Jonah Berggren, Classical tilting and tau-tilting theory.

Abstract: Tau-tilting theory arose as a generalization of classical tilting theory allowing one to realize a cluster-like structure in the module category of any finite dimensional algebra. Moreover, the classical tilting poset of a finite dimensional algebra A is a subposet of its tau-tilting poset. We show conversely that the tau-tilting poset of any finite-dimensional algebra A is obtained as the classical tilting poset of its duplicated algebra. Recall that the duplicated algebra of a finite dimensional algebra A is obtained by taking two copies of A and relating them via the bimodule DA. This generalizes earlier work by Assem, Brüstle, Schiffler, and Todorov, who showed that when A is hereditary, the classical tilting of the duplicated algebra of A agrees with the cluster category of A. This work is joint with Khrystyna Serhiyenko.


Ben Blum-Smith, Multilinear products on invariant rings.

Abstract: The great nineteenth century invariant theorists were concerned with describing their invariants and covariants in terms of data we now recognize as a presentation of a commutative algebra: generators and algebraic relations. However, toward this goal, they made elaborate use of a wealth of additional structure, with names now somewhat obscure: transvectants, the Aronhold process, the Cayley omega process, and more.
    In this talk, we describe a family of n-ary products, for any n, possessed by any nontrivial ring of invariants of a linear group action. This family includes most of the 19th century tools as special cases. The products are linear in their individual arguments; the construction involves an interplay between the invariant ring and the tensor algebra. We show that equipping invariant rings with this added structure often allows to dramatically reduce the number and degrees of generating sets.


Amanda Burcroff, Broken lines and compatible pairs for rank 2 quantum cluster algebras.

Abstract: There have been several combinatorial constructions of universally positive bases in cluster algebras, and these same combinatorial objects play a crucial role in proofs of the famous positivity conjecture for cluster algebras. In rank 2, two notable examples are the greedy basis constructed by Li-Lee-Zelevinsky using compatible pairs on Dyck paths and the theta basis constructed by Gross-Hacking-Keel-Kontsevich using broken lines on scattering diagrams. While these two bases have been shown to coincide via an algebraic proof, a combinatorial proof has remained elusive. I will talk about partial progress toward this end in joint work with Kyungyong Lee. In the more general setting of rank 2 quantum cluster algebras, we construct a quantum-weighted bijection between certain broken lines and compatible pairs. For the quantum Kronecker cluster algebra, we extend this bijection to all broken lines appearing in the theta basis.


Shujian Chen, Combinatorics of signed exceptional sequences.

Abstract: In 2017, Igusa and Todorov introduced signed exceptional sequences and gave a bijection between signed exceptional sequences and ordered partial clusters. In this talk, we will talk about two combinatorial aspects of signed exceptional sequences. The first one is to address its connection to reflection groups, in particular, to the notion of one-way reflections defined by Josuat-Vergès and Biane. The second one is to give a combinatorial model for signed exceptional sequences based on chord diagrams. This is based on joint work with Kiyoshi Igusa.


Xueqing Chen, Some properties of generalized cluster algebras of geometric types.

Abstract: We study the lower bound algebras generated by the generalized projective cluster variables of acyclic generalized cluster algebras of geometric types. We prove that this lower bound algebra coincides with the corresponding generalized cluster algebra under the coprimality condition. As a corollary, we obtain the dual PBW bases of these generalized cluster algebras. Moreover, we show that if the standard monomials of a generalized cluster algebra of geometric type are linearly independent, then the directed graph associated to the initial generalized seed of this generalized cluster algebra does not have 3-cycles. This is a joint work with M. Ding, J. Huang and F. Xu.


Ted Chinburg (expository lecture), The module theory of special extensions, units and numbers.

Abstract: In this talk I will describe how representation theory plays an essential role in Iwasawa theory, which began as the study of the growth rates of class numbers in towers of number fields. I will describe some classical and some new results connecting such growth rates to special units in the towers and to rational numbers arising from L-series values.  The role of representation theory is to phrase the main theorems and conjectures in the subject in terms of invariants of modules for profinite group rings.


Ted Chinburg (delivered on behalf of Frauke Bleher), How close are cup products on curves over finite fields to the Weil pairing?

Abstract: This is joint work with Frauke Bleher. Suppose k is a finite field, C is a smooth projective geometrically irreducible curve over k, and n is a positive integer not divisible by the characteristic of k. In this talk I will consider cup products of elements of the first etale cohomology group H^1(C,Z/n) with  elements of the first etale cohomology group H^1(C,\mu_n). Over the algebraic closure \bar{k} of k, such cup products are connected to values of the Weil pairing on the n-torsion of the Jacobian of \bar{C} by using a fixed isomorphism from Z/n onto the n-th roots of unity in \bar{k}. Over finite fields k, such cup products are more subtle due to the fact that they take values in the group H^2(C,\mu_n) = Pic(C)/n Pic(C) rather than Z/n. In this talk, I will first describe a general formula for the cup products on C using work of McCallum and Sharifi. I will then discuss how one can relate this cup product to the Weil pairing in case C is an elliptic curve. Finally, I will discuss what happens when the genus of C is bigger than 1.


Souvik Dey, TBA

Abstract: TBA


Theo Douvropoulos, TBA

Abstract: TBA


Yuriy Drozd, Nodal orders and action of groups

Abstract: We consider Bäckström and nodal orders over arbitrary discrete valuation rings. A construction which gives all nodal orders is given. We also prove that if a finite group acts on a nodal order and the order of the group is invertible, the crossed product is also nodal. Analogous result is also valid for skewed gentle rings.


Sofia Franchini, Torsion pairs in the completion of Igusa-Todorov discrete cluster categories.

Abstract: From an infinity-gon having a finite number of accumulation points, Igusa and Todorov defined a discrete cluster category which generalises the classical cluster category of type A. Igusa-Todorov category is 2-Calabi-Yau and triangulated, and has a nice geometric model: its indecomposable objects can be identified with arcs of the infinity-gon, and the morphisms can be understood from arc crossings. Paquette and Yıldırım introduced a larger category, the completion, whose indecomposable objects can be regarded as arcs, or limits of arcs, of the infinity-gon. Many important classes of subcategories of the Igusa-Todorov category and its completion can be classified using arc combinatorics. For instance, torsion pairs and t-structures in the Igusa-Todorov category were classified by Gratz-Holm-Jørgensen and Gratz-Zvonareva. In this talk we discuss the classification of torsion pairs, t-structures, and co-t-structures in the Paquette-Yıldırım  completion.


Vitor Gulisz, n-Abelian categories through functor categories.

Abstract: We discuss how to view the axioms of an n-abelian category through its functor categories, which are abelian. This point of view allows the use of classical homological algebra to understand higher homological algebra. As an application, we show how to generalize the axioms “every monomorphism is a kernel” and “every epimorphism is a cokernel” of an abelian category to n-abelian categories. Furthermore, we establish a correspondence for n-abelian categories with additive generators, which extends the Higher Auslander Correspondence.


Emily Gunawan, Pattern-avoiding polytopes and Cambrian lattices via the type A Auslander-Reiten quivers.

Abstract: For each Coxeter element c in the symmetric group, we define a pattern-avoiding Birkhoff subpolytope whose vertices are the c-singletons. We show that the normalized volume of our polytope is equal to the number of longest chains in a corresponding type A Cambrian lattice. Our work extends a result of Davis and Sagan which states that the normalized volume of the convex hull of the 132 and 312 avoiding permutation matrices is the number of longest chains in the Tamari lattice, a special case of a type A Cambrian lattice. Furthermore, we prove that each of our polytopes is unimodularly equivalent to the (Stanley’s) order polytope of the heap H of the longest c-sorting word. The Hasse diagram of H is given by the Auslander - Reiten quiver for certain quiver representations. Our result gives an affirmative answer to a generalization of a question posed by Davis and Sagan. This talk is based on joint work with Esther Banaian, Sunita Chepuri, and Jianping Pan.


Tony Guo, Derived delooping levels and finitistic dimension.

Abstract: The finitistic global dimension conjecture (findim conjecture) has been unsolved for over 60 years since its inception in 1960 by Rosenberg and Zelinsky. Several invariants were developed to bound and analyze the finitistic dimensions (findim), the most recent of which is the delooping level (Gelinas, '20). In this talk, we will improve upon the delooping level by introducing three new invariants, all of which can be strictly better bounds. We will also discuss some applications to tilting theory, injective generation, and contravariantly finite subcategories.


Tony Iarrobino, Limits of Artinian Gorenstein algebras.

Abstract: See N. Abdallah, J. Emsalem, A. Iarrobino, and J. Yam\'{e}ogo: Limits of graded Gorenstein algebras of Hilbert function (1,3^k,1), European Journal of Math.Vol. 10:9, 42 pages, Jan 2024. doi 10.1007/s40879-023-00714-0.


David Jorgensen, Multiplicities in triangulated categories.

Abstract: We introduce notions of multiplicity for pairs of objects in certain triangulated categories. We draw analogies to the classical theory of multiplicity for graded modules over a graded ring. Various applications will also be discussed.


Ray Maresca, Exceptional collections of modules, simple-minded collections, and silting objects.

Abstract: In this talk, we will construct bijections between equivalence classes of simple-minded collections, equivalence classes of silting objects, and exceptional collections of modules over hereditary algebras. We will show how to attain all possible silting objects, simple-minded collections, and hence algebraic t-structures, from the set of all possible exceptional collections of modules. Finally, we will extend to hereditary algebras a result by Keller and Vossieck that states that all silting objects in type A come from tilting objects in some way.


Frantisek Marko, Presentation of rational Schur algebras

Abstract: We present rational Schur algebra S(n,r,s) over an arbitrary ground field K as a quotient of the distribution algebra Dist(G) of the general linear group GL(n) by an ideal I(n,r,s) and provide an explicit description of the generators of I(n,r,s).


Charles Paquette, Bricks and tau-rigid modules

Abstract: For a finite dimensional algebra A, let ind(A) denote the set of isoclasses of indecomposable modules in mod(A). We consider the subsets brick(A) and i\tau-rigid(A) of ind(A) consisting of bricks and \tau-rigid modules, respectively. We show that every brick over A is tau-rigid if and only if A is locally representation directed. In this case, we get ind(A) = brick(A) = i\tau-rigid(A). This extends a result obtained by P. Draexler. We also study the geometric counterpart of this phenomenon at the level of irreducible components of module varieties. Namely, we study indecomposable, brick and generically \tau-reduced irreducible components and explain what happens when some of these classes coincide. A special attention is given to E-tame algebras (which include tame algebras). This is a report on a joint work with Kaveh Mousavand.
 

David Pauksztello, Is convex geometry trying to teach us homological algebra?

Abstract: Arising in cluster theory, the g-vector fan is a convex geometric invariant encoding the mutation behaviour of clusters. In representation theory, the g-vector fan encodes the mutation theory of support tau-tilting objects or, equivalently, two-term silting objects. In this talk, we will describe a generalisation of the g-vector fan which in some sense “completes” the g-vector fan: the heart fan of an abelian category. This convex geometric invariant encodes many important homological properties: e.g. one can detect from the convex geometry whether an abelian category is length, whether it has finitely many torsion pairs, and whether a given Happel-Reiten-Smalo tilt is length. This talk will be a report on joint work with Nathan Broomhead, David Ploog and Jon Woolf.


Matthew Pressland, Quasi-coincidence of positroid cluster structures via categorification.

Abstract: I will give a brief introduction to the categorification of cluster structures on positroid varieties in the Grassmannian. A typical such variety carries a pair of cluster structures which are abstractly isomorphic, but for which different functions are cluster variables. I will explain a proof, via representation theory, that these two structures quasi-coincide, so that they do at least have the same cluster monomials. This confirms a conjecture by Muller and Speyer.


Fan Qin, Analogs of dual canonical bases for cluster algebras from Lie theory.

Abstract: The (quantized) coordinate rings of many interesting varieties from Lie theory are (quantum) cluster algebras. We construct the common triangular bases for these algebras. Such bases provide analogs of the dual canonical bases, whose existence has been long expected in cluster theory. For symmetric Cartan matrices, they are positive and admit monoidal categorification after base change.
Moreover, we will see that the coordinate rings of double Bott-Samelson cells are categorified by representations of quantum affine algebras.


Adrien Segovia, Combinatorics of the lattice of d-torsion classes of the higher Auslander algebras of type A.

Abstract: A combinatorial description of the (higher) d-torsion classes are known for the higher Auslander algebras of type A, and such of their intersection lattices [A characterisation of higher torsion classes, August and al.]. It was observed that unlike the classic case of torsion classes, these lattices are not semidistributive in general. Using the combinatorial description, we proved that these lattices are join-semidistributive and left modular. As both these properties together implies EL-shellability, it gives insight on the topology of their order complexes. These results are obtained as a special case of a general construction that we define on partially ordered sets.


Emre Sen, TBA

Abstract: TBA


Ryan Schroeder, On Modules whose submodules are determined by their dimension vectors.

Abstract: Let A be an algebra over a field k. An A-module M is said to be grin if each submodule is uniquely determined by its dimension vector. This leads to an F-polynomial with coefficients only 0 or 1. In this talk, we give a classification of grin modules provided that M is nilpotent, as well as briefly discuss relationships with cluster algebras and projective geometry.


Hugh Thomas, Flow polytopes and gentle algebras.

Abstract: A recent paper by von Bell, Braun, Bruegge, Hanely, Peterson, Serhiyenko and Yip made an important connection between the tau-tilting theory of (some) gentle algebras and the flow polytopes of (some) oriented graphs. We deepen the connection and relax the conditions on the oriented graphs. This is joint work with a large working group of faculty, postdocs, and students at UQAM.


Blas Torrecillas Jover, Galois theory for cowreaths. Applications to quasi-Hopf algebras

Abstract: Motivated for the study of Galois theory for quasi-Hopf algebra, we develop the theory for cowreaths in monoidal categories. We study cleft extensions and their connections with wreath algebras. We will present several situations where these algebras appear in a natural way: crossed product by a coalgebra, generalized crossed products and quasi-Hopf bimodules. Finally we study Frobenius and separable Galois cowreaths.


Jie Xiao, Lie algebras arising from two-periodic projective complex and derived categories.

Abstract: Let A be a finite-dimensional C-algebra of finite global dimension and consider the category of finitely generated right A-modules. By using of the category of two-periodic projective complexes C2(P), we construct the motivic Bridgeland’s Hall algebra for A, where structure constants are given by Poincaré polynomials in t, then construct a C-Lie subalgebra g = n⊕h at t = −1, where n is constructed by stack functions about indecomposable radical complexes, and h is by contractible complexes. For the stable category K2(P) of C2(P), we construct its moduli spaces and a C-Lie algebra ˜ g = ˜ n⊕˜ h, where ˜ n is constructed by support-indecomposable constructible functions, and ˜ h is by the Grothendieck group of K2(P). We prove that the natural functor C2(P) → K2(P) together with the natural isomorphism between Grothendieck groups of A and K2(P) induces a Lie algebra isomorphism g ∼ = ˜ g. This makes clear that the structure constants at t = −1 provided by Bridgeland in [5] in terms of exact structure of C2(P) precisely equal to that given in [30] in terms of triangulated category structure of K2(P). This is based on the joint work with J. Fang and Y. Lan.


James Zhang, Poisson valuations.

Abstract: We introduce the notation of a Poisson valuation and use it to study automorphism, isomorphism, and embedding problems for several classes of Poisson algebras/fields. This is joint work with Hongdi Huang, Xin Tang, and Xingting Wang.