Graduate Seminar (Paderborn)

The talks happen on Mondays from 11:00 to 12:00 in the classroom D2.314 and we stream them via zoom:
https://uni-paderborn-de.zoom-x.de/j/66180880183?pwd=FoCQiQaFusJRhZ6aKJbDSBz7gpOB1P.1
Meeting ID: 661 8088 0183, Password: gradsempb

25.11 - Daniel Perniok
Title: Hall algebras and quantum groups
Abstract:
Given an abelian category with finite Hom and Ext groups we can associate with it the so called Hall algebra which encodes the structure of extensions between objects in the category. After defining this associative algebra we calculate it in small examples. For this we consider the category of representations of a certain quiver (or equivalently the module category of a certain finite dimensional algebra) that can be understood using only techniques from linear algebra. Finally we talk about Lie algebras, (deformations of) their universal enveloping algebras and how the observations from the sample calculations extend to general results. This builds a bridge between representations of quivers (respectively coherent sheaves on weighted projective lines) and quantum groups.

22.04 - Stepan Maximov
Title: Regular decompositions and integrable systems
Abstract:
An important way of obtaining integrable systems is the r-matrix method. Generalized r-matrices can be constructed using Lie algebra decompositions: they can be viewed as generating series of decompositions.
In this talk, we look at certain decompositions, called regular, of the Lie algebra of formal Laurent series g((x)) and present partial classification results. The r-matrices corresponding to such decompositions give rise to particularly nice Gaudin models.

29.04 - Milan Niestijl
Title: Representation theory of infinite-dimensional Lie groups
Abstract:
After introducing infinite-dimensional Lie groups, we shall discuss the study of their unitary representations. In particular, I will indicate the main challenges that appear in the infinite-dimensional context, and what type of results one might hope for. To that end, a so-called 'positive energy' condition is introduced, which isolates a mathematically interesting class of representations and is motivated from physics. We will explore the consequences of this condition, and present some classification results.

06.05 - Carsten Peterson
Title: The Poincare series of a Coxeter group and its applications
Abstract:
The Chevalley-Solomon theorem provides a formula for the Poincare series of a Coxeter group G, namely the generating function for the number of elements of each length in the group (the length of an element is the length of its shortest representative as a product of Coxeter generators). In this talk we shall present a proof of this theorem which involves studying the action of G on its "Coxeter complex", which is a simplicial complex encoding the combinatorics of the group, and the associated action of G on the chain complex of the Coxeter complex. We shall also discuss why this formula is useful in computing the orders of finite groups of Lie type (i.e. semisimple algebraic groups over finite fields) and the volume of certain naturally defined subsets of buildings.

13.05 - Alexandre Maksoud
Title: Modular forms and congruences
Abstract:
This is a friendly introduction to the topic of modular form, with an emphasis on congruences mod p between their coefficients. We'll start with Ramanujan's famous congruence mod 691 and we'll explain how congruences are related to special values of L-functions.

27.05 - Charly Schwabe
Title: Elliptic Weyl groups
Abstract:
After recalling some basic theory about finite and affine root systems, we will introduce their generalization, the extended affine root systems. We will then classify the elliptic Weyl groups associated to them and investigate a special element called Coxeter transformation. We shall finally use this Coxeter element to motivate a connection to representation theory (and the logo of the CRC).

3.06 - Daniel Perniok
Title: Representation theory of quivers
Abstract:
Originating in fundamental linear algebra problems, the representation theory of quivers has proved useful in many other areas such as representation theory of Lie algebras and Lie groups, geometric group theory or categorification of quantum groups. In this talk we focus on basic definitions and comprehensible examples. Finally we consider the Kronecker quiver and how it gives a different perspective to the study of coherent sheaves on the projective line.

10.06 - Praful Rahangdale
Title: Poisson and related structures
Abstract:
Poisson structure is a Foliation on the manifold, which partitions it into leaves. These leaves have a Symplectic structure, and transverse to the leaves, there is a Lie algebra structure, so Poisson geometry amalgamates Foliation theory, Symplectic geometry, and Lie theory.
The original motivation for studying Poisson structure comes from classical mechanics. Phase spaces of classical mechanical systems are modelled on Poisson manifolds, and the dynamics of the system are governed by a distinguished function called the Hamiltonian of the system. The time evolution of the system is given by the flow of the vector field corresponding to the Hamiltonian. The system is constrained to submanifolds, which are the level sets of functions which Poisson commute with the Hamiltonian, and we hope to constrain the dynamics enough to completely solve the problem.
In this talk, we shall first review the basics of Poisson geometry. Then, we will discuss the local description of a Poisson manifold according to Weinstein splitting theorem, as a product of a symplectic manifold with another manifold transverse to it whose Poisson structure vanishes at the point. I will introduce symplectic realizations of Poisson manifolds, and we will see that any Poisson manifold can be viewed as the quotient of a symplectic manifold. Finally, we will discuss Lie bialgebras, the algebraic counterpart of Poisson Lie groups, and the one-to-one correspondence between connected, simply connected Poisson Lie groups and finite-dimensional Lie bialgebras.

17.06 - Kyungmin Rho
Title: An introduction to homological mirror symmetry
Abstract:
Recent developments in homological mirror symmetry have significantly connected symplectic geometry, algebraic geometry, and representation theory. Fukaya categories of Riemann surfaces have emerged as powerful tools for dealing with representation theory of tame categories. These categories include categories of matrix factorizations, derived categories of coherent sheaves of singular complex curves and derived categories of some finite-dimensional algebras. In this talk, we will explore these extensive fields by examining concrete examples and their correspondences.

24.06 - Abhijit Aryampilly Jayanthan
Title: Étale and profinite-étale vector bundles on elliptic curves
Abstract:
Over an algebraically closed field, vector bundles on elliptic curves were classified by Atiyah in 1957. He showed that for a chosen rank and degree, the indecomposable vector bundles are in one to one correspondence with the closed points of the elliptic curve. Further, every vector bundle can be written as a direct sum of its indecomposable subbundles, completing this classification.
The universal profinite-étale cover of an elliptic curve (over some p-adically complete algebraically closed field) is represented by an perfectoid space. With the help of Atiyah's classification, along with some results by Ben Heuer, one is able to classify the collection of all étale (equivalently Zariski) vector bundles which become trivial on some profinite-étale cover of the elliptic curve.

01.07 - Mingyu Ni
Title: O-minimality and Hodge Theory
Abstract:
It was noticed in 1980s that many properties of semi-algebraic sets could be derived from several simple axioms, which is now known as o-minimal structures. These structures provide a candidate of the framework to develop the notion of tame topology, as envisaged by Grothendieck in 1984, and the techniques have profound applications to number theory and arithmetic geometry. More recently, after the work of B. Bakker, Y. Brunebarbe, B. Klingler, and J. Tsimerman, o-minimal structures have become an important tool in the study of Hodge theory. In this expository talk I would start with basic definitions and examples of o-minimal structures, with a view from and towards algebraic geometry, then discuss the particular applications to Hodge theory.

08.07 - Cancelled

15.07 - Lukas Klawuhn
Title: Why everyone should know association schemes: An introduction to Delsarte Theory
Abstract:
Interesting combinatorial structures can often be characterised as special subsets of association schemes. In his PhD thesis, Philippe Delsarte developed powerful linear programming techniques to prove non-existence and uniqueness results for such structures. Ideas of this type were fundamental in the work for which Marina Viazovska was awarded the Fields medal in 2022. This talk will begin with an overview of Delsarte theory.
The association scheme of the symmetric group is well known. We will introduce it, and then study generalised permutations. They act on the set {1,2,...,n}, whose elements are coloured with one of r possible colours. We consider different notions of transitivity and interpret these algebraically in the appropriate association scheme . We will give existence results showing that there exist transitive sets of generalised permutations that are small compared to the size of the group. Many of these results extend results previously known for the symmetric group.
No particular knowledge of association schemes will be required to appreciate this talk.

21.10 Carlo Kaul (Different room: D3.339)
Title: Modularity of Elliptic Curves and Fermat's Last Theorem
Abstract:
In this talk, I want to give an overview of ideas surrounding the Modularity Theorem (partially) proven by Wiles and Taylor-Wiles in 1994 and show how this result can be used to deduce Fermat's Last Theorem. In particular, I will explain how Galois representations are associated to elliptic curves and give a quick overview on how to construct the Galois representation attached to modular forms in the étale cohomology of moduli spaces with coefficients. Lastly, I want to survey some results generalizing the Modularity Theorem in the realm of the Langlands program.

28.10 - Daniel Kahl (Different room: D3.339)
Title: Euclidean buildings and shift dynamics
Abstract:
In this talk we introduce Euclidean buildings and a shift operator on the set of sectors of the building. We equip the set of sectors with a metric and study some properties of the shift operator on this metric space.

04.11 - Claudius Heyer
Title: What is a 6-Functor Formalism?
Abstract:
The yoga of six operations was introduced by Grothendieck in order to show that several phenomena in the étale cohomology of schemes can be formally deduced from a small set of axioms. Since then these six operations have been constructed in many other contexts such as D-modules, motives and rigid-analytic geometry. But only recently has there been a formal definition of a 6-functor formalism, mainly due to Liu--Zheng and then further simplified by Mann in his PhD thesis. In this talk I will introduce the six functors and their relations, and then explain how they are encoded in the modern definition of a 6-functor formalism. As an application we will see how Poincaré duality and the Künneth formula follow easily from this formalism.

11.11 - Claudius Heyer
Title: What is a Fourier–Mukai functor?
Abstract:
This is a follow-up on my last talk on 6-functor formalisms. I will introduce the 2-category of kernels, which is an incredibly powerful tool attached to any 3-functor formalism. We will see how its morphisms can be viewed as Fourier--Mukai functors. Moreover, one can use this 2-category to encode certain geometric notions (e.g., smooth, étale, or proper). The same idea enables us to study some finiteness properties of sheaves (e.g., universally locally acyclic, compact, or invertible) and how they relate to different kinds of duality. If time permits, I will explain some recent applications to the representation theory of locally profinite groups, which arose in joint work with Lucas Mann.

18.11 - Sebastian Bischof
Title: Roots of flat groups and Bruhat-Tits buildings
Abstract:
For a long time, van Dantzig’s theorem (the set of compact open subgroups forms a neighborhood basis of the identity) was the only deep result for general totally disconnected locally compact (tdlc) groups. In his groundbreaking paper from 1994, Willis introduced the scale of an automorphism α of a tdlc group G as
s(α) := min{[α(U ) : α(U ) ∩ U ] | U ≤ G compact and open}.
We call a compact open subgroup U minimizing for α if the minimum is attained at U . A subgroup H ≤ (G) is called flat, if there exists a subgroup U which is minimizing for all α ∈ H . Willis has shown that to any flat group H one can associate a set Φ(H ) of roots where a root is a suitable homomorphism ℓ : H → Z. In this talk we will give a strategy how to compute Φ(H ) for a given flat group H . Moreover, we will consider the case of Bruhat-Tits buildings in more detail and compare the two notions of roots.

Abhijit Aryampilly Jayanthan, Stepan Maximov