In this talk we will talk about the relationship between the stratifying systems defined by Erdmann and Sáenz and the τ-tilting theory introduced by Adachi Iyama and Reiten. In the first part of the talk we will start the talk explaining how the properties of the mutation process on tau-tilting pairs enables us to build at least one stratifying from every τ-rigid module.

In the second part of the talk we will change gears slightly and speak about Cartan matrices as an invariant for stratifying systems, as it was recently proposed by Marcos, Mendoza and Marcos. In particular we will speak how the Cartan matrix for a stratifying systems induced by a τ-rigid module can be computed using the g-vectors of the said τ-rigid module.

This is a report on joint work with Octavio Mendoza and Corina Sáenz.

https://arxiv.org/abs/1904.11903

https://arxiv.org/abs/2111.11376

Unlike one-parameter persistence modules, for which we have the barcode, persistence modules with two or more parameters do not admit a complete discrete invariant, and thus incomplete invariants must be used to study the structure of such modules in practice. The Hilbert function and the multigraded Betti numbers are among the simplest such incomplete invariants. Although these two invariants are already being used in applications, it is a priori unclear whether they satisfy a stability result analogous to the stability of the one-parameter barcode. Stability results are essential for the interpretability and consistency of practical methods. I will present joint work with Steve Oudot in which we prove stability results for multigraded Betti numbers and for the Hilbert function. I will also discuss ongoing work in which we prove the stability of finer invariants coming from different exact structures on a category of multiparameter persistence modules.

Paper: https://arxiv.org/abs/2112.11901

Examples of structure-preserving functors between extriangulated categories, so-called extriangulated functors, include the canonical functor from an abelian category to its derived category, and the quotient functor from a Frobenius exact category to its stable category. The first, for example, is structure-preserving in the sense that short exact sequences are sent to distinguished triangles in a functorial way. In higher homological algebra, we also see examples of structure-preserving functors, but not covered by the current terminology. E.g. n-cluster tilting subcategories sitting inside an ambient abelian category. In an attempt, with R. Bennett-Tennenhaus, J. Haugland and M. H. Sandøy, to place these kinds of more general situations in a formal framework, we have been led to a new perspective on extriangulated (or, more generally, n-exangulated) functors. The aim of my talk is to explain this.

Gentle algebras form a class of finite dimensional algebras introduced by Assem and Skowroński in the 80’s. Indecomposable modules over such an algebra admit a combinatorial description in terms of strings and bands, which are walks in the associated gentle quiver (satisfying some further conditions), thanks to the work of Butler and Ringel. A subcategory C of modules is said to be Jordan recoverable if a module X in C can be recovered from the Jordan forms, at each vertex, of a generic nilpotent endomorphism. This data is encoded by a tuple of integer partitions.

After we have introduced some definitions and set the context, the main aim of the talk is to explain the notion of Jordan recoverability through various examples, and to highlight a combinatorial characterization of when that property holds for some special subcategories of modules. This result is extending the work of Garver, Patrias and Thomas in Dynkin types. If time allows, we may discuss some open questions related to this result and, in particular, exhibit new ideas to characterize all the subcategories of modules that are Jordan recoverable in the A_n case.

This is a part of my Ph.D. work supervised by Hugh Thomas.

The Castelnuovo-Mumford regularity of a graded module provides a measure of how complicated its minimal free resolution is. In work with Rajchgot, Ren, Robichaux, and St. Dizier, we noted that the regularity of Matrix Schubert Varieties can be easily obtained by knowing the degree of the corresponding Grothendieck polynomial. Furthermore, we gave explicit, combinatorial formulas for the degrees of symmetric Grothendieck polynomials. In this talk, I will present a combinatorial degree formula for arbitrary Grothendieck polynomials. This is joint work with Oliver Pechenik and David Speyer.

In this talk I will discuss the notion of an idempotent (co)algebra (or perhaps more descriptively, idempotent *dg* (co)algebra). The two-sided bar complex of an algebra gives an especially important class of examples, but there is a plethora of other examples appearing throughout mathematics. I will describe how the endomorphism algebra of an idempotent (co)algebra naturally admits the structure of a Gerstenhaber algebra, generalizing a well known structure on Hochschild cohomology.

Integrable systems first came to prominence as a geometric abstraction of structure in classical mechanics. Despite these origins, integrable systems have been found to interact meaningfully with pure mathematics. Modern examples include the role such systems play in the Langlands program, mirror symmetry, and quantum cohomology. On the other hand, Mishchenko-Fomenko systems represent another paradigm of integrable systems in pure mathematics. They exhibit the kind of Lie-theoretic symmetry that allows difficult geometric problems to be posed and solved entirely in algebraic terms. The algebraic incarnations of Mishchenko-Fomenko systems are the so-called shift-of-argument algebras, which enjoy connections to geometry and representation theory.

I will give a non-technical overview of the themes mentioned above. Some emphasis will be placed on work in progress with Iva Halacheva and Valerio Toledano Laredo.

Let k be an algebraically closed field and A a finite dimensional k-algebra. The universal localization of A with respect to a set of morphisms between finitely generated projective A-modules always exists. Moreover, when A is hereditary, Krause and Stovicek proved that the universal localizations of A are in bijection with various natural structures.

In this talk, based on joint work with Peter Jorgensen, I will introduce the higher analogue of universal localizations, that is universal localizations of d-homological pairs with respect to certain wide subcategories, and show a (partial) generalisation of Krause and Stovicek result in the higher setup.

A celebrated result by Keller and Reiten says that 2-Calabi–Yau tilted algebras are Gorenstein and stably 3-Calabi–Yau, in particular Jacobian algebras over an algebraically closed field satisfy this. Jacobian algebras are 2-Calabi-Yau tilted as proven by Amiot. These results originated several conjectures in the opposite direction: Are all 2-Calabi-Yau tilted algebras Jacobian? (Amiot 2011 - Kalck Yang 2020). We show that the converse holds in the monomial case: a 1-Gorenstein monomial algebra that is stably 3-Calabi–Yau has to be 2-Calabi–Yau tilted, moreover it has to be Jacobian. This result can be explained fully in a 50 mins seminar, so I aim to do that.

Abstract: Indecomposable modules over string algebras were classified by Butler and Ringel, and take exactly one of two forms: string modules, defined by walks in the quiver; or band modules, given by cyclic walks together with a representation of the Laurent polynomial ring. Clannish algebras, introduced by Crawley-Boevey, are a generalisation of string algebras – where one specifies a set of special loops, each bounded by some quadratic polynomial. An analogue of Butler and Ringel’s result was given where the class of string (or band) modules splits into so-called asymmetric and symmetric subclasses. Said symmetry is given by reflecting the walk about a special loop, and symmetric strings and bands are parameterised using appropriate replacements for the Laurent polynomial ring.

Both string algebras and clannish algebras were defined over a field, and the quadratics bounding special loops were assumed to have distinct roots in this field. This talk will be about ongoing joint work with Bill Crawley-Boevey, where we generalise the classification for clannish algebras. For the rings we consider we replace this field with a division ring equipped with a set of automorphisms, indexed by arrows of the quiver, and we allow irreducible quadratics to bound the special loops. The resulting notion of a semilinear clannish algebra recovers a generalisation of string algebras considered by Ringel, where one allows the map associated to an arrow to be semilinear with respect to its automorphism.

One approach to studying the representation theory of Lie algebras and their associated quantum groups is through combinatorial shadows known as crystals. While the original representations carry an action of the braid group, their crystals carry an action of a closely related group known as the cactus group. I will describe how we can realize this combinatorial action both geometrically, as a monodromy action coming from a family of ‘’shift of argument’’ algebras, as well as categorically through the structure of certain equivalences on triangulated categories known as Rickard complexes. Parts of this talk are based on joint work with Joel Kamnitzer, Leonid Rybnikov, and Alex Weekes, as well as Tony Licata, Ivan Losev, and Oded Yacobi.

Macdonald polynomials are distinguished symmetric functions that have played important or useful roles in a wide range of fields: combinatorics, enumerative geometry, integrable systems, probability, knot theory, etc. Defined by Haiman, wreath Macdonald polynomials are generalizations of Macdonald polynomials wherein the symmetric groups are replaced with their wreath products with a fixed cyclic group Z/rZ. I will discuss work in progress where, using the quantum toroidal algebra of rank r, one can derive analogues of some standard parts of Macdonald theory: orthogonality, evaluation formulas, and difference operators. This is joint work with Daniel Orr and Mark Shimozono.

This is a report on joint work in progress with Nicholas Williams.

Maximal green sequences were introduced by Keller in the studies of connections between cluster algebras and quantum dilogarithm identities. In a broader sense, such sequences are given by maximal chains in lattices of torsion classes tors A in module categories of finite-dimensional algebras A. Recently, lattices of torsion classes have been a subject of intensive research. Demonet-Iyama-Reading-Reiten-Thomas proved that for a quiver Q of Dynkin type ADE, the lattice of torsion classes of its path algebra realizes the Cambrian lattice, while the lattice tors Π for the preprojective algebra is isomorphic to the weak Bruhat order on the corresponding Weyl group. The Cambrian map, introduced by Reading in combinatorial terms, can thus be interpreted as a morphism of lattices of torsion classes.

Higher versions of Bruhat and Cambrian orders in type A first appeared in late 80s in works of Manin-Shekhtman and Kapranov-Voevodsky, respectively. Kapranov and Voevodsky also defined a family of maps from higher Bruhat orders to higher Tamari-Stasheff orders (the latter are the higher versions of Cambrian orders in type A). More recently, second Bruhat orders showed up in works by Elias on (monoidal) categories of Soergel bimodules. I will explain how to realize a version of the second Bruhat order for a quiver Q as an order on equivalence classes of maximal green sequences for the corresponding preprojective algebra and the second Cambrian order as a similar order for the path algebra of Q. The second Cambrian map can then be interpreted as the "second level" of the Demonet-Iyama-Reading-Reiten-Thomas map. In type A this provides, in a sense, an additive categorification of the corresponding Kapranov-Voevodsky map, which allows for a representation-theoretic interpretation of its fibers. If time permits, I will also explain how one can interpret second Cambrian orders in terms of polytopes and toric varieties associated with certain subword complexes.

Associated to any acyclic skew-symmetrizable matrix B and a symmetrizer, Geiss, Leclerc and Schröer have defined a finite-dimensional algebra H over any field. Many geometric constructions for acyclic quivers carry over to this situation by using complex numbers. They show that in finite types, the non-initial cluster variables (of the cluster algebra associated to B) are exactly the locally free Caldero—Chapoton functions of indecomposable locally free rigid H-modules and conjecture it to be true in general. We verify this conjecture in rank 2 by showing that the locally free F-polynomials of certain modules under reflection functors satisfy the same recursion of the F-polynomials of cluster variables. This is joint work with Daniel Labardini-Fragoso.

A 2-group is a higher categorical analogue of a group, while a smooth 2-group is a higher categorical analogue of a Lie group. An important example is the string 2-group in the sense of Schommer-Pries. We study the notion of principal bundles for smooth 2-groups, and investigate the moduli "space" of such objects. In particular, in the case of flat principal bundles for a finite 2-group over a Riemann surface, we prove that the moduli space gives a categorification of the Freed-Quinn line bundle. This line bundle has as its global sections the state space of Chern-Simons theory for the underlying finite group. We can also use our results to better understand the notion of geometric string structures (as previously studied by Waldorf and Stolz-Teichner).

The talk will not assume background knowledge on 2-groups or Chern-Simons theory. It is based on joint work with Dan Berwick-Evans, Laura Murray, Apurva Nakade, and Emma Phillips.

Geometric models associated to triangulations of Riemann surfaces arose in the context of cluster algebras and have since been used as an important tool to study representation theory of algebras and provide connections with algebraic geometry and symplectic geometry.

Significant applications of geometric models include a description of extensions and a classification of support tau-tilting modules over gentle algebras. Gentle algebras are a particular subclass of string algebras, which are of tame representation type, meaning it is often possible to get a global understanding of their representation theory.

In this talk I will describe the module category of a gentle algebra via partial triangulations of unpunctured surfaces, explain how to extend this model to a geometric model of the module category of any string algebra and use this model to obtain a classification of support tau-tilting modules. This is based on joint work in progress with Karin Baur.

Exact Borel subalgebras of quasihereditary algebras emulate the role of "classic" Borel subalgebras of complex semisimple Lie algebras. Not every quasihereditary algebra A has an exact Borel subalgebra. However, a theorem by Koenig, Külshammer and Ovsienko establishes that there always exists a quasihereditary algebra Morita equivalent to A that has a (regular) exact Borel subalgebra. Despite that, an explicit characterisation of such "special" Morita representatives is not directly obtainable from Koenig, Külshammer and Ovsienko's work. In this talk, I shall present a numerical criterion to decide whether a quasihereditary algebra contains a regular exact Borel subalgebra and I will provide a method to compute all Morita representatives of A that have a regular exact Borel subalgebra. We shall also see that the Cartan matrix of a regular exact Borel subalgebra of a quasihereditary algebra A only depends on the composition factors of the standard and costandard A-modules and on the dimension of the Hom-spaces between standard A-modules. I will conclude the talk with a characterisation of the basic quasihereditary algebras that admit a regular exact Borel subalgebra.

Representations of the circle generalize representations of Ãn quivers. We will briefly motivate the study of such representations and show how to decompose an arbitrary pointwise finite-dimensional representation. We’ll also talk about isomorphism classes of indecomposable representations. This is joint work with Eric J. Hanson.

In this talk I will present recent applications of representation theory to the study of neural networks in artificial intelligence. First, a neural network can be taken as a representation-like object to which we can apply isomorphisms of quiver representations that preserve what a neural network computes. Second, we can encode the decisions and computations of a neural network on a single sample of data in terms of a stable double-framed thin quiver representation, and since the output of a neural network is independent of the representative in the isomorphism class, it makes sense to consider these "data quiver representations" in a moduli space of stable thin representations.

Understanding the representation theory of a wild quiver Q is an unfeasible task. But if one restricts one’s attention to representations constructed from a fixed 'model', one may hope that only a small number of indecomposable representations can be ‘realized’. In this talk, I will discuss a few models for generating quiver representations from ‘data', computational experiments, and relevant theorems emerging from recent work with U. Bauer, S. Oppermann, and J. Steen.

Semi-invariant pictures (or wall and chamber structures) arise naturally when considering stability conditions for finite dimensional algebras. For algebras which are not tau-tilting finite, these semi-invariant pictures contain accumulation points. In this talk, we describe a new semi-invariant picture which captures the local structure near such an accumulation point. For tame hereditary algebras, we further show that this new semi-invariant picture can be described (both geometrically and via tau-tilting theory) from those of certain Nakayama algebras. This is joint work with Kiyoshi Igusa, Moses Kim, and Gordana Todorov.

For a quiver Q, denote by J(Q) the ideal of the path algebra KQ generated by the arrows.

A central role in Iyama's higher dimensional Auslander–Reiten theory is played by n-cluster tilting modules. However, such modules are not so easy to find. In this talk, I will present a simple criterion that characterises all bound quiver algebras of the form KQ/J(Q)^2 that admit an n-cluster tilting module for some n>1. This criterion is based only on the shape of Q.

A Postnikov diagram consists of a collection of strands in the disc, with combinatorial restrictions on their crossings. Such diagrams were used by Postnikov and others to study weakly separated collections in certain matroids called positroids. In this talk I will explain how the diagram determines a cluster algebra structure on a suitable subvariety of the Grassmannian, and simultaneously provides a (Frobenius) categorification of this cluster algebra.

After reviewing discrete cluster categories of type A and their completions, we will see how the properties of these categories (which do not have a Serre functor, and hence not 2-Calabi-Yau) still allow one to define a cluster character map on some infinite dimensional but finitely presented representations. This is joint work with Emine Yildirim.

Motivated by wanting to extend the functorial approach of Auslander to exact categories, we introduce the category of admissibly presented functors mod_{adm}(E) for an exact category E. In particular, we characterize exact categories equivalent to categories of the form mod_{adm}(E), and we show that they have properties similar to module categories of Auslander algebras. For a fixed idempotent complete category C, we also use this construction to show that exact structures on C are in bijection with certain resolving subcategories of mod C, and we compare this with the bijection to certain Serre subcategories of mod C due to Enomoto.

Motivated by a new conjecture on tau-tilting infinite algebras, we study minimal tau-tilting infinite algebras as a modern counterpart of minimal representation infinite algebras. This talk begins by a discussion of some fundamental similarities and differences between these two families of algebras. Then we relate our studies to the classical tilting theory. In particular, for each minimal tau-tilting infinite algebra A, we show that the mutation graph of tilting A-modules is infinite and n-regular at almost all vertices, where n is the rank of Grothendieck group of A.

This is a report on my joint work with Charles Paquette.

The g-vector fan of a finite-dimensional algebra encodes information about several of its properties, including its tau-tilting theory and its set of stability conditions. In this talk, I will present known and conjectural properties of this fan, focusing on the case of a tame algebra.

This is based on a joint work with Toshiya Yurikusa.

Frieze patterns, introduced by Coxeter, are infinite arrays of numbers where neighbouring numbers satisfy a local arithmetic rule. Under a certain finiteness assumption, they are in one-to-one correspondence with triangulations of polygons [Conway–Coxeter] and they come from triangulations of annuli in an infinite setting [Baur–Parsons–Tschabold]. I will discuss a relationship between pairs of infinite friezes associated with a triangulation of the annulus and how one determines the other in an essentially unique way. We will also consider module categories associated with triangulated annuli where infinite friezes may be recovered using a specialised CC-map.

This is joint work with Karin Baur, Karin Jacobsen, Maitreyee Kulkarni, and Gordana Todorov.

I will talk about an ongoing work with Ian Le (Australian National University).

We look at the categorification of cluster algebras in the context of Jensen-King-Su and try to relate this representation theoretical approach to combinatorics of web basis which is introduced by Kuperberg and studied by Fomin-Pylyavskyy in their work on cluster structures of certain invariant rings.

The aim of this talk is to review the utility of studying framed versions of moduli spaces of quiver representations. We first review the general construction of framed moduli spaces, and discuss several classes of examples (for example, acyclic quivers and quiver Grassmannians, m-loop quivers and explicit normal forms). Turning to the topology of framed quiver moduli spaces, we state a formula for their Betti numbers, and exhibit a coupled system of functional equations relating Euler characteristic of framed an unframed moduli spaces. Finally, we study the geometry of the Hilbert-Chow map from framed to unframed moduli spaces, and derive a formula for the intersection Betti numbers of unframed moduli spaces from this.

Classical frieze patterns are combinatorial structures which relate back to Gauss’ Pentagramma Mirificum, and have been extensively studied by Conway and Coxeter in the 1970’s.

A classical frieze pattern is an array of numbers satisfying a local (2 × 2)- determinant rule. Conway and Coxeter gave a beautiful and natural classification of SL(2)-friezes via triangulations of polygons. One way to generalise the notion of a classical frieze pattern is to ask of such an array to satisfy a (k × k)-determinant rule instead, for k at least 2, leading to the notion of higher SL(k)-friezes. While the task of classifying classical friezes yields a very satisfying answer, higher SL(k)-friezes are not that well understood to date.

In this talk, we’ll discuss how one might start to fathom higher SL(k)-frieze patterns. The links between frieze patterns and cluster combinatorics encoded by triangulations of polygons in the k=2 case suggests a link to Grassmannian varieties under the Plücker embedding and the cluster algebra structure on their homogeneous coordinate rings. We find a way to exploit this relation for higher SL(k)-friezes and provide an easy way to generate SL(k)-friezes via Grassmannian combinatorics.

This talk is based on joint work with Baur, Faber, Serhiyenko and Todorov.

In this talk, I'll describe our work towards providing an infinite rank version of the Grassmannian cluster categories introduced by Jensen, King and Su. We show that there is a structure preserving bijection between Plücker coordinates and the generically free modules of rank 1 in our Grassmannian category of infinite rank. In the k=2 case, this allows us to model our category, and its cluster combinatorics, using triangulations of an infinity-gon. This is joint work with Man-Wai Cheung, Eleonore Faber, Sira Gratz and Sibylle Schroll.

In this talk I will completely determine the derived functors (both left and right) of both torsion and cotorsion functors defined on the module category of an arbitrary artin algebra. The obtained results hold even in greater generality. No prior knowledge of (co)torsion or functor categories is assumed.

The talk is based on joint work with Jeremy Russell.

In higher homological algebra, (d+2)-angulated categories are higher analogues of triangulated categories. They primarily appear as d-cluster-tilting subcategories of triangulated categories closed under d-suspension. We give a complete classification of such d-cluster-tilting subcategories of the derived category of a gentle algebra by using the geometric model given by Opper-Plamondon-Schroll. It turns out that, up to derived equivalence, they occur only in Dynkin type A; in other words we have a puzzling lack of d-cluster-tilting subcategories associated to gentle algebras.

This is joint work with Johanne Haugland and Sibylle Schroll.

Cluster categories and cluster algebras can be described via triangulations of surfaces or via Postnikov diagrams. In type A, such triangulations lead to frieze patterns or SL_2-friezes in the sense of Conway and Coxeter. We explain how infinite frieze patterns arise and how Grassmannians give rise to SL_k-friezes.

In this talk I will explain how to construct a new co-t-structure from a given co-t-structure (A,B) and a complete cotorsion pair in its extended co-heart C=Σ^2A∩B. This construction is similar to the Happel-Reiten-Smalo tilt of a t-structure with respect to a torsion pair in its heart and it induces a bijection between intermediate co-t-structures and complete cotorsion pairs. For a finite dimensional algebra this introduces a new component to the numerous existing bijections studied in tau-tilting theory and in silting theory, that is the bijections between intermediate algebraic t-structures, functorially finite torsion pairs, two-term silting complexes, and so on.

Based on joint work with David Pauksztello.

In the seminal paper where Ziegler developed the model theory of modules, he introduced, using model theoretical language, the notion of partial morphism. Later, Monari Martinez characterized this notion in module theoretical terms. In the talk, we will give a categorical characterization of partial morphisms in module categories, which will allow us to extend them to any exact category. Then, we will see how all properties of partial morphisms in module categories extend to exact categories, and we will establish their relationship with cophantom morphisms and injective objects. Finally, we will discuss how partial morphisms can be used to characterize injective hulls, and we will give a condition that implies the existence of such hulls in an exact category.

In 2015, F Dupont and G Palesi defined quasi-cluster algebras from triangulations of non-orientable surfaces. We now associate quivers to these triangulations and define a mutation rule to encode flips of arcs in non-orientable surfaces. We will show some properties of these quivers and their mutation rule; in particular, we will demonstrate that they generalize the classical definitions of quiver and mutation.

Joint work with Linda He, Ruiyang Huang, Hanyi Luo and Kayla Wright.