# BiPb Research Seminar

A joint research seminar with researchers from Bielefeld and Paderborn on current topics in Algebra, Number Theory and Geometry takes place 6 times a year. It usually is scheduled on **Firday afternoons and alternates between Bielefeld and Paderborn**. Due to the COVID pandemic the seminar currently takes plase as a virtual seminar via zoom.

*Organizers:*

Kai-Uwe Bux (Uni Bielefeld)

Tobias Weich (Uni Paderborn)

### Seminar Program

11.06.2021 | zoom (contact Tobias Weich for login details) |

Sprecher: | Christopher Voll |

Titel: | Representation zeta functions of integral quiver representations |

Abstract: | An integral representation of a finite quiver (i.e.\ directed graph) is an assignment of a $\mathbb{Z}$-module (of finite rank, say) to each vertex of the quiver and an $\mathbb{Z}$-module homomorphism to each arrow. A subrepresentation is just an assignment of submodules of the modules for each vertex, which are compatible with the homomorphisms. The representation zeta function associated with an integral representation is the Dirichlet generating series enumerating the representation's subrepresentations. In this generality, these functions arise naturally in various contexts calling for the enumeration of (tuples of) lattices invariant under linear operators. Examples and applications include Dedekind and other ideal zeta functions, Solomon's zeta functions and P-partition generating functions. I will report on recent joint work (https://arxiv.org/abs/2006.12346) with Seungjai Lee (Seoul National University) on representation zeta functions associated with integral nilpotent representations. Results I shall explain include a self-reciprocity result for representation zeta functions associated with "homogeneous" such representations, as well as connections with combinatorial algebra. |

25.06.2021 | zoom (contact Tobias Weich for login details) |

Sprecher: | Igor Burban |

Titel: | The classical Yang-Baxter equation and algebraic geometry |

Abstract: | Classical Yang-Baxter equation (CYBE) plays an important role in the modern theory of classical integrable systems. In my talk, I am going to give a survey of main properties of solutions of CYBE and explain which algebraic, analytic and geometric structures naturally arise in its framework. In particular, I shall explain how solutions of CYBE can be classified in terms of appropriate coherent sheaves of Lie algebras on elliptic curves and their degenerations and illustrate this approach by concrete examples. |

09.07.2021 | zoom (contact Tobias Weich for login details) |

Sprecher: | Claudia Alfes-Neumann |

Titel: | Indefinite Theta functions and period polynomials |

Abstract: | In this talk I will give a short introduction to the theory of theta functions and indefinite theta functions (mostly covering work of Vignéras and Zwegers). Moreover, I will introduce period polynomials of modular forms and their zeta polynomials and report on a connection to Ehrhart polynomials established by Ono, Rolen and Sprung. |

11.10.2019 | 14:15 -- Room T2-149 -- Uni Bielefeld |

Sprecher(in): | Tobias Weich |

Titel: | Von Ruelle Resonanzen zu Ruelleschen Zeta Funktionen |

Abstract: | In diesem Vortrag werde ich zuerst den Begriff einer Ruelle Resonanz als Polstelle einer meromorph fortgesetzten Resolvente einführen und den Zusammenhang zum exponentiellen Mischen der dynamischen Systeme erklären. In einem zweiten Teil des Vortrages möchte ich dann darauf eingehen, wie die selben meromorphen Resolventen genutzt werden können um dynamische Zeta Funktionen meromorph fortzusetzen. |

| 14:15 -- Room V2-200 -- Uni Bielefeld |

Sprecher(in): | William Crawley-Boevey |

Titel: | Clannish algebras revisited |

Abstract: | We are concerned with classifying the finitely generated indecomposable modules for a finite-dimensional associative algebra, or more generally a ring, or some related situation, such as objects in a derived category. There are a number of situations where classifications have been obtained in terms of so-called strings and bands. This includes string algebras, clannish algebras (introduced by the speaker in 1989), Dedekind-like rings and nodal algebras. I shall review some of this work, with examples from geometry, topology and arithmetic. In addition, I aim to describe some improvements to my earlier work on clannish algebras. |

Sprecher(in): | Fabian Januszewski |

Titel: | A cohomological approach to characters of Lie groups |

Abstract: | Global characters were introduced by Harish-Chandra as an important tool in the study of the representation theory of real reductive groups. While they share many properties of characters of finite and compact groups, their original definition and theory is inherently analytic, despite the fact that Harish-Chandra's approach to the classification of irreducible admissible representations is essentially algebraic. In this talk, I will give a homological definition of characters in this setting which on the one hand extends to larger categories of representations than Harish-Chandra's does, and on the other hand extends to other setups where the field of complex numbers is replaced by an arbitrary field or ring. The original motivation for such a theory comes from number theory and mathematical physics, where such contexts have appeared naturally. |

| 14:15 -- Room D1-320 -- Uni Paderborn |

Sprecherin: | Margit Rösler |

Titel: | Dunkl operators and special functions associated with root systems |

Abstract: | Classical special functions such as the Gaussian hypergeometric function are known to play an important role in the analysis on symmetric spaces of rank one. There are multivariable generalizations of these functions within the rational and trigonometric Dunkl theories, which allow far-reaching generalizations of the spherical analysis on Riemannian symmetric spaces. |

Sprecherin: | Barbara Baumeister |

Titel: | Dual approach to Coxeter and Artin groups and applications |

Abstract: | In the talk I will introduce into the broad field of non-crossing partitions. Specially I will focus on the dual approach to Coxeter and Artin groups and give an application to representation theory of algebras. |

10.01.2020 | 14:30 -- Room V2-200 -- Uni Bielefeld |

Sprecher: | Henning Krause |

Titel: | Local versus global for representations of algebras |

Abstract: | We consider some classes of finite dimensional algebras and discuss the classification of thick subcategories for their module categories. Typical examples are path algebras of quivers or group algebras of finite groups. This leads naturally to the study of derived categories. When a cohomology ring is acting, we may pass from global to local and obtain in some interesting cases a stratification of the module category. |

Sprecher: | Kai-Uwe Schmidt |

Titel: | Highly nonlinear functions |

Abstract: | The nonlinearity of a Boolean function in n variables is its Hamming distance to the set of all affine Boolean functions in n variables. Boolean functions with large nonlinearity are difficult to approximate by affine Boolean functions, which is of significant interest in cryptography. The largest possible nonlinearity of a Boolean function in n variables also equals the covering radius of the [2^n,n+1] Reed-Muller code, whose determination is subject to a famous conjecture from the 1980s. In this talk, I will survey the history of this conjecture and then explain how the conjecture can be proved using a mixture of number-theoretic and probabilistic arguments. I will also discuss generalisations of this conjecture. |