Participant Talks

A limited number of participants will have the opporutunity to give twenty-minute talks (or longer) presenting their research. If you would like to give a talk, please state so in your registration form and we will get in touch with you. After the participants talks have been selected, their titles and abstracts will be displayed on this page.

Minimal classes and maximal subgroups, Oliver Braun

(Oliver's slides can be found on his homepage)

In this talk we discuss an algorithmic way to determine the conjugacy classes of maximal finite subgroups of certain infinite groups. Specifically we study the full module-automorphism groups $\mathrm{GL}(L)$ of lattices $L$ over the ring of integers of an imaginary quadratic number field $K$. The finitely many isomorphism types of such lattices are parametrized by the ideal class group of $K$. Our approach, which is a refinement of G. Voronoi's algorithm to determine all perfect Hermitian forms on $L$, allows us to verify that - in some cases - non-isomorphic lattices have non-isomorphic automorphism groups. These results originate from my master thesis, written under the supervision of Prof. Gabriele Nebe at RWTH Aachen University.

Calculating theta series of lattices over totally real number fields, David Dursthoff

(David's slides can be found on his homepage)

In my talk I will construct theta series of lattices over totally real number fields. A theta series of a lattice with totally positive quadratic form is a generating function for the number of vectors of the same length. If the lattice is even and unimodular the theta series turns out to be a Hilbert modular form and has a nice expansion in two variables. Using trace lattices (over the integers) we calculate the first coefficients in this expansion for some lattices over $\mathbb{Q}[\sqrt{2}]$ and $\mathbb{Q}[\sqrt{3}]$.

Rediscovering a matroid in the representations of its automorphism group, Jens Eberhardt

Matroids generalize various concepts of independency, for example, a vector matroid $M=M[A]$ describes which multisets of columns of a matrix $A$ are independent. We observe that an intertwining matrix of a monomial representation affords a matroid and a subgroup of its automorphism group naturally. In this talk I will discuss the inverse problem. Given a matroid $M$ and $G \leq Aut(M)$, one tries to find a monomial representation $S\rightarrow \mathbb{C^*} \wr G$ together with an intertwining matrix $A$, such that $A$ represents $M$. I will present a theorem, which restricts the search to representations of a covering group of $G$, and allows to decide the problem algorithmically. As an application, we will take a look at some classical geometric configurations arising in the representation theory of their automorphism groups.

Quasiprojectivity of toric varieties via convex continuous piecewiese linear functions, Michal Farnik

(Michal's slides can be found here)

Projectivity and quasi-projectivity are one of the most important concepts in algebraic geometry. Verifying whether an abstract algebraic variety is quasi-projective is often nontrivial. We will discuss a solution to this problem for toric varieties. A toric variety can be encoded by a fan (finite set of cones) in a real vector space. We will show how to use piecewise linear functions to describe ample divisors on a toric variety and classify all possible embeddings into projective space. For non-quasiprojective toric varieties we will show how to determine all maximal with respect to inclusion quasiprojective open subvarieties.

Deciding kernel membership of the sheafification functor on toric varieties, Sebastian Gutsche

In this talk I will give a constructive description of coherent sheaves over toric varieties. For this I will use the notion of Serre quotient of computable categories. A category A is called computable if all existential quantifiers are constructive. Given a computable category $\mathcal{A}$ and a thick subcategory $\mathcal{C} \subset \mathcal{A}$ the Serre quotient category $\mathcal{A}/\mathcal{C}$ is computable if for an $M \in \mathcal{A}$ the membership $M \in \mathcal{C}$ is decidable. Given a normal toric variety $X$ with Cox ring $S$ we have $\mathfrak{Coh} ( X ) \cong S-\text{grmod} / \text{ker } \textrm{Sh}$, where $\textrm{Sh}:\ S-\text{grmod} \rightarrow \mathfrak{Coh} ( X )$ denotes the sheafification functor. This means $\mathfrak{Coh} ( X )$ is computable if it is decidable whether the sheafification $\widetilde{M}$ of a module $M$ is zero. Given such a graded module $M \in S-\text{grmod}$ it is $\Gamma ( U_\sigma, \widetilde{M} ) = ( M_{x^{\widehat{\sigma}}} )_0$, where $U_\sigma \subset X$ is the affine subvariety associated to a maximal cone $\sigma \in \Sigma$, and $x^{\widehat{\sigma}}$ the corresponding generator of the irrelevant ideal. In my talk I will present a combinatorial algorithm to present $( M_{x^{\widehat{\sigma}}} )_0$ as the cokernel of a morphism of free $(S_{x^{\widehat{\sigma}}})_0$-modules. The algorithm is completely combinatorial, and it does not compute $S_{x^{\widehat{\sigma}}}$. Using this algorithm, one can decide whether the sheafification of a graded module over the Cox ring is trivial, so the category $\mathfrak{Coh}( X )$ is computable as a Serre quotient.

Algorithmic aspects of integral representation theory of finite groups, Tommy Hofmann

(Tommy's slides can be found on his homepage)

In representation theory of finite groups it is well known that a complex character of a finite group $G$ can already be obtained from a representation $G \to \operatorname{GL}_n(K)$, where $K$ is a finite extension of $\mathbf Q$, i.e., $K$ is a number field. It is then only natural to ask whether we can adjust this representation such that our matrices are elements of $\operatorname{GL}_n(\mathcal O_K)$, where $\mathcal O_K$ denotes the ring of integers of $K$. This is even a nontrivial problem for small groups like the quaternion group of order eight. So far only ad hoc methods are known to decide this question for particular representations. By translating this question into the setting of orders and lattices we will show how this problem can be settled using modular data of the representation and the class group of $K$.

Counting rational points on toric varieties via Cox rings, Marta Pieropan

(Marta's slides can be found here)

Let $X$ be a smooth complete split toric variety over a number field $k$, with anticanonical sheaf generated by global sections. Manin's conjecture predicts an asymptotic formula for the number of $k$-rational points of $X$ with bounded anticanonical height that depends, up to a multiplicative constant, only on the rank of the Picard group of $X$. This talk presents a proof of the above conjecture based on a parametrization of $X(k)$ via the universal torsor of $X$.

Homology groups of unit groups of orders, Sebastian Schönnenbeck

(Sebastian's slides can be found on his homepage)

In homology theory for groups it is often very useful to have a free resolution of the trivial G-module readily available. However for an infinite group it is fairly complicated to compute such a resolution. In this talk we will discuss an algorithm to solve this problem when G is the unit group of a maximal order in a simple rational algebra. In this case one may consider the action of the given group on the space of positive definite quadratic forms which is equipped with a G-invariant cell structure. Following an idea of G. Ellis it is possible to combine the cellular chain complex of this decomposition with resolutions for certain finite stabilizers to form the desired free resolution. In the case where the algebra is a matrix ring over an imaginary quadratic number field we will have a look at an implementation of the algorithm and the computed results. This talk will give a short summary of the results of my master thesis which I am currently writing under the supervision of G. Nebe.