# Computer Algebra

## Schedule

### All talks will take place in room 48-210.

 Montag Dienstag Mittwoch Donnerstag Freitag 09:00-10:00 Bettina Eick:The Isomorphism problem for torsion free nilpotent groups Michael Stoll:The Generalized Fermat Equation x^2 + y^3 = z^{11} Janko Böhm:Computing GIT-Fans with Symmetry Pinar Kilicer:Some problems related to CM theory 10:00-10:40 Christian Eder:Parallel Gröbner Basis Computation Markus Kirschner:Definite hermitian lattices with class number one Sebastian Gutsche:Categories, algorithms, and programming Claus Fieker:Class Groups is Large Degree Fields Theory and some Computations 10:40-11:15 Kaffepause in 48-436 11:15-12:15 Gabriele Nebe:A mathematical framework of network coding Anne Frühbis-Krüger:Computing the vanishing topology of determinantal ICMC2 singularities Samuele Anni:Congruence Graphs Steffen Koenig:How to describe diagram algebras 12:15-14:00 Mittagspause 14:00-15:00 Stefan Wewers:Semistabile Reduktion und L-Reihen von Kurven Christopher Voll:Enumerating lattices invariant under nilpotent algebras of endomorphisms Exkursion nach Speyer (Rückkehr: ca. 22:00 Uhr) Yue Ren:Tropical Geometry in Singular 15:00-15:40 Meinolf Geck:A new construction of semisimple Lie algebras and Chevalley groups Ernst Mayr:Radicals of Term Replacement Systems Nils Amend:Restrictions of Reflection Arrangements and the K(\pi, 1) Property 15:40-16:15 Kaffepause in 48-436 Kaffepause in 48-436 16:15-17:15 Thorsten Theobald:Imaginary projections of polynomials Jürgen Hausen:Computational Theory of Mori dream spaces Konstantin Jakob:The Classification of Rigid Irregular G_2-Connections 17:15-18:00 Pavel MetelitsynEine Datenbank für Differentialoperatoren vom Calabi-Yau Typ sowie dazugehörige Software Treffen der Projektleiter in 48-438 Ute Spreckels:On the Order of CM Abelian Varieties over Finite Prime Fields

## List of Abstracts

Nils Amend: Restrictions of Reflection Arrangements and the K(\pi, 1) Property

Suppose that G is a finite unitary reflection group acting on the complex vector space V and let A = A(G) be the associated reflection arrangement. It has been a long standing conjecture that in this case the complement of A in V is a K(\pi, 1) space, with the last six open cases being settled by Bessis in 2006. We will have a look at the situation for restrictions of reflection arrangements to elements of their intersection lattice. In particular, we will focus on the restrictions of the infinite family A(G(r, p, l)).

Samuele Anni: Congruence Graphs

The theory of congruences of modular forms is a central topic in contemporary number theory, lying at the basis of the proof of Mazur's theorem on torsion in elliptic curves, Fermat's Last Theorem, and Sato-Tate, amongst others.
Congruences are a display of the interplay between geometry and arithmetic. In order to study them, in a joint work with Vandita Patel (University of Warwick), we are constructing graphs encoding congruence relations between newforms.
These graphs have extremely interesting features: they help our understanding of the  structure of Hecke algebras, and they are also a new tool in the study of numerous conjectures.
In this talk I will describe these new objects, show examples and explain the possible applications.

Janko Böhm: Computing GIT-Fans with Symmetry

The GIT-fan is a combinatorial structure which describes all reasonable
quotients obtainable from the action of an algebraic group on an algebraic variety. In the case of an algebraic torus acting on an affine variety, based on work of Berchtold and Hausen, Keicher has developed an algorithm for computing the GIT-fan. The algorithm relies
on Groebner basis and polyhedral computations. Due to the complexity, in important examples from algebraic geometry a direct application of the algorithm is not feasible. In this talk, I will describe how to make these examples accessible by the use of efficient algorithms for computing saturations and by taking into account actions of symmetry groups.

This is joint work with Simon Keicher and Yue Ren.

Christian Eder: Parallel Gröbner Basis Computation

We present a new open-source implementation of Faugere‘s F4 Algorithm. The implementation is based on the fast linear algebra library GBLA that is dedicated to matrices arising in Gröbner Basis computation. The current framework is able to construct and to reduce the matrices in parallel using OpenMP. We present timing comparisons with other CAS implementations and give an outlook to new features in the near future.

Bettina Eick: The Isomorphism problem for torsion free nilpotent groups

Claus Fieker: Class Groups is Large Degree Fields Theory and some Computations

Class groups are core invariants of number fields, they control the multiplicative structure of those fields, indeed, for example it is well known that he class group is trivial iff the maximal order is a PID. Furthermore, computationally, deciding if an ideal is principal is based on a successful computation of the class group as well.
While computational class groups have a long history, peaking in the early 90s with Buchmann's (Cohen, Diaz y Diaz and Olivier) subexponential algorithm, they are currently again in the lime-light: the security of various modern cryptographic primitives relies on the difficulty of finding small generators of principal ideals in certain (very) large degree fields. The practical difficulty here lies in the fact, that several of the hidden implicit constants governing the runtimes of the used algorithms are exponential in the degree of the field.
I will given an overview over the newly developed techniques that at least theoretically overcome this problem. I will illustrate what currently is possible and what is still open.
Large portions of the new algorithms are implemented in the newly developed Hecke project. I will demonstrate what is available and explains what still needs to be done.

Anne Frühbis-Krüger: Computing the vanishing topology of determinantal ICMC2 singularities

The topology of the Milnor fibre of a smoothable singularity provides valuable information about the singularity itself; for complete intersection singularities it is known to be a bouquet of spheres. The most accessible case, where this is no longer the case, are Cohen-Macaulay codimension 2
singularities in $({\mathbb C}^5,0)$. For this case, we present an algebraic approach to compute this topological data and explain the conclusions about  the geometry of the singularity which we can draw from it.

Meinolf Geck: A new construction of semisimple Lie algebras and Chevalley groups

We present a new, and quite elementary construction of semisimple Lie algebras and Chevalley groups. This is based on recent results of Lusztig concerning the "canonical basis" of the

Sebastian Gutsche: Categories, algorithms, and programming

In the talk I present Cap, which is a realization of categorical programming written in Gap. Cap makes it possible to compute complicated mathematical structures, e.g., spectral sequences. This can be achieved using only a small set of basic algorithms given by the existential quantifiers of Abelian categories, e.g., composition, kernel, direct sum.

In this talk I will explain the concept of categorical programming. As an example for the computational possibilities of Cap, I will describe an algorithm to compute the purity filtration of a graded module and a coherent sheaf. The purity filtration is the filtration arising from the bidualizing Grothendieck spectral sequence. We will see that using categorical programming in Cap, the same algorithm can be applied to both contexts. This is joint work with Sebastian Posur.

Jürgen Hausen: Computational Theory of Mori dream spaces

Mori dream spaces are algebraic varieties with a finitely generated Cox ring. A well known class of examples are the toric varieties. The strong finiteness properties of Mori dream spaces allow to extend many features of toric geometry to this class, which finally leads to computational
approaches. After a brief general introduction, we give a survey of recent algorithms, results and open problems in the field.

Konstantin Jakob: The Classification of Rigid Irregular G_2-Connections

We outline the explicit construction of rigid irregular connections with differential Galois group G_2, the simple exceptional algebraic group. The construction is carried out using the Katz-Arinkin algorithm which involves middle convolution and Fourier transform of connections. Using methods of differential Galois theory we can completely classify all irreducible rigid connections of this type.

Pinar Kilicer: Some problems related to CM theory

I will first talk about the CM class number one problem and give an answer to this problem for quartic and sextic CM fields. Then I will talk about the construction of CM curves of genus 3 over $\QQ$. Finally, I will talk about the embedding problem and give a solution for CM curves of genus 3.

Markus Kirschmer: Definite hermitian lattices with class number one

The local global-principle states that two hermitian spaces over some number field $K$ are isometric if and only if they are isometric over every completion of $K$.

The genus of a lattice $L$ in an hermitian space consists of those lattices which are isometric to $L$ locally everywhere.
Every genus decomposes into finitely many isometry classes.
The number of isometry classes in a genus is called its class number.
Hence the genera with class number one are precisely those lattices for which the local-global principle holds.

For indefinite lattices, the class number can be expressed a-priori in terms of some local invariants.
For definite lattices, such a description is not possible. However, up to similarity, there are only finitely many genera with a given class number.

In my talk, I will present the classification of all definite hermitian lattices of class number one.

Steffen Koenig: How to describe diagram algebras

Diagram algebras is a common name for algebras like Brauer algebras and their quantisations (BMW- and q-Brauer algebras), for partition algebras, Temperley-Lieb algebras and other algebras that are defined by a basis of diagrams and a graphical way of multiplying diagram
algebras. Such algebras occur for instance in invariant theory, representation theory, combinatorics, topology and mathematical physics. The challenge typically is to find properties and structure, to describe particular representations, to compute numerical invariants such as decomposition numbers and to classify semisimple or symmetric algebras.
Tools include cellular bases and cellular stratifications, double centraliser properties, permutation modules and Schur algebras as well as explicit experiments and computations.

Pavel Metelytsin: Eine Datenbank für Differentialoperatoren vom Calabi-Yau Typ sowie dazugehörige Software

We present the third version of the interactive database of differential operators of Calabi-Yau type and a package for computer algebra system Maple which can be used to manipulate such operators with a short demonstration. We also share some ideas about integration of mathematical databases into computer algebra systems.

Gabriele Nebe: A mathematical framework of network coding.

Yue Ren: Tropical Geometry in Singular

In this talk, we will discuss some features of Singular in tropical geometry, which was made possible thanks to a low-level interface to both gfanlib and polymake. We will touch upon some theoretical challenges that arose when trying to t tropical concepts into the computational framework of Singular, as well as some new algorithms that emerged from the rich

computational toolbox oered by it.

Ute Spreckels: On the Order of CM Abelian Varieties over Finite Prime Fields

Fix a CM abelian variety $A$ over a number field $F$. We ask for the probability that the number of points on the abelian variety $A\mod p$ is prime as $p$ varies. This question has first been treated by Koblitz for elliptic curves, motivated by use of elliptic curves with prime orders for cryptography. We state a conjectural formula for arbitrary dimension and present numerical evidence.

Michael Stoll: The Generalized Fermat Equation x^2 + y^3 = z^{11}

Generalizing Fermat's original problem, equations of the form $x^p + y^q = z^r$, to be solved in coprime integers, have been quite intensively
studied. It is conjectured that there are only finitely many solutions in
total for all triples $(p,q,r)$ such that $1/p + 1/q + 1/r < 1$ (the
hyperbolic case'). The case $(p,q) = (2,3)$ is of special interest, since
several solutions are known. To solve it completely in the hyperbolic case,
one can restrict to $r = 8,9,10,15,25$ or a prime $\ge 7$. The cases $r = 7,8,9,10,15$ have been dealt with by various authors. In joint work with Nuno
Freitas and Bartosz Naskrecki, we are now able to solve the case $r = 11$ and
prove that the only solutions (up to signs) are $(x,y,z) = (1,0,1), (0,1,1), (1,-1,0), (3,-2,1)$. We use Frey curves to reduce the problem to the
determination of the sets of rational points satisfying certain conditions on
certain twists of the modular curve $X(11)$. A study of local properties of
mod-11 Galois representations cuts down the number of twists to be considered.
The main new ingredient is the use of the Selmer group Chabauty' techniques
developed recently by the speaker to finish the determination of the relevant
rational points.

Thorsten Theobald: Imaginary projections of polynomials

We introduce the imaginary projection $\mathcal{I}(f)$ of a multivariate polynomial $f \in \mathbb{C}[\mathbf{z}]$ as the projection of the variety of $f$ onto its imaginary part.
Since a polynomial $f$ is stable if and only if $\mathcal{I}(f) \cap \mathbb{R}_{>0}^n \ = \ \emptyset$, the notion offers a novel geometric view underlying stability questions of
polynomials.

We show that the connected components of the complement of the
imaginary projections are convex, thus opening a central connection
to the theory of amoebas and coamoebas. Building upon this, we
establish structural properties of the components of the complement,
such as lower bounds on their maximal number, prove a complete
classification of the imaginary projections of quadratic polynomials
and characterize the limit directions for polynomials of arbitrary
degree.

Based on joint work with Thorsten Jörgens and Timo de Wolff.

Stefan Toman / Ernst Mayr: Radicals of Term Replacement Systems

Binomial ideals, a subclass of polynomial ideals, can be
identified with certain term replacement systems. This correspondence
can be used to apply tools from both worlds for proofs as it was for
instance done by Mayr and Meyer in 1982 in the proof that the word
problem for binomial ideals is EXPSPACE-complete. While many
operations on binomial ideals like sum, product, intersection, and
saturation can be easily translated to term replacement systems there
is no known corresponding operation to the computation of radical
ideals. In this talk we will propose a definition of radicals of those
term replacement systems and explore the structure of binomial ideals
that allows for this definition.

Christopher Voll: Enumerating lattices invariant under nilpotent algebras of endomorphisms

I will report on recent results on Dirichlet-type generating series enumerating lattices which are invariant under a nilpotent algebra of endomorphisms. Examples of such series include the ideal zeta functions enumerating ideals in nilpotent (Lie) rings. Under certain homogeneity conditions on the algebra of endomorphisms, these series satisfy local functional equations. I will explain how this result is complemented and illustrated by theoretical and computational work of Tobias Rossmann. Relevant preprints include arXiv:1606.04515 and arXiv:1606.00760.

Stefan Wewers: Semistabile Reduktion und L-Reihen von Kurven