The concept of $S$-characters of finite groups was introduced by Zhmud' (1995) as a generalization of transitive permutation characters. Any non-trivial $S$-character takes a zero value on some group element. By a deep result depending on the classification of finite simple groups a non-trivial transitive permutation character even vanishes on some element of prime power order. J-P. Serre asked whether this generalizes to $S$-characters. We provide a translation of this question into the language of polyhedral geometry and thereby construct many counterexamples, using the capabilities of the computer algebra system OSCAR. The work on S-characters is joint with Thomas Breuer und Gunter Malle. OSCAR is joint work with The OSCAR Development Team, currently lead jointly with Simon Brandhorst, Claus Fieker, Tommy Hofmann and Max Horn

For a wide range of subcategories of the module category of a finitely generated algebra, we show a variant of the inductive step of the second Brauer-Thrall conjecture. That is, if there are infinitely many non-isomorphic indecomposable modules of the same finite dimension in the subcategory, then there are infinitely many dimensions that each admit infinitely non-isomorphic indecomposable modules in the subcategory. This also implies a variant of the first Brauer-Thrall conjecture in this context. The subcategories in question are the constructible subcategories, which are those that consist of all modules that vanish on a finitely presented functor. A key ingredient of the proof is a new connection between the Ziegler spectrum and schemes of finite dimensional modules that allows for a geometric approach. An important step is to find a suitable curve inside a constructible subset of the scheme. This result is contributed by Andres Fernandez Herrero.

The notion of strong generation can be used to extract properties of a geometric or algebraic object from its derived category. For example, the global dimension of a Noetherian ring is equal to the minimum number of steps it takes to generate every object from the regular module. A result of Neeman shows that a Noetherian separated scheme is regular and of finite dimension if and only if its perfect derived category admits a strong generator. However, unlike the algebraic situation, there is no candidate generator playing the role of the regular module. This means that the problem of extracting the precise dimension of a scheme using this technique remains open. Orlov has conjectured that the dimension of a smooth projective variety is equal to the smallest generation time ranging across all possible generators. We take a different approach, motivated from the algebraic situation, by taking our candidate generator to be all of the indecomposable injectives. We’ll present some results showing that one can bound the dimension of the scheme in this way.

In this talk, I will adress the question of classification of commutative and cocommutative (a.k.a. abelian) Hopf algebras over a field. From an algebro-geometric point of view, these are abelian group schemes, and thus any classification must be at least as hard as the classification of not necessarily finitely generated abelian groups. In characteristic 0, it is in fact about equally hard, but in characteristic p, the question becomes significantly more complicated. As abelian Hopf algebras form an abelian category, their classification can be reduced to the classification of modules over certain rings. Under stronger assumptions, namely the presence of a grading, a perfect ground field, and p-torsion pure-injective Hopf algebras, I will present a complete classification via certain string modules along with inroads to a classification without the p-torsion assumption.