general type, if the canonical bundle K_M is big. In the first case it is easy to find points on M, in the second case there is no P^1 through a general point of M.
The question whether a variety is of one of these types is especially important for moduli spaces, such as the moduli of curves M_g. If a moduli space M is unirational then we can in principle write down a dominant family depending on free parameters. In case of general type, any set of parameters satisfy a system of algebraic equations. We plan to investigate various refined moduli spaces of curves such as M_{g,d}^r = {(C,g_d^r)} of curves C together with an r-dimensional linear system g_d^r of divisors of degree d on C. In case the moduli space is unirational we want to provide a computer algebra code which chooses points at random, which will be useful for further experimental investigations.
Bifurcations and Singularities of Algebraic Differential Equations
Project Leaders: Werner M. Seiler (Kassel), Andreas Weber (Bonn)
We will analyse bifurcations and singularities of algebraic systems of ordinary differential equations with particular emphasis on questions concerning the existence of oscillations. Exploiting previous results that the study of bifurcations for normal systems of ordinary differential equations leads to questions in real algebraic geometry, we will develop efficient algorithmic methods for parametric bifurcation analysis and use them both for experimental mathematical investigations of low dimensional systems and for "real world applications" like the analysis of controllers for humanoid locomotions or the analysis of chemical reaction systems of non-trivial size. Then we will extend these results to non-normal systems, i. e. to differential algebraic equations (DAEs), and study both the effect of singularities on bifurcations and the relationship between bifurcations of DAEs and singularities of associated systems of partial differential equations. Another major goal consists of making all the developed algorithmic methods available in an integrated form in a common software environment.
Algorithmic and Experimental Arithmetic Geometry
Project Leader: Michael Stoll (Bayreuth)
This proposal contains two projects related in various ways to the arithmetic of algebraic varieties and to the study of their rational points in particular.
The aim of the first project is to devise and implement an algorithm that produces a nice set of equations for a given variety; this is important for doing further computations with it. In many cases, further computations would be infeasible without such a nice model.
The second project is more specifically concerned with rational points on curves of higher genus. It is known that on each such curve there can be only finitely many rational points, but so far there is no algorithm that determines this set explicitly. We will improve and extend existing algorithms covering certain cases. We will then use them to gather statistical information from many curves to get some more precise idea on the behavior of their rational points. On the other hand, we plan to study possible approaches to a general algorithm that finds the set of rational points on any given curve.
Monodromy Algorithms in Singular
Project Leader: Duco van Straten (Mainz)
The proposal fits into the DFG priority programme SPP1489, which aims at combining computer algebra methods from different areas of mathematics, and to apply the resulting algorithms to central problems of theoretical and practical interest.
It is our aim to develop and implement algorithms to compute and analyse the monodromy representation for families of varieties depending on a parameter t. Rather than striving for complete generality, we propose to start with hypersurfaces and use special features like the appearance of special singularities in a fibre. The proposed algorithms combine Gröbner-basis calculations in polynomial rings and localisations, algorithms for primary decomposition, Gröbner-basis calculations in Weyl-algebras, normal-form algorithms, numerical techniques and algorithms from algorithmic group theory. It is our intention to develop these algorithms inside the SINGULAR system.
Algorithms for the Computation of Canonical Forms and Groups of Automorphisms of Linear Codes over Finite Rings and Related Objects
Project Leaders: Alfred Wassermann (Bayreuth), Axel Kohnert (Bayreuth)
Since the discovery of codes over finite rings which are better than codes over finite fields (it is possible to correct more errors using the same number of bits) there is an increased interest in codes over rings. In the proposed project we want to develop several algorithms to handle such codes. An important step is the computation of a canonical form of a linear code over a finite chain ring. As a byproduct this will also allow us to compute the group of automorphisms of a given code, and it will allow us to check whether two given codes are equivalent. We also want to use these algorithms to study similar objects like cryptographic functions and point-sets in a finite projective geometry. Having at hand a good algorithm we will also be able to classify certain codes, which allows to provide a complete list of all 'different' codes.
One further application is the intended database of codes over rings, given in canonical form, as a further contribution to the international tables of error-correcting codes from our group. It extends our classification methods from codes over fields to codes over rings and should comprise the present knowledge on such codes. We want in addition to provide a test on pairwise equivalence to recognize cases where different constructions give isomorphic objects.
Semistable reduction and wild quotient singularities
Project Leaders: Stefan Wewers (Hannover), Irene Bouw (Ulm)
The main theme of the proposed project is to use techniques related to semistable reduction of curves to study resolution of singularities, in particular wild quotient singularities in dimension two. A central case arises from the study of smooth projective curves over a p-adic field. Here we search to relate a suitable semistable model of the curve to a regular model via a quotient construction. This construction gives rise to quotient singularities, and we propose new methods for resolving them. More generally, the goal is to extend this construction also to surfaces in equal characteristic p, and to higher dimension.
We plan to use the results of this study to develop practical algorithms to compute semistable models of curves. We also expect to obtain a better insight into the structure of the quotient singularities arising in our work, and thus be able to algorithmically compute their resolution, in cases that were so far inaccessible to brute force calculation.
Algorithmic and experimental aspects of modular Galois representations over finite and modulo prime powers
Project Leader: Gabor Wiese (Luxembourg)
Recent breakthroughs in Arithmetic Geometry and various topical conjectures in the spirit of the Langlands programme establish and postulate deep correspondences between certain geometric objects: modular and automorphic forms and certain number theoretic objects: Galois representations. The geometric side is often amenable to calculations and by the explicit nature of the correspondences also number theoretic objects become computationally accessible.
The objectives of this proposal concern the investigation of these geometric and arithmetic objects either directly or through the correspondence as well as an extension of the correspondence to new cases. A special emphasis is placed on these aspects over finite fields and modulo prime powers. Concrete questions deal with the modularity as well as level and weight optimisation of Galois representations modulo prime powers and the explicit determination of the local Galois representations attached to classical and Hilbert modular forms at all primes.
The methods to be employed are experimental, algorithmic and theoretical and progress is expected from the interplay of these. For the experimental study, algorithms will be developed and implemented in computer algebra systems. These new computer tools will be of service to other researchers as well.
Coordinator Project
Project Leaders: Wolfram Decker (TU Kaiserslautern), Anne Frühbis-Krüger (Hannover), Frank Lübeck (Aachen)
The goal of the DFG Priority Programme SPP 1489 is to further the algorithmic and experimental methods in different areas of mathematics, to combine the methods where needed, and to apply the resulting algorithms to central questions of theoretical and practical interest. This requires the close cooperation of the different groups, the consistent training of young researchers, and the effective transfer of knowledge. The coordinator project aims at fostering and bringing together the individual efforts of the members of the programme and will, thus, be of importance for the priority programme as a whole. In detail, we propose
  • to create and maintain an open source platform which allows software from different areas to interact, which creates an infrastructure for enhancing each individual system, and which is custom-made for the research projects of the programme,
  • to arrange a programme-wide system of schools, workshops and conferences which is modeled on the successful activities of European networks in algebraic geometry, and which targets the needs of researchers at different stages of their career,
  • to set up and maintain a webserver whose main goals are to facilitate the management of the programme and to disseminate results, software, and data bases created by the programme.