A conference in celebration of the work of Bill Crawley-Boevey – Programme

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Shown date applies to first talk in each column.

Shown date applies to first talk in each column.

1 Sep | |
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10:00 | Bernhard Keller |

11:00 | social break |

12:00 | Pierre-Guy Plamondon |

13:15 | Gustavo Jasso |

2 Sep | |
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08:00 | Amnon Neeman |

09:00 | Hiroyuki Minamoto |

10:00 | social break |

10:30 | Tamas Hausel |

3 Sep | |
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10:00 | Andrew Hubery |

11:00 | social break |

12:00 | Raphael Bennett-Tennenhaus |

13:15 | Rosanna Laking |

6 Sep | |
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12:30 | Špela Špenko |

13:30 | Harm Derksen |

14:30 | social break |

15:00 | Victor Ginzburg |

7 Sep | |
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12:00 | Ulrich Bauer |

13:15 | Karin Baur |

8 Sep | |
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10:00 | Alastair King |

11:00 | social break |

12:00 | Jenny August |

13:15 | Wassilij Gnedin |

9 Sep | |
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09:00 | Igor Burban |

10:00 | Julian Külshammer |

11:00 | social break |

12:00 | Sibylle Schroll |

10 Sep | |
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10:00 | Karin Erdmann |

11:00 | social break |

12:00 | Ben Davison |

13:15 | Pavel Etingof |

schedule may be subject to change

Timetable also available as iCalendar file.

Higher Auslander-Reiten theory provides “higher” versions of many of the classical concepts in representation theory, such as abelian categories, torsion classes and the Auslander-Reiten translation. In this talk, I will present joint work with J. Haugland, K. Jacobsen, S. Kvamme, Y. Palu and H. Treffinger where we explore the connection between $n$-torsion classes and $\tau_n$ tilting theory (where $n=1$ is the classical setup). We show that a higher analogue of the classical bijection between torsion classes and support tau-tilting pairs exists, as well as providing a useful characterisation of $n$-torsion classes.

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Topological persistence is a cornerstone of applied topology, enabling the use of methods from algebraic topology in novel and powerful techniques of data analysis. In my talk, I will illustrate the central role and the historical development of persistent homology in different incarnations, connecting recent developments in topological data analysis with classical results in critical point theory and the calculus of variations. Presenting recent work with M. Schmahl and A. Medina-Mardones [1], I will explain how modern persistence theory, in particular recent fundamental structural results by Crawley-Boevey et al. [2,3], provide a new and clarifying perspective on Morse‘s theory of functional topology, which has been instrumental in the first proof of the existence of unstable minimal surfaces by Morse and Tompkins.

[1] Bauer, U., Medina-Mardones, A. M., & Schmahl, M. (2021). *Persistence in functional topology and a correction to a theorem of Morse*. arXiv preprint arXiv:2107.14247.

[2] Crawley-Boevey, W. (2015). “Decomposition of pointwise finite-dimensional persistence modules”. *J. Algebra Its Appl.*, **14**(05), 1550066.

[3] Chazal, F., Crawley-Boevey, W., & de Silva, V. (2016). “The observable structure of persistence modules”. *Homol. Homotopy Appl.*, **18**(2), 247-265.

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Surface combinatorics have been instrumental in describing algebraic structures such as cluster algebras and cluster categories, gentle algebras, etc. In this talk, I will present some of these and then concentrate on combinatorial approaches to cluster structures on the coordinate ring of the Grassmannians.

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The model theory of modules involves interpreting model theoretic notions in terms of module theory. For example, an injective module embedding is pure provided solutions to certain formulas (called pp-formulas) are reflected. A module is $\Sigma$-pure-injective provided any set-indexed coproduct of it is pure-injective: that is, injective with respect to pure embeddings. There are various well-known ways to characterise both pure-injective and $\Sigma$-pure-injective modules.

In this talk, I will begin by replacing the category of modules with a compactly generated triangulated category. The notions of purity in this setting were defined by Krause, and the canonical model theoretic language here was defined by Garkusha and Prest. I will then present some ways to characterise $\Sigma$-pure-injective objects here, analogous to the module category setting. Time permitting, I will try to say something about the proof, and motivate the consideration of endocoperfect objects.

This talk is based on the arxiv preprint 2004.06854.

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Nodal orders are appropriate non-commutative generalizations of the ring
$\mathbb{k}[\![ x, y]\!]/(xy)$. They are characterized by the property to be the only orders of tame (and even derived-tame) representation type.
A non-commutative (projective) nodal curve is a ringed space $\mathbb{X} = (X, \mathcal{A})$, where $X$ is a conventional (projective) curve (over a field $\mathbb{k}$) and $\mathcal{A}$ is a sheaf of nodal orders on $X$.

I shall make a review of results concerning the tameness of the derived category of coherent sheaves $D^b\bigl(\mathsf{Coh}(\mathbb{X})\bigr)$ of a non-commutative nodal curve $\mathbb{X}$. For any such $\mathbb{X}$ there exists a derived-tame finite dimensional $\mathbb{k}$-algebra $\Lambda$ of global dimension two and an exact fully faithful functor
$\mathsf{Perf}(\mathbb{X}) \longrightarrow D^b(\Lambda-\mathsf{mod}).
$
This correspondence will be illustrated by concrete examples and some applications will be given.

My talk is based on joint works with Yuriy Drozd.

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The Hall algebra $A_Q$ built out of the Borel-Moore homology of the stack of representations of a preprojective algebra $\Pi_Q$ contains one half of the Kac-Moody Lie algebra $g_Q$ associated to $Q$, and indeed one half of the Yangian of $Q$. Moreover, this BM homology is entirely situated in even cohomological degrees, and can be shown to contain the universal enveloping algebra of the BPS Lie algebra associated to $Q$, a cohomologically graded Lie algebra, concentrated entirely in even degrees, which recovers all of the Kac polynomials of $Q$ via the taking of characteristic polynomials.

In this talk I will review these results, and then introduce a construction of partially fermionized versions of the above algebras. These new algebras also arise as cohomological Hall algebras, this time built out of the vanishing cycle cohomology of certain central extensions of $\Pi_Q$ considered by Etingof and Rains. These central extensions are determined by a parameter in the Cartan algebra of $g_Q$, which also determines the extent of fermionization that the new algebra exhibits, when compared to $A_Q$.

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For a fixed quiver and a given dimension vector, we can consider the ring of semi-invariants on the representation space.
We will give degree bounds for generators of this ring that are polynomial in the dimension vector. There are also interesting applications to theoretical computer science. This is joint work with Visu Makam.

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Hybrid algebras, introduced in [Hy], form a large class of tame symmetric algebras. As extremes they include weighted surface algebras (which are periodic of period four), but also Brauer graph algebras (which are special biserial). Intermediate versions have ‘clannish’ representations, as in [CB-II] and [CB-III]. With these we get new information on stable Auslander-Reiten components of hybrid algebras.

[CB-II], [CB-III] W. W. Crawley-Boevey, “Functorial filtrations II: Clans and the Gelfand problem. III: Semidihedral algebras”, *J. London Math. Soc.* **40**(1989) 9-30 and 31-39.

[Hy] K. Erdmann, A. Skowroński, *Hybrid algebras*. arXiv 2103.05963.

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I will report on a joint work with K. Coulembier and V. Ostrik. We show that a symmetric tensor category in characteristic $p>0$ admits a fiber functor to the Verlinde category (semisimplification of $Rep(Z/p)$) if and only if it has moderate growth and its Frobenius functor (an analog of the classical Frobenius in the representation theory of algebraic group) is exact. For example, for $p=2$ and $3$ this implies that any such category is (super)-Tannakian. We also give a characterization of super-Tannakian categories for $p>3$. This generalizes Deligne's theorem that any symmetric tensor category over C of moderate growth is super-Tannakian to characteristic $p$. At the end I'll discuss applications of this result to modular representation theory.

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Let X be an algebraic symplectic manifold and A a formal deformation quantization of O_X. We show that the existence of a quantization of a coherent sheaf E on X to an A-module E_A forces the vanishing of homogeneous components of the Chern character ch(E) in a certain range of degrees.
In the special case where the support of E is a (possibly singular) Lagrangian subvariety, this implies that the image of ch(E) in the Borel-Moore homology reduces to the support cycle of E. To prove our vanishing result we obtain a formula for the Chern character of the quantized A-module E_A in the negative cyclic homology in terms of the ordinary Chern character of E.

Let X be a conical symplectic resolution and B the algebra of global sections of a filtered quantization of X. We use the vanishing result to prove that the characteristic cycles of finite dimensional simple B-modules are linearly independent.

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At the beginning of the nineties, Rickard showed that the derived representation theory of an $\mathsf{R}$-free Noetherian algebra $\Lambda$ over a complete local ring $\mathsf{R}$ is closely related to that of its quotient $\Lambda/\mathfrak{m} \Lambda$, where $\mathfrak{m}$ denotes the maximal ideal of the local ring $\mathsf{R}$.
More precisely, he established a bijection between certain tilting complexes of the algebra $\Lambda$ and tilting complexes of the algebra $\Lambda/\mathfrak{m}\Lambda$.

The broader notion of silting complexes
has attracted much interest recently, after Aihara and Iyama have shown that silting complexes can always be mutated to produce new ones.

My talk is motivated by the question how the silting theory of the ring $\Lambda$ differs from that of
one of its quotients $\overline{\Lambda}$ or its tensor product $\Lambda \otimes \Gamma$ with another $\mathsf{R}$-algebra $\Gamma$.

It turns out that there is a bijection between silting complexes of the $\mathsf{R}$-free Noetherian algebra $\Lambda$ and silting complexes of its quotient $\Lambda /\mathfrak{a} \Lambda$ for any proper ideal $\mathfrak{a}$ of the ring $\mathsf{R}$, as well as an embedding of silting complexes in the context of faithfully flat base change.

Using a variation of the silting bijection, the problem to classify silting complexes of certain derived-wild rings, which include some affine preprojective algebras and all Brauer graph algebras, can be reduced to the classification problem of silting complexes of certain derived-tame rings, which are nodal orders.

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I will discuss three occurances of Frobenius algebras. First in joint work with Letellier and Villegas on quiver representations over Frobenius algebras. Second is an analogue of McWilliams identity for dual codes over finite graded commutative Frobenius algebras. Finally in joint work with Hitchin we study multiplicity algebras of the Hitchin integrable system on certain Lagrangians, which in the very stable case are graded commutative Frobenius algebras.

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The Deligne-Simpson Problem arises from the study of regular differential equations on the Riemann sphere, and via the Riemann-Hilbert Correspondence can be expressed in terms of connections on locally-free sheaves on the projective line.

We will review Bill Crawley-Boevey's work on the Deligne-Simpson Problem, including his proof of the additive version, his results on the multiplicative version, as well as our joint work towards completing the classification of the irreducible solutions. This important achievement of Crawley-Boevey draws on his earlier work concerning deformed preprojective algebras, their multiplicative analogues (joint with P. Shaw), as well as his reformulation of the problem in terms of connections on sheaves on weighted projective lines.

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Exact categories were introduced by Quillen in 1973 as part of his
foundational work on higher algebraic K-theory. Exact categories are
ubiquitous in representation theory of rings and algebras: they arise
as module categories, categories of Gorenstein projective modules,
categories of cochain complexes, etc. Much more recently, in 2015,
Barwick introduced the class of exact infinity-categories in order to
prove an infinity-categorical version of Neeman's Theorem of the Heart.
Rouhgly speaking, Barwick's exact infinity-categories are a
simultaneous generalisation of Quillen's exact categories and of
Lurie's stable infinity-categories (the latter can be thought of as
enhancements of Verdier's triangulated categories).

In this talk, I will explain the role of Barwick's exact infinity-
categories in representation theory of rings and algebras. I will also
explain how Cisinki's results on infinity-categorical homotopical
algebra can be leveraged to study localisations of exact
(infinity-)categories and their derived infinity-categories.

This is a report on joint work in progress with Sondre Kvamme
(Uppsala), Yann Palu (Amiens) and Tashi Walde (TU Munich).

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In 2010, Claire Amiot conjectured that algebraic 2-Calabi-Yau categories
with cluster-tilting object must come from quivers with potential. This
would extend a structure theorem obtained with Idun Reiten in the case
where the endomorphism algebra of the cluster-tilting object is
hereditary. Many other classes of examples are also known. We will
report on recent progress in the general case.

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In the context of dimer models on a disc, I will explain how treating perfect matchings as modules for the associated dimer algebra can shed some light on the cluster combinatorics of Grassmannians.

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One of Bill's earliest contributions to representation theory is the theorem that for a tame algebra all but finitely many modules of each dimension lie in homogeneous tubes. Its proof relies on the theory of bocses (bimodules over a category with coalgebra structure). In joint work with Koenig and Ovsienko we used this theory, together with A-infinity-Koszul duality, to construct an exact Borel subalgebra for every quasi-hereditary algebra up to Morita equivalence. In this talk I will present a uniqueness result of these subalgebras. This is joint work with Vanessa Miemietz.

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In this talk we will consider a large class of t-structures in compactly generated triangulated categories called cosilting t-structures. In particular, we will consider those determined by pure-injective cosilting objects, which have the desirable property that the heart is a Grothendieck abelian category. Examples of such t-structures in the derived category of a finite-dimensional algebra include those induced by derived equivalences with Grothendieck categories and those “lifted” from bounded t-structures in the bounded derived category (in the sense of Marks-Zvonereva). In this talk we will define and explore a mutation operation on such t-structures that naturally extends the mutation operation on silting t-structures in the bounded derived category of a finite-dimensional algebra. This is a report on joint work with Lidia Angeleri Hügel, Jan Stovicek and Jorge Vitória.

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This talk is a report on a joint work with M. Herschend in which we study a class of central extensions of the preprojective algebras under the name “quiver Heisenberg algebras (QHA)”.

We note that QHA is special case of “central extensions of the preprojective algebras” introduced by Etingof-Rains,
which is a special case of $N=1$ quiver algebras introduced by Cachazo-Katz-Vafa, which is obtained as a pull-back of
the deformation family of the preprojective algebas introduced by Crawley-Boevey-Holland.
In the previous talks about QHA, I gave a description of QHA as a modules over the path algebra KQ, which implies the dimension formula by Etingof-Rains in the case where Q is Dynkin.

The central aim of this talk is to provide an understaunding of the description.
Recall that left almost split map is nothing but a left approximation with respect to the radical functor $rad$.
Thus, universal Auslander-Reiten triangle given in the previous talks tells that degree 1-part of QHA provides left minimal approximation with respect $rad$ of KQ-modules.

The main theorem of this talk tells that degree n-part of QHA provides left minimal approximation with respect to the n-th power of radical functor $rad^{n}$ of KQ-modules. From this result the above mentioned description of QHA immediately follows.

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A metric on a category assigns lengths to morphisms, with the triangle inequality holding. This notion goes back to a 1974 article by Lawvere. We'll begin with a quick review of some basic constructions, like forming the Cauchy completion of a category with respect to a metric.

And then will begin a string of surprising new results. It turns out that, in a triangulated category with a metric, there is a reasonable notion of Taylor series, and an approximable triangulated category can be thought of as a category where many objects are the limits of their Taylor expansions. And then come two types of theorems: (1) theorems providing examples, meaning showing that some category you might naturally want to look at is approximable, and (2) general structure theorems about approximable triangulated categories.

And what makes it all interesting is (3) applications. These turn out to include the proof of a conjecture by Bondal and Van den Bergh, a major generalization of a theorem of Rouquier's, and a short, sweet proof of Serre's GAGA theorem.

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The g-vector fan of an algebra is a polytopal object defined from tau-tilting theory. In this talk, we will study the case when the fan has finitely many cones: we will see that, under certain conditions on the algebra, the fan is related to a set of generators of the Grothendieck group of the extriangulated category of 2-term complexes of projective modules. We will then give applications to the theory of finite type cluster algebras. Finally, we will look at some cases where the fan is not finite, and in particular at tame algebras. This is a report on joint works with A.Padrol, Y.Palu, V.Pilaud and T.Yurikusa.

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In this talk on joint work in progress with Ed Green and Eduardo Marcos we will introduce varieties of finite dimensional modules over the path algebra of a quiver with relations. Modules in the same variety have the same dimension vector and share certain homological properties. The construction of the module varieties is closely related to the construction of varieties of algebras defined in earlier work with Ed Green and Lutz Hille.

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The derived category of an algebraic variety might be a source of its myriad new (categorical) symmetries. Some are predicted by homological mirror symmetry, as representations of the fundamental group of complex structures of its mirror pair. These finally lead to differential equations. We will explain that in the case of (a class of) toric varieties we get GKZ hypergeometric systems. This is a joint work with Michel Van den Bergh.

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