# Read PDF Computing in algebraic geometry: A quick start using SINGULAR

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.

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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. 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 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.

## Computing in Algebraic Geometry

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. However, often their calculation is infeasible for large examples, even though algorithms and computers improved. Hence, this parameter will be treated for different classes of ideal often occuring in practice e.

Thomas in the s, 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.

## algebraic geometry - Computing singular locus - Mathematics Stack Exchange

We also expect applications to the algebraic analysis of numerical methods for solving nonlinear PDEs. 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. 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. Although constituting powerful tools, their efficiency suffers from several drawbacks such as coefficient blowups during computation, non-adaptiveness and difficulties in parallelizing the computation.

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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.

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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. 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.

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. 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.

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. Another major goal consists of making all the developed algorithmic methods available in an integrated form in a common software environment.

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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.

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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 SPP, 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.

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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.