Program

Breakfast is served 8:00-9:00, Lunch 12:30-13:30, Dinner 18:30-19:30

Auslander-Reiten theory was introduced in the 1970s to representation theory of algebras. Since then it has become one of the cornerstones of the subject due to its numerous theoretical and computational applications both inside and outside the field. Higher Auslander-Reiten theory was introduced by Iyama in the early 2000s as a variant of the classical theory that is adapted to the study of 'higher-dimensional' objects that arise in algebra and geometry. Nowadays, Iyama's higher Auslander-Reiten theory is a very active topic of research in representation theory of algebras with connections and applications to commutative algebra, algebraic geometry, symplectic geometry, combinatorics, etc.

The purpose of this lecture series is to discuss the role that higher structures (differential graded algebras and A-infinity algebras) play in Iyama's higher Auslander-Reiten theory, as well as their crucial role in the recent proof of the Donovan-Wemyss Conjecture in the context of the three-dimensional Homological Minimal Model Program. Rather than discussing the theory in detail, we will mostly focus on the relationship between higher Auslander-Reiten theory and Calabi-Yau categories.

Geometric surface models in the representation theory of finite dimensional algebras have their origin mainly in the theory of cluster algebras and their categorifications. In particular, the representation theory of a class of quadratic monomial algebras, the so-called gentle algebras, is closely governed by surface combinatorics. Indeed, geometric surface models not only encode their module categories but also their derived categories. Furthermore, it has been shown by Haiden-Katzarkov-Kontsevich and Lekili-Polishchuk that, based on the associated surface model, the bounded derived category of a homologically smooth and graded gentle algebra can be viewed as the partially wrapped Fukaya category of a surface with stops, opening up an interesting interplay between algebra and geometry. In this lecture series we will introduce graded gentle algebras and show how to construct their geometric surface model. We then show how they appear in the context of partially wrapped Fukaya category of surfaces with stops and explore the representation theoretic consequences of the geometric surface combinatorics. In particular, we will see that we can determine silting objects on the surface and this enables us to construct a complete derived invariant which was conjectured and constructed by Lekili and Polishchuk. Time permitting, we will also see how we can use the surface geometry to construct a family of counter-examples to a conjecture of Bondal and Polishchuk on the transitivity of the braid group action on full exceptional sequences. Much of the content of these lectures is based on joint works with Wen Chang, Fabian Haiden, Haibo Jin, Sebastian Opper, Pierre-Guy Plamondon, and Zhengfang Wang.

Various well-known algebras that appear in representation theory can be studied geometrically via Fukaya categories. Among these are gentle algebras, higher Auslander algebras, (derived) preprojective algebras, zigzag algebras, NCCRs for cDV singularities, (derived) contraction algebras (the list goes on - come to the lectures to find out!). Such geometric realizations give insight into the representation theory of these algebras. For example, symmetries of derived categories of these algebras become manifest as they get induced by the action of the symplectomorphim groups. In this lecture series, we will focus on computations of Fukaya categories that relate to representation theory. Some of these are based on my previous joint works with Polishchuk, Ueda, Dyckerhoff and Jasso. More recent aspects involving equivariant Fukaya categories and contraction algebras are based on joint works with Segal and Evans.

Higher Auslander correspondence establishes a bijection between the set of equivalence classes of d-cluster tilting subcategories of Artin algebras and Morita classes of d-dimensional Auslander algebras. The aim of this talk is to give a symplectic interpretation to this correspondence. This result relies on a realisation of the algebras on both sides of the bijection as endomorphism algebras of a collection of generators of Fukaya-Seidel categories of Lefschetz fibrations.

Let \((A, \leq_A)\) be a quasi-hereditary algebra over an algebraically closed field \(k\). For such algebras, König defined in [1] the concept of an exact Borel subalgebra, which is inspired by Borel subalgebras of Lie algebras.

In general, not every quasi-hereditary algebra admits an exact Borel subalgebra [1, Example]. However, in [2] König, Külshammer and Ovsienko showed that for every quasi-hereditary algebra there is a Morita-equivalent quasi-hereditary algebra which admits an exact Borel subalgebra. Their proof crucially uses the A-infinity structure on the Yoneda algebra \(\operatorname{Ext}_A^*(\Delta, \Delta)\) where \(\Delta\) is the direct sum of the standard modules.

Suppose \(G\) is a finite group acting on \(A\) via automorphisms. In this setting, a much studied object is the skew group algebra \(A*G\). Supposing a natural compatibility condition of \(\leq_A\) with the group action, we will show that there is an induced partial order \(\leq_{A*G}\) such that \((A*G, \leq_{A*G})\) is a quasi-hereditary algebra. Moreover, if \(B\) is an exact Borel subalgebra of \(A\) such that \(g(B)=B\) for all \(g\in G\), then we will prove that \(B*G\) is an exact Borel subalgebra of \((A*G, \leq_{A*G})\). This part of the talk is based on the preprint [3].

Time permitting, we will also present current work in progress on how to use the construction of [2] to show that there is a Morita equivalent quasi-hereditary algebra \((R, \leq_R)\) which is equipped with a \(G\)-action compatible with this Morita-equivalence and which admits an exact Borel subalgebra $B$ such that \(g(B)=B\) for all \(g\in G\).

[1] Steffen König; Exact Borel Subalgebras of Quasi-Hereditary Algebras, I; Mathematische Zeitschrift, 1995.
[2] Julian Külshammer, Steffen König and Sergiy Ovsienko; Quasi-Hereditary Algebras, Exact Borel Subalgebras, A-\(\infty\)-Categories and Boxes; Advances in Mathematics, 2014.
[3] Anna Rodriguez Rasmussen; Quasi-Hereditary Skew Group Algebras; https://arxiv.org/abs/2305.06825, 2023.

Motivated by homological mirror symmetry, we study a series of singular surfaces (called log del Pezzo). I will describe the category arising in the B side, using the McKay correspondence and explicit birational geometry. If time permits, I will discuss some preliminary results obtained on the A side, which relate to results on string junctions from the physics literature. The description of the B side is joint with Giulia Gugiatti, while the work on the A side is in collaboration with Giulia Gugiatti and Matt Habermann.

We use twisted Hodge diamonds in order to compute Hochschild cohomology and construct non-Fourier-Mukai functors as A-n-deformations of closed immersions. In particular we observe that there are geometric shadows of these non-geometric deformations and how one can identify them.

Higher hereditary algebras play a central role in Iyamas higher Auslander-Reiten theory. This class of algebras is divided into the n-representation finite and the n-representation infinite algebras, and the (in)finite-ness is in turn detected by the corresponding higher preprojective algebras. In the n-representation infinite case, the preprojective algebra is (n+1)-Calabi-Yau. It is therefore interesting to ask which (n+1)-Calabi-Yau algebras arise as higher preprojective algebras, and of which n-representation infinite algebras. Motivated by McKay Correspondence, we investigate this question for skew-group algebras constructed from a finite subgroup of \(SL_n\) acting on the polynomial ring in n variables. We explain a general approach to this question, and the open problems involved in a general answer. Then we give a complete answer for the case of abelian subgroups of \(SL_3\), which is the n=2 case of type \(\tilde{A}\). Lastly, we mention what this has to do with Gorenstein quotient singularities.