Representation Theory of Algebras and its ApplicationsHSM Special Topic School

Jul 6, 2026, 8:00 AM → Jul 10, 2026, 8:00 PM Europe/Berlin

Endenicher Allee 60/1-016 - Lipschitzsaal (Mathezentrum)

Pavel Barikin (Hausdorff Center for Mathematics)

In recent years, the representation theory of finite-dimensional algebras has gained increasing prominence, driven by its deep connections and applications to a variety of active areas of mathematics. These include the theory of cluster algebras, topological data analysis, representation theory of posets, and algebraic combinatorics, among others.

With this Special Topics School on "Representation Theory of Algebras and its Applications", we aim to bring together experts, young researchers, and graduate students to explore current developments in the field. The school will offer a series of introductory lecture courses on some of these recent connections, complemented by problem sessions.

Our goal is to foster an interactive and collaborative atmosphere, providing participants with insights into ongoing research directions. The school will also offer opportunities for participants to present their own work, exchange ideas, and build connections within the representation theory community.

The deadline for the application for participation is May 31^st, 2026.

Our Global Mobility Fellowships provide participation opportunities for selected researchers and PhD students from countries of the Global South in the Special Topic Schools of the Hausdorff School for Mathematics. For more information, please visit our website.

Lecture series by:

• Johanne Haugland (NTNU)
• Volodymyr Mazorchuk (Uppsala University)
• Khrystyna Serhiyenko (University of Kentucky)
• Hugh Thomas (UQAM)

Additional talks by:

• Antonia Kekkou (University of Utah)
• Azzurra Ciliberti (Ruhr-Universität Bochum)
• Jonas Nehme (Unviersity of Bonn)
• Marvin Plogmann (Universität zu Köln)

Minitalk Speakers:

• Carlo Klapproth (University of Stuttgart)
• Cyril Matousek (Aarhus University)
• Daniel Perniok (University of Paderborn)
• Emily Poelders (University of Bonn)
• Adina Veronica Remor (Federal University of Paraná)
• Annoy Sengupta (IIT Kanpur)

Tutors for the Lecture Series:

• Viktória Klász (University of Bonn)
• Judith Marquardt (Université Grenoble Alpes)
• Iacopo Nonis (University of Leeds)
• Mads H. Sandoy (NTNU)

Scientific Organizers:

• Viktória Klász (University of Bonn)
• Markus Kleinau (University of Bonn)
• Rene Marczinzik (University of Bonn)

Abstracts School Representation Theory.pdf

Additional Information

Schedule School Representation Theory.pdf

Useful information.pdf

Mon, July 6

8:15 AM Self-Registration

1 Syzygy categories over Iwanaga-Gorenstein algebras

Syzygies are a fundamental topic of study in both commutative and non-commutative algebra. They are defined as submodules of projectives and appear as kernels of projective resolutions that are used to approximate every module. It is then natural to study the category of all syzygies, particularly when the entire module category cannot be explicitly understood.

In this series of talks, we will explore syzygy categories over Iwanaga-Gorenstein algebras of Gorenstein parameter 1, particularly focusing on the class of 2-Calabi-Yau tilted algebras. In this case, it is known that the syzygy category is triangulated and equivalent to the singularity category of an algebra as well as the category of maximal Cohen-Macauley modules. It is an open question to characterize algebras of finite CM type. As a first step in this direction, we will investigate the behavior of syzygy categories when passing from an algebra A to its quotient A/Ae_iA by an idempotent e_i, which corresponds to deleting a vertex from the quiver of A. Consequently, we describe various equivalent characterizations for these two algebras to have equivalent syzygy categories. We will then focus on the special class of Cohen-Macauley finite algebras, called dimer tree algebras and their skew group algebras, and provide a geometric model for their syzygy categories in terms of their associated checkerboard polygons.

Speaker: Prof. Khrystyna Serhiyenko (University of Kentucky)

10:00 AM Break

2 Koszul algebras in representation theory

Koszul algebras were first defined by Priddy in 1970. These algebras have been extensively studied and arise naturally in various fields of mathematics, such as algebraic geometry, noncommutative geometry, topology and number theory. In this lecture series, we give an introduction to Koszul algebras and the role they play in representation theory. Many central classes of algebras in representation theory turn out to be Koszul. Examples include hereditary algebras, gentle algebras, quadratic monomial algebras, polynomial algebras and exterior algebras, as well as certain preprojective algebras and trivial extensions.

A main reason for the importance of Koszul algebras is their duality theory, as studied in the influential paper "Koszul duality patterns in representation theory" by Beilinson, Ginzburg and Soergel. To any Koszul algebra, there is an associated Koszul dual algebra. A key topic in the mini-course is the Koszul duality equivalence and how it reflects the strong connection between a Koszul algebra and its dual. As a motivating example, we look at how Koszul duality manifests in the geometric model for gentle algebras due to Opper, Plamondon and Schroll.

A core perspective in the lecture series is the notion of higher Koszul algebras, or n-T-Koszul algebras, which yields a natural connection to Iyama's higher Auslander–Reiten theory. This framework builds on a generalization of T-Koszul algebras, as introduced by Madsen and Green, Reiten and Solberg. We discuss a higher version of classical Koszul duality and sketch some applications for n-hereditary algebras. This part of the lecture series builds on joint work with Mads H. Sandøy, who is also responsible for the exercise classes in the course.

Speaker: Dr Johanne Haugland (NTNU)

3 Regular Sequences in Triangulated Categories

Regular sequences are a fundamental tool in commutative algebra. In this talk, we introduce regular sequences in R-linear triangulated categories, where R is a graded-commutative ring, and give some examples. As an application of this definition, we show that the length of regular sequences provides lower bounds on levels. This is joint work with Janina C. Letz and Marc Stephan.

Speaker: Antonia Kekkou (University of Utah)

12:00 PM Lunch Break

4 Representations of lattices, Auslander regularity, and Coxeter permutations

In this minicourse, I will be focusing on the representation theory of lattices (that is to say, of their incidence algebras). This turns out to be an interesting setting in which to explore Auslander regularity and related notions, in part because they make contact with certain already studied notions within lattice theory, which I will also discuss. This minicourse assumes no prior knowledge of lattice theory, and the representation theory background required is fairly minimal.

I will explain the definition of Auslander regularity, and then present the result of Iyama and Marczinzik showing that a lattice is Auslander regular if and only if it is distributive. I will also explain how Auslander regularity (of incidence algebras of lattices) can be read off from the Coxeter matrix. This leads naturally to the notion of Coxeter permutation, and the rowmotion operation on distributive lattices. Finally, I will discuss the extension of rowmotion to semidistributive lattices, the way this manifests in terms of the Coxeter matrix, and how this can also be understood in representation-theoretic terms.

Speaker: Prof. Hugh Thomas (Université du Québec à Montréal)

3:00 PM Break

5 Exercise Class

There will also be exercises in rooms N0.007 and N0.008.

Speaker: Prof. Khrystyna Serhiyenko (University of Kentucky)

6 Exercise Class

There will also be exercises in rooms N0.007 and N0.008.

Speaker: Dr Johanne Haugland (NTNU)

Tue, July 7

7 Homological properties of category O

In this series of lectures I will try to describe
various homological properties of BGG category O
and their relation to the Kazhdan-Lusztig
combinatorics of Hecke algebras.

The plan is as follows: the first lecture will
be about Hecke algebras and Kazhdan-Lusztig
combinatorics. The second lecture will be
about basics of category O and its connection with
Hecke algebras. Finally, the third lecture
will focus on various homological properties,
in particular, on projective dimension of
structural modules and Auslander regularity.

The excercise classes will treat small rank
cases in detail, including sl_2 (type A_1)
and sl_3 (type A_2).

Speaker: Prof. Volodymyr Mazorchuk (Uppsala University)

10:00 AM Break

8 Syzygy categories over Iwanaga-Gorenstein algebras

Speaker: Prof. Khrystyna Serhiyenko (University of Kentucky)

9 A Caldero-Chapoton map for the derived category of gentle algebras

The derived category of a gentle algebra A can be described via the geometry of surface dissections, where indecomposable complexes of A-modules correspond to possibly infinite graded arcs on the surface, and morphisms between them are encoded by crossings between the associated arcs. Moreover, their mapping cones are given by the resolution of these crossings.
We introduce a Caldero-Chapoton map in this setting. Specifically, we associate a Laurent polynomial to every finite indecomposable complex, and show that skein relations hold whenever the corresponding arcs cross in the interior. For each complex, its Caldero-Chapoton map specializes to the corresponding element of the Grothendieck group. Furthermore, if A is hereditary, the algebra generated by these functions is an ordinary cluster algebra.
This is joint work in progress with Esther Banaian, Ilaria Di Dedda, Khrystyna Serhiyenko, Yadira Valdivieso-Díaz and Kayla Wright.

Speaker: Azzurra Ciliberti (Ruhr Universität Bochum)

12:00 PM Lunch Break

10 Koszul algebras in representation theory

Speaker: Dr Johanne Haugland (NTNU)

3:00 PM Break

11 Exercise Class

There will also be exercises in room N0.007.

Speaker: Prof. Volodymyr Mazorchuk (Uppsala University)

12 Exercise Class

There will also be exercises in room N0.007.

Speaker: Prof. Hugh Thomas (Université du Québec à Montréal)

6:00 PM Get-Together

Wed, July 8

13 Mini Talks

11:00 AM Break

14 Open Problem Sessions

12:20 PM Lunch Break

Thu, July 9

15 Representations of lattices, Auslander regularity, and Coxeter permutations

Speaker: Prof. Hugh Thomas (Université du Québec à Montréal)

10:00 AM Break

16 Homological properties of category O

Speaker: Prof. Volodymyr Mazorchuk (Uppsala University)

17 Categorification of the quantum electrical algebra

We introduce the quantum electrical algebra, which is a quantum version of the electrical Lie algebra governing electrical networks. We will then provide a categorification of these algebras via electrical KLR algebras mimicking the classical construction of categorification of the positive half of quantum groups. In addition, this electrical KLR algebra bears close connections to the representation theory of the periplectic Lie superalgebra. (jt. with Catharina Stroppel)

Speaker: Jonas Nehme (University of Bonn)

12:00 PM Lunch Break

18 Syzygy categories over Iwanaga-Gorenstein algebras

Speaker: Prof. Khrystyna Serhiyenko (University of Kentucky)

3:00 PM Break

19 Exercise Class

There will also be exercises in room 1.008.

Speaker: Prof. Khrystyna Serhiyenko (University of Kentucky)

20 Exercise Class

There will also be exercises in room 1.008.

Speaker: Prof. Volodymyr Mazorchuk (Uppsala University)

7:00 PM Conference Dinner 'Tuscolo'

Fri, July 10

21 Koszul algebras in representation theory

Speaker: Dr Johanne Haugland (NTNU)

10:00 AM Break

22 Representations of lattices, Auslander regularity, and Coxeter permutations

Speaker: Prof. Hugh Thomas (Université du Québec à Montréal)

23 On extended hearts of t-structures

For a finite-dimensional algebra A, the category mod(A) can be
recovered inside the bounded derived category of A as the full
subcategory of objects whose cohomology is concentrated in degree 0. A
natural enlargement is obtained by considering, for a fixed n>0, those
objects whose cohomology is concentrated in degrees 1-n,...,0. This
subcategory is called the n-extended heart of the standard t-structure.
For n=1, one recovers the abelian category mod(A), while for n>1 the
extended heart is no longer abelian. Nevertheless, it retains many
features reminiscent of module categories; for example, it admits
Auslander--Reiten sequences. I will explain why extended hearts are
useful for studying finite-dimensional algebras and their derived analogues.

Speaker: Marvin Plogmann (Universität zu Köln)

12:00 PM Lunch Break

24 Homological properties of category O

Speaker: Prof. Volodymyr Mazorchuk (Uppsala University)

3:00 PM Break

25 Exercise Class

There will also be exercises in room 0.006.

Speaker: Dr Johanne Haugland (NTNU)

26 Exercise Class

There will also be exercises in room 0.006.

Speaker: Prof. Hugh Thomas (Université du Québec à Montréal)