Last updated: April 22, 2025, 21:58 EST

Abstracts, Slides, Videos, and Schedule

of Distinguished Lectures - 2025

Grant Barkley, Torsion classes for the preprojective algebra of the cycle quiver.

Abstract: Let Pi be the preprojective algebra of an orientation of the n-cycle. We show that the map sending a torsion class T to the set of dimension vectors of the real/rigid bricks in T is a lattice quotient from the lattice of torsion classes of Pi to the lattice of biclosed sets in the positive real roots of an affine type A root system. This gives a combinatorial description of the torsion classes (of nilpotent modules) for Pi in terms of translation-invariant total orders of the integers. In joint work with Colin Defant, we show how this also gives a new description of the torsion classes (of nilpotent modules) for the oriented n-cycle quiver in terms of 312-avoiding translation-invariant total orders.

Frauke Bleher, Triple Massey products for smooth projective curves of arbitrary genus.

Abstract: This is joint work with T. Chinburg and J. Gillibert. 

Massey products originated in the study of higher linking numbers and generalize cup products. In this talk I will consider Massey products of elements of the first etale cohomology group of a smooth projective variety X over a field F with coefficients in Z/ell, when ell is a prime number that is invertible in F.  When X has dimension 0, Minac and Tan showed that such triple Massey products vanish. The next natural case to consider is when X is a smooth projective curve over F. For elliptic curves, I will provide a classification for the non-vanishing of triple Massey products under various natural assumptions. The main tool is the representation theory of etale fundamental groups into upper triangular unipotent matrix groups. For higher genus curves, we additionally need to use a fair amount of topology on mapping class groups and pure braid groups to obtain sufficient criteria for the non-vanishing of triple Massey products.

Amanda Burcroff, Generalized positivity for cluster scattering diagrams. 

Abstract: Cluster algebras are celebrated for their intriguing positivity properties. Two distinct proofs of this positivity have emerged, one through the combinatorics of Dyck paths, and another via scattering diagrams.  Scattering diagrams originate from mirror symmetry and had previously lacked a combinatorial description.  Combining these approaches, we find a directly computable, manifestly positive, and elementary but highly nontrivial formula describing rank 2 scattering diagrams. Using this, we prove the Laurent positivity of generalized cluster algebras of all ranks, resolving a conjecture of Chekhov and Shapiro from 2014.  This is joint work with Kyungyong Lee and Lang Mou.

Ted Chinburg, Lattices arising from number fields.

Abstract: In this talk I will discuss some lattices in Euclidean space that arise from the integers O_N of number fields N with small root discriminants relative to their degrees.  Two standard questions about lattices are to understand their successive minima and their fundamental domains.  I'll discuss some work with Xuxi Ding showing that the successive minima of O_N as above are all nearly the same, but their directions are not random.  I'll then describe some work with Frauke Bleher, Xuxi Ding, Nadia Heninger, Daniele Micciancio and Adam Suhl on bounding the size of fundamental domains for an infinite family of O_N.  

Calin Chindris, The Jordan type of a multiparameter persistence module.

Abstract: In this talk, we present an approach to defining invariants of persistence modules over a poset without relying on their decomposition into direct sums of indecomposables. Our methodology is inspired by the theory of modules of constant Jordan type for elementary abelian p-groups. Let P be a poset and S a sequence of finite subsets of P. We define the Jordan type of a P-persistence module M at S as the Jordan type of a nilpotent operator T, which is constructed from M and S.

The nilpotent operator T is functorial in M, and this functoriality allows us to define the Jordan filtered rank invariant of M. We first show that these invariants are strictly finer than the classical rank invariants. Furthermore, we demonstrate that the Jordan filtered rank invariants are complete for persistence modules over finite zigzag posets.

For P-persistence modules M and N, we also prove that the erosion distance between their Jordan filtered rank invariants is bounded from above by the interleaving distance between M and N. This is based on joint work with Min Hyeok Kang and Dan Kline.

Manuel Cortés-Izurdiaga, Flat objects in finitely accessible additive categories.

Abstract: In module categories, flat objects can be characterized via purity: a module is flat if and only if every epimorphism onto it is pure. The appropriate context for defining purity more generally is that of finitely accessible additive categories—that is, additive categories that admit direct limits, possess a set of isomorphism classes of finitely presented objects, and in which every object can be expressed as a direct limit of finitely presented ones.

A foundational result by Crawley-Boevey shows that every finitely accessible additive category is equivalent to the full subcategory of flat functors in a functor category—that is, the category of additive functors from a small category to the category of abelian groups.

In this talk, we will explore how the categorical properties of the subcategory of flat functors relate to the torsion theory they cogenerate. In particular, we will show that if the category of flat functors has enough flat objects, then the associated torsion theory is Jansian.

Theo Douvropoulos, Deformations of restricted reflection arrangements.

Abstract: The Shi and Catalan arrangements associated to a Weyl group W are deformations of its reflection arrangement that have been studied in combinatorics, representation theory, and geometry: Their dominant regions are labelled by non-nesting partitions (closely related to non-crossing partitions and clusters in the assosiated cluster algebra); the Shi regions carry a W-action that gives a combinatorial model for the W-module-structure of finite dimensional Cherednik algebras; the Catalan regions encode non-isomorphic crepant resolutions of quotient singularities associated to W.

We define analogues of the Shi and Catalan deformations for restrictions of Weyl arrangements (which are not always crystallographic) extending the combinatorial and representation theoretic properties. This is in part joint work with Olivier Bernardi.

Yuriy Drozd, Lattices and cohomologies of the alternative group of order 4.

Abstract: A complete description of lattices over the alternative group of order 4 is given and all their cohomologies are calculated.

Vitor Gulisz, 0-Abelian categories.

Abstract: About 10 years ago, Jasso introduced the notion of an n-abelian category, which is analogous to an abelian category, but which is defined in terms of n-kernels and n-cokernels instead of kernels and cokernels. In this context, n is an arbitrary positive integer, and the case n = 1 recovers the concept of an abelian category. However, there seems to be one case that we have been missing all this time, namely, when n = 0. In this talk, we will address this problem and present a definition of a 0-abelian category, which will be based on 0-kernels and 0-cokernels. Moreover, we will discuss a few basic results on 0-abelian categories, including a result that is analogous to the long exact sequence in abelian categories.

Tony Guo, Some applications of the delooping and derived delooping levels 

Abstract: The delooping level, which was introduced in 2020, is an upper bound of the big finitistic dimension (Findim). We motivate and define the derived delooping level (ddell) as an improvement of the delooping level both in terms of an upper bound of Findim and satisfying more enjoyable properties. We also discuss some applications of the two invariants including a symmetry condition arising from a special construction of triangular matrix algebras and their behavior in monomial algebras.

Tony Iarrobino, Equations for Jordan type strata in spaces of commuting nilpotent matrices. 

Abstract: (work joint with Mats Boij and Leila Khatami)

We determine equations for Jordan type loci in the space of nilpotent matrices commuting with a given nilpotent Jordan block matrix  (Linear Algebra and Applications 710 (2025) 183-202)  "

Ellen Kirkman, TBA.

Abstract: TBA

John Klein, Local invariants of mixed states of multiple qubits

Abstract:  For an n-qubit system, a complex-valued rational function on the space of mixed states which is invariant with respect to the action of the group of local symmetries may be viewed as a detailed measure of entanglement.

I will explain how one computes  the field such invariant rational functions. In particular, I will show that this field is purely transcendental over the complex numbers and has transcendence degree 4^n - 3n-1.

Bernard Leclerc, Cluster structures on spaces of bands.

Abstract: We introduce new geometric objects, called (G,c)-bands, associated with a simple simply-connected and simply-laced algebraic group G, and a Coxeter element c in its Weyl group. We show that bands of a given type are the rational points of an infinite dimensional affine scheme, whose ring of regular functions has a cluster algebra structure. We also show that two important invariant sub-algebras of this ring are cluster sub-algebras. These three cluster structures have already appeared as Grothendieck rings of certain categories of representations of quantum affine algebras, their Borel sub-algebras, and shifted quantum affine algebras. Schemes of (G,c)-bands provide a common geometric setting in which these Grothendieck rings can be studied and related to each other. This is a joint work with Luca Francone.

Ray Maresca, The Hom-Ext quiver as a super quiver. 

Abstract: We will introduce the Hom-Ext quiver and explore its applications to exceptional sets. We will see how it can be used to count the number of exceptional orderings of an exceptional set and classify exceptional sets up to derived autoequivalence. At the end, we will introduce super quivers, see that all Hom-Ext quivers are super quivers, and realize exceptional sets as representations of (functors from) super quivers.

Aria Masoomi, Irreducible representations of quantum flag varieties at roots of unity.

Abstract: In this work, we focus on classifying irreducible representations of the Quantum function algebra at a root of unity for partial flag varieties G/P, where G is a simply connected, semisimple algebraic group over a field K of characteristic 0, and ε is a primitive ℓ-th root of unity for ℓ an odd positive integer, and ℓ ≥ 3. Our approach involves descending the action of O_ε(G/P) to a specific Normal Quantum Solvable algebra and employing techniques to classify the irreducible representations of such algebras.

Cyril Matoušek, Metric completions from finite dimensional algebras

Abstract: Neeman's novel method of creating new triangulated categories from old ones involves assigning a metric to a triangulated category and constructing its completion analogously to the completion of a metric space. In this talk, we will explore those metric completions in the context of triangulated categories arising from nice enough algebras. In particular, we provide a concrete description of all completions of bounded derived categories of hereditary finite dimensional algebras of finite representation type. This talk is based on pre-print arXiv:2409.01828v2.

Miranda Moore, Twists, higher dimer covers, and web duality.

Abstract: We give a description of the twist map (a cluster algebra automorphism on Grassmannian coordinate rings) in terms of higher dimer covers on plabic graphs and Fraser--Lam--Le’s web duality. This generalizes results of Marsh--Scott and Elkin--Musiker--Wright. In the particular case of Gr(3,12) and Gr(4,12), we explicitly compute dual web bases for certain graded pieces of the coordinate rings. This is based on joint work with Esther Banaian, Elise Catania, Christian Gaetz, Gregg Musiker, and Kayla Wright.

Chloe Napier, Extensions in the preprojective algebras.

Abstract: In 2001, Fomin and Zelevinsky introduced cluster algebras which can be used to describe many important varieties from Lie theory. Leclerc gives a cluster structure on coordinate rings of Richardson varieties using the representation theory of preprojective algebras. While his construction is very algebraic, we take a more combinatorial approach. We are aiming to find a combinatorial description for when modules in the preprojective algebra have extensions, which corresponds to describing when cluster variables are compatible. We will partially answer this question for certain types of modules and discuss future goals for further development.

Iacopo Nonis, tau-exceptional sequences for representations of quivers over local algebras.

Abstract: Exceptional sequences were first introduced in triangulated categories by the Moscow school of algebraic geometry. Later, Crawley-Boevey and Ringel studied exceptional sequences in the module categories of hereditary finite-dimensional algebras. Motivated by tau-tilting theory introduced by Adachi, Iyama, and Reiten, Jasso’s reduction for tau-tilting modules, and signed exceptional sequences introduced by Igusa and Todorov, Buan and Marsh developed the theory of (signed) tau-exceptional sequences – a natural generalization of (signed) exceptional sequences that behave well over arbitrary finite-dimensional algebras.

In this talk, we will study (signed) tau-exceptional sequences over the algebra Λ=RQ, where R is a finite-dimensional local commutative algebra over an algebraically closed field, and Q is an acyclic quiver. I will explain how (signed) tau-exceptional sequences over Λ can be fully understood in terms of (signed) exceptional sequences over kQ."

Fan Qin, Based cluster algebras of infinite ranks. 

Abstract: We introduce based cluster algebras of infinite rank. By extending cluster algebras arising from double Bott-Samelson cells to the infinite rank setting, we recover certain infinite rank cluster algebras connected to monoidal categories of representations of (shifted) quantum affine algebras. Several conjectures follow as a result.

Ralf Schiffler, On maximal almost rigidity over gentle algebras.

Abstract: For a module T over a finite dimensional algebra, the classical notion of rigidity is defined by the condition that every short exact sequence of the form 0 → T → E → T → 0 splits. We weaken this condition and say that T is almost rigid if for all indecomposable summands T', T'' of T and all short exact sequences 0 → T' → E → T'' → 0, either the sequence splits, or the module E is indecomposable. 

For every gentle algebra, there are geometric models for its module category as well as for its derived category in terms of surfaces with dissections. In both situations, the concept of maximal almost rigiditiy corresponds to the triangulations of the surface. 

In this talk, we will consider two different approaches to almost rigidity over gentle algebras, one for the derived category and one for the module category. For Dynkin type A, we shall also consider a third approach via a non-standard exact structure.

Andrea Solotar, tau-Hochschild homology and Han's conjecture.

Abstract: I will introduce the tau-Hochschild cohomology of a finite dimensional

algebra and show that its global dimension is related to the finiteness of the tau-

Hochschild (co)homology

Hugh Thomas, Harder-Narasimhan polytopes and weighted exceptional sequences.

Abstract: TBA

Jose Velez-Marulanda, Exact weights and path metrics for triangulated categories and the derived category of persistence modules.

Abstract: We define exact weights on a pretriangulated category to be nonnegative functions on objects satisfying a subadditivity condition with respect to distinguished triangles. Such weights induce a metric on objects in the category, which we call a path metric. Our exact weights generalize the rank functions of J. Chuang and A. Lazarev for triangulated categories and are analogous to the exact weights for an exact category given by the first author and J. Scott and D. Stanley. We show that (co)homological functors from a triangulated category to an abelian category with an additive weight induce an exact weight on the triangulated category. We prove that triangle equivalences induce an isometry for the path metrics induced by cohomological functors. In the perfectly generated or compactly generated case, we use Brown representability to express the exact weight on the triangulated category. We give three characterizations of exactness for a weight on a pretriangulated category and show that they are equivalent. We also define Wasserstein distances for triangulated

categories. Finally, we apply our work to derived categories of persistence modules and to representations of continuous quivers of type A.

Sarah Witherspoon, Twisted tensor product algebras, cohomology, and deformations.

Abstract: Many rings of interest can be described as twisted tensor products of other rings, for example, quantum polynomial rings, skew group algebras, Ore extensions, Drinfeld doubles, and more. Other rings may be viewed as deformations of these. We aim to understand cohomology and deformations. In this talk, we first introduce twisted tensor product algebras and examples, and explain some homological techniques for handling them. Then we focus on the special case of group actions/Hopf algebra actions on Koszul algebras, and a homological approach to graded deformations of the corresponding twisted tensor product algebras. This includes many settings of interest, for example, symplectic reflection and Drinfeld Hecke algebras, infinitesimal Hecke algebras (via actions of Lie algebras), and quantized symplectic oscillator algebras (via actions of quantum groups).