Last updated: May 09, 2026, 21::19 EST

Abstracts, Slides, and Videos

of Distinguished Lectures - 2026

Aaron Agulnick, Higher-order autocorrelations over number fields.

Abstract: The higher-order autocorrelations of a function are a family of statistics about its Fourier transform. In applications, e.g. X-ray crystallography, it is useful to understand how much of this data one needs in order to recover the original function. It is a surprising fact that the algebraic properties of the field in which the function takes its values plays a central role; for instance, all previous results restrict to rational-valued functions. I will describe joint work with Milen Yakimov giving an upper bound on the order of autocorrelation data needed to recover a function valued in a real quadratic number field, and give an extension to arbitrary (not-totally-complex) number fields. Furthermore, I will show how the problem of reconstructing a function from its autocorrelation data can be formulated as a polynomial optimization problem, compatibly with the extension from quadratic number fields to those of higher degree.

Jonah Berggren, Flows on gentle algebras.

Abstract: I will define the polyhedron of “flows” on a gentle algebra G, generalizing unit flow polytopes of certain directed acyclic graphs. The tau-tilting theory of G gives rise to triangulation and subdivision results on its polyhedron of flows. Moreover, a quotient map from the polyhedron of flows to the Grothendieck group of G allows these results to give new insights to g-vector fans of gentle algebras. In particular, we are able to prove a notion of g-convexity of gentle algebras through defining “g-polyhedra” and give new insights into the complements of g-vector fans of gentle algebras.

Frauke Bleher, On the radii of Voronoi cells of rings of integers.

Abstract: This is joint work with Ted Chinburg, Xuxi Ding, Nadia Heninger and Daniele Micciancio. 

Two important measures of the size of a number field K are its degree and its discriminant. In this talk I will discuss a third measure given by the covering radius of the ring of integers O_K of K. This is the radius of the Voronoi cell of O_K, which is the set of points inside the real vector space W that are at least as close to the origin as they are to any non-zero element of O_K, where W arises from the real and complex embeddings of K. Put differently, this radius is the minimal radius of a fundamental domain for O_K as a lattice inside W. A number of famous results have to do with infinite families of number fields whose discriminants grow slowly with their degree. I will discuss analogous results for the covering radius. This work is motivated by cryptography.

Thomas Brüstle, Generalized rank computation via minimal subsets.

Abstract: We study a functorial notion of rank for representations of small connected categories and relate it to the multiplicity of a distinguished indecomposable object: Given a small connected category C, we consider C-modules M from C to vector spaces over a field K and show that the rank of the canonical map from the limit to the colimit of M coincides with the multiplicity of the “identity” module K_C as a direct summand. This extends classical results of Kinser for tree quivers and Chambers–Letscher for posets to small categories.

Motivated by computational questions in multiparameter persistence theory, we investigate when this multiplicity can be computed after restricting along a suitable subcategory. We prove that restriction along a functor F from J to C preserves this multiplicity precisely when F is both initial and final, thus characterizing such embeddings in representation-theoretic terms. In the case where C is a poset satisfying mild finiteness conditions, we construct minimal full subposets whose embeddings are initial and final, and hence control the multiplicity of K_C. These constructions yield smaller indexing categories—often of finite representation type—on which the problem becomes tractable. This generalizes results by Dey and Lesnick.

This is a report on joint work with Justin Desrochers and Samuel Leblanc. 

Amanda Burcroff, Eventual sign coherence. 

Abstract: The sign coherence of c-vectors is one of the fundamental theorems of cluster algebras with principal coefficients.  Gekhtman and Nakanishi posed the Asymptotic Sign Coherence Conjecture for cluster algebras with arbitrary coefficients, which says sign coherence should eventually hold in any sufficiently generic infinite mutation sequence.  We prove that for cluster algebras from quivers of arbitrary rank, their conjecture holds with probability 1 for a random mutation sequence.  Our results also establish the conjecture in full generality for many families of quivers.  This is joint work with Scott Neville.

Jon Carlson, Trace  ideals.

Abstract: This is joint work with Srikanth Iyengar, Janina Letz and Haydee Lindo. We look at the images of trace maps in cohomology and derived categories in commutative algebra and group representations.

Sean Carroll, Euler characteristics via Gorenstein approximations.

Abstract: We present a general construction of eventually periodic projective resolutions for modules over quotients of rings of finite global dimension by a regular central element. Our approach builds on a construction of Shamash, combined with the iterated mapping cone technique, to systematically ‘purge’ homology from a complex. The resulting periodicity allows us to define a generalized version of a matrix factorization. Moreover, we prove that the homotopy category of these matrix factorizations is triangle equivalent to the stable module category of Gorenstein projective modules over the quotient ring. This equivalence is used to show that any module, satisfying certain finiteness conditions, admits a Gorenstein projective approximation, in the sense of Auslander-Buchweitz. These ideas are applied to integral group rings of groups with finite virtual cohomological dimension, yielding a new approach for calculating their rational Euler characteristic. We demonstrate the computability of our method through explicit calculations for several families of groups, including hyperbolic triangle groups and mapping class groups of the punctured plane.

Vyjayanthi Chari, Monoidal categorification of cluster algebras

Abstract: We discuss some recent results with Matheus Brito on certain subcategories of finite--dimensional representations of quantum affine algebras. These categories are monoidal and we show that the corresponding Grothendieck rings have a structure of a cluster algebra of type $A$. The cluster structure reflects the representation theory in the sense that cluster monomials correspond to irreducible representations and the cluster variables to prime representations. 

Qiyue Chen, Stability scattering diagrams and quiver coverings.

Abstract: Given a covering of a quiver (with potential), we show that the associated Bridgeland stability scattering diagrams are related by a restriction operation under the assumption of admitting a nice grading. We apply this to quivers with potential associated to marked surfaces. In combination with recent results of the second and third authors, our findings imply that the bracelets basis for a once-punctured closed surface coincides with the theta basis for the associated stability scattering diagram, and these stability scattering diagrams agree with the corresponding cluster scattering diagrams of Gross-Hacking-Keel-Kontsevich except in the case of the once-punctured torus.

Xueqing Chen, Fundamental relations in quantum cluster algebras.

Abstract: Let A_q  be an arbitrary quantum cluster algebra with principal coefficients. We give the fundamental relations between the quantum cluster variables arising from one-step mutations from the initial cluster in A_q. Immediately and directly, we obtain an algebra homomorphism from the corresponding (untwisted) quantum group to A_q. This is a joint work with M. Ding, J. Huang and F. Xu.

Ted Chinburg, Unramified elementary abelian two extensions of number fields.

Abstract:  In 1801 Gauss developed what is now called genus theory by studying the arithmetic of quadratic forms.  In modern terms, this has to do with the maximal unramified elementary abelian two-extension F^{max} of a quadratic field F = Q(\sqrt{d}).  In this talk I will discuss methods for constructing F^{max} for an arbitrary number field F.  The size of the relative degree [F^{max}:F] is relevant to cryptography, since the larger this is the more useful F^{max} is likely to be for disguising data.  I will discuss some results and conjectures about the size of [F^{max}:F] as well as the size of [F^{unit}:F] when F^{unit} is the maximal subfield of F^{max} generated by taking square roots of units of F.  If there is time, I'll explain how fast Fourier techniques can be used to do arithmetic quickly in F^{max}.  This is joint work with F. Bleher, X. Ding, D. Micciancio and A. Suhl. 

Harm Derksen, Invariants of finite groups acting on skew fields.

Abstract: The free skew field can be viewed as a noncommutative quotient field of the free algebra.  I will discuss joint results with Jurij Volcic on invariant skew fields of finite group actions. The invariant skew field is always finitely generated. We also proved a non-commutative analog of Noether’s problem: If the action is linear, then the skew field of invariants is again free, but there are also examples of non-linear actions for which this skew field is not free. 

Justin Desrochers, Stabilization of the spread global dimension.

Abstract: The representation theory of posets holds great potential for applications in topological data analysis. In traditional persistent homology, a data set is associated with a representation M of a totally ordered poset, and topological information is extracted from the indecomposable summands of M. However, in data sets with varying densities or time dependence, it is natural to consider two-dimensional grid posets.

The representation theory of higher dimensional grids is less understood, as there is no complete description of the indecomposables. One approach to this problem is to study resolutions with respect to distinguished classes of relative projective objects. In particular, resolutions by spread modules play an important role in the study of two-dimensional persistence.

In this talk, I will introduce the notions of resolutions and approximations in this context, with an emphasis on spread resolutions over grid posets. I will present an upper bound on the spread global dimension of any finite poset, and discuss how this result extends to certain infinite posets.

This presentation is based on recent work by Benjamin Blanchette, Eric Hanson, Luis Scoccola, and me.

Christof Geiß, Shifted quantum affine algebras and cluster algebras.

Abstract: This is a report on joint work with David Hernandez and Bernard Leclerc. To each Cartan C matrix of finite type and a given coweight μ we have a (non-twisted) shifted quantum affine algebra U^μ, and we may consider its category O^μ of representations, which is similarly defined as the classical BGG-category O. The Grothendieck group K_0(O^{sh}) of the sum of all those categories O^μ has a natural cluster structure of infinite rank. The initial seeds of this cluster algebra structure are defined in terms of nice quivers and solutions of the quantum integrable  QQ-system. We conjecture that all cluster monomials are the class of a real simple object in O^{sh}.  If C is of type A_1, this cluster algebra is of type A^\infty_\infty, and our conjecture is easy to verify.

Benjamin Grant, Topologizing infinite quivers and their mutations.

Abstract: We offer a topological perspective toward infinite quivers and their mutations. Specifically, we describe two topological spaces whose points are (labeled) countably infinite quivers and demonstrate that mutations act by homeomorphisms on these spaces. Several natural properties of quivers are discussed and their topological complexities as subspaces are examined. One may also consider infinite sequences of mutations and the subspaces of quivers on which a given sequence converges/diverges; we completely characterize the (non-)density of these subspaces in terms of combinatorial properties of the sequence. Lastly, using a classical model-theoretic construction, we point out the existence of a special, highly symmetric infinite quiver satisfying a remarkable dichotomy between its finite and infinite mutations.

Tony Guo, Delooping and derived delooping level under algebraic and combinatorial contexts. 

Abstract: Developments in homological conjectures go hand in hand with the study of representation theory. Plenty of research has been done on the representation theory of algebraic constructions such as triangular matrix rings and tensor products. In return, our understanding of these new objects allows us to derive corresponding information regarding their homological dimensions. Similarly, we can leverage our knowledge of algebras whose representation theory (at least partially) can be interpreted combinatorially. In this talk, we focus on the behavior of the delooping level and the derived delooping level in algebraic and combinatorial contexts and their implications on homological dimensions.

Ivo Herzog, The category of coherent functors.

Abstract: Maurice introduced the category of coherent functors in the 1960's and he proved that they form an abelian category. I will give a historical account of some of the early developments of the subject, but also consider the category in light of a recollement structure that may be imposed. This point of view identifies three important subcategories of coherent functors and we will describe how the respective categories of injective functors on these three categories are related.

Ivan Horozov, Euler characteristics of a family of congruence subgroups of GL(m,Z).

Abstract: The congruence subgroups Gamma1(m,p) that we consider here are subgroups of GL(m,Z) that fix the vector (0,…,0,1) mod p, where p≥5 is a prime. We present a method and many computations of homological Euler characteristics of GL(m,Z) and Gamma1(m,p) with coefficients in any highest weight representation V. By homological Euler characteristics we mean the alternating dimensions of cohomology of the group with coefficient in V. We compute the homological Euler characteristics for Gamma1(2,p), and Gamma1(3,p) with coefficients in any finite dimensional highest weight representation. Also we compute the homological Euler characteristics for of Gamma1(4,p) and Gamma1(5,p) with coefficients in the trivial and the determinant representations. We give application to cohomology of Gamma1(3,p) with trivial and with determinant representation. We also give an alternative method for computing the cohomology of GL(4,Z).

Osamu Iyama, Auslander–Gorenstein algebras and fractionally Calabi–Yau algebras

Abstract: Commutative rings have long been a rich source of ideas in representation theory. For example, there is a hierarchy of commutative Noetherian rings: regular rings, Gorenstein rings, and Cohen–Macaulay rings. It is well-known that, in the non-commutative setting, this hierarchy admits a much richer structure (even for finite dimensional algebras over fields). Among others, Auslander–regular and Auslander–Gorenstein rings provide more refined classes in this hierarchy. One of the principles initiated by Auslander is that such classes of rings often admit representation-theoretic realizations. For instance, the higher Auslander correspondence gives a bijection between d-Auslander algebras and d-cluster tilting modules.

In this talk, after recalling some basic facts, I will explain a new construction of higher Auslander algebras using twisted fractionally Calabi–Yau algebras. As an application, we show that twisted fractionally Calabi–Yau algebras are precisely the stable higher Auslander algebras of higher representation-finite algebras. This is a joint work with Aaron Chan and Rene Marczinzik.

Maximilian Kaipel, The Auslander algebra of k[x]/(x^t): Exceptional vs. tau-exceptional sequences.

Abstract: Exceptional sequences are classical objects in algebraic geometry and representation theory. They are particularly useful when they generate their ambient triangulated category, in which case we call them complete. While complete exceptional sequences always exist for hereditary algebras, they generally fail to exist for other finite dimensional algebras.

Motivated by this observation and the generalisation of classical tilting theory to tau-tilting theory, Buan—Marsh introduce tau-exceptional sequences. For these sequences of modules, complete versions always exist. However, while exceptional and tau-exceptional sequences coincide for hereditary algebras, essentially no relationship between these two notions is known beyond this setting.

In my talk I will present first results that establish a connection between these two types of sequences for Auslander algebras of k[x]/(x^t), where t is some positive integer. In particular, I will show that complete exceptional sequences form a proper subset of complete tau-exceptional sequences and compare their mutation theories.

Ben Kaufman, The extension conjecture and Koszul algebras with coherent Koszul dual.

Abstract: A strengthening of the strong no loop conjecture proved by Igusa, Liu, and Paquette, the extension conjecture asserts that for a finite dimensional algebra and a simple module S, Ext^1(S,S) nonzero implies that Ext^i(S,S) is nonzero for infinitely many i.  The extension conjecture has been established for Koszul algebras with Noetherian Koszul dual by Bouhada, Huang, Lin, and Liu using techniques of Lenzing.  However, this result also holds in the case that the Koszul dual is merely coherent.  In this talk we will discuss a recent result characterizing finite dimensional Koszul algebras with coherent Kozul dual in terms of linearity properties of syzygies.  We will also present some additional examples of non Koszul algebras with loops and exhibiting extension gaps - values of i for which Ext^i(S,S)=0.

Ryan Kinser, Symmetric quivers and symmetric spaces.

Abstract: I'll briefly summarize a story arc that connects equivariant geometry/commutative algebra of quiver representations with the classical theory of Schubert varieties and symmetric spaces. This story starts with a beautiful insight of Zelevinsky in the early 1980s motivated by representations of p-adic groups, developed further by Bobinski-Zwara and others in early 2000s, culminating (at least over the complex numbers) in recent work of myself with Jenna Rajchgot and Martina Lanini.

Ellen Kirkman, Artin-Schelter algebras from dual reflection groups.

Abstract: Let A be an Artin-Schelter regular algebra that is graded by a finite group G, and let A_e denote the identity component of A.  We call G a dual reflection group for A when under the coaction by G, the ring of invariants, A_e is also Artin-Schelter regular. We view the property that both A and A_e are Artin-Schelter regular as a generalization of the Shephard-Todd-Chevalley Theorem on the invariants of a commutative polynomial ring under the action of a reflection group. Using necessary conditions for such a pair (A,G) to exist, for certain groups G of order 16 we have constructed several interesting 4-dimensional Artin-Schelter algebras A and studied their geometric and algebraic properties and the relationship between these properties. This is work with Peter Goetz, W. Frank Moore, and Kent Vashaw.

Shiping Liu, Auslander-Reiten theory via Nakayama duality in abelian categories.

Abstract: Using the Nakayama duality induced by a Nakayama functor, we provide a novel and concise account of the existence of Auslander-Reiten dualities and almost split sequences in abelian categories with enough projective objects or enough injective objects. As an example, we establish the existence of almost split sequences ending with finitely presented modules and those starting with finitely copresented modules in the category of all modules over a small endo-local Hom-reflexive category. Specializing to algebras given by (not necessarily finite) quivers with relations, we further investigate when the categories of finitely presented modules, finitely copresented modules and finite dimensional modules have almost split sequences on either or both sides.

Cyril Matoušek, Completing derived categories of tame algebras and weighted projective lines.

Abstract: There are relatively few methods for constructing a new triangulated category from an existing one. The standard options are to take a triangulated subcategory or Verdier’s quotient. One of the recent, more sophisticated methods is to build Neeman’s metric completion out of directed colimits of sequences which converge with respect to a metric on the triangulated category. Moreover, particular choices of a metric on the triangulated category can recover the standard constructions (subcategories, quotients) as completions. In this talk, we use this insight to explicitly describe the completions of bounded derived categories of well-behaved hereditary categories – such as modules over hereditary finite dimensional tame algebras, hereditary commutative noetherian rings, or coherent sheaves of weighted projective lines.

Blake Mattson, Hopf actions on representations of cyclic oriented affine quivers.

Abstract: There has been much work done on representations of quivers, as well as on Hopf actions on a quiver's path algebra. I blend these two areas of study by classifying the representations of cyclic oriented affine quivers which are equivariant with respect to certain Hopf actions on their path algebras. Additionally, in the finite case, I classify the representation type.

Chloe Napier, Extensions between modules defined by lattice paths in the preprojective algebra.

Abstract: In 2001, Fomin and Zelevinsky introduced cluster algebras which appear as coordinate rings of many varieties. We study cluster algebras coming from Richardson varieties. Leclerc gives a cluster structure on Richardson varieties using the representation theory of preprojective algebras. While this construction is very algebraic, we take a more combinatorial approach. The main goal is to find a combinatorial description for when certain cluster variables are compatible, or equivalently when modules defined by lattice paths in the preprojective algebra have trivial extensions. We extend the known results from Geiss, Leclerc, and Schröer that answer this question in the case of the Grassmannian by using various methods. We introduce the notion of extending a module, describe how add/remove operators applied to pairs of modules affects the extension space between them, and provide homological and combinatorial conditions that determine when two arbitrary modules have trivial extension. 

Scott Neville, Mutation invariants for exceptional sequences.

Abstract: Exceptional sequences associated to quiver representations are closely related to seeds in cluster algebras. Both objects have (seemingly different) mutation operations. We will see they are closely related, and thus show how new mutation invariants for quivers give mutation invariants for exceptional sequences.

Fan Qin, Cluster algebras via Auslander-Reiten theory. 

Abstract: In this talk, I will explain how Auslander–Reiten theory plays a central role in the categorical understanding of cluster algebras and in the construction of their bases. I will review some classical results as well as recent developments.

Nathan Reading, Para-exceptional sequences and the McCammond-Sulway lattice.

Abstract: The noncrossing partition poset in a Coxeter group is isomorphic to the inclusion order on exceptional subcategories of the module category of a corresponding connected hereditary algebra $\Lambda$, and maximal chains in the noncrossing partition poset are in bijection with complete exceptional sequences.  The noncrossing partition poset in an infinite Coxeter group can fail to be a lattice.  In affine type, McCammond and Sulway constructed a larger lattice that contains the noncrossing partition poset and that furthermore is a combinatorial Garside structure.  In joint work with Eric Hanson, we construct a lattice, isomorphic to McCammond and Sulway's lattice, using the representation theory of $\Lambda$ and give a representation-theoretic proof that it is a combinatorial Garside structure.  To construct the lattice, we introduce para-exceptional sequences and para-exceptional subcategories in the module categories of tame hereditary algebras.  This talk has two goals:  To describe the representation-theoretic construction of the lattice and its connection to the McCammond-Sulway lattice and to describe combinatorial models in the classical affine types (some joint with Laura Brestensky) that realize the poset of exceptional subcategories and the lattice of para-exceptional subcategories.

Claus Michael Ringel, Tilting with Maurice. II. Homological theory of representations.

Abstract: At the memorial meeting in 1995, I gave a lecture with this title: "Tilting with Maurice". It was devoted to fights about the relevance of concepts in representation theory. Some participants complained afterwards that they had expected (say as a final pun) a discussion about Auslander's contribution to tilting theory. At that time, I felt that tilting theory was my business, not his. But I was wrong. Auslanders's ideas (quite different from mine) have turned out to be essential for further developments: for example, tau-tilting is strongly based on his visions. 

The second lecture will focus attention on Auslander's aim of providing a homological theory of representations, as seen already in his early work in representation theory: the Paris lectures and the Queen Mary lectures. Surprisingly, some of his suggestions have yet been neglected, for example his proposal to look at what he called the t-torsion-free modules. Graphical visualisation (the agemo quiver, the dominance quiver) may stimulate further interest. Maurice himself never relied on visual means - it seems that this has to be my task.

Both lectures will start with historical recollections. This then will be followed by a report on recent investigations of various mathematicians which correspond to Auslander's ideas. The first lecture will concentrate on the role of bricks in module categories, the second on parity features of homology groups, always with emphasis on suitable examples.

Andrew Salch, An Auslanderian view on stable homotopy groups of spheres.

Abstract: I will describe some old and some new calculations of stable homotopy groups of spheres by means of calculating the cohomology of certain profinite groups, the Morava stabilizer groups. In this approach to stable homotopy groups of spheres, what one sees again and again are some of the same themes that also appeared in Auslander's work, in particular: representations of associative algebras, the relationship between ring-theoretic properties of such an algebra and various homological-algebraic properties of its representations, and relative homological-algebraic invariants of representations. No prior knowledge of stable homotopy theory will be assumed; instead, I hope to explain some state-of-the-art computational results in the subject in a way which I hope will seem natural to algebraists who work on representation theory and/or homological algebra.

Ralf Schiffler, On Cohen-Macaulay modules over Iwanaga-Gorenstein algebras.

Abstract: Cohen-Macaulay modules form an important class of modules that is very well studied in commutative algebra. In the non-commutative setting of Iwanaga-Gorenstein algebras, these modules were promoted in the seminal work of Auslander and Reiten and Buchweitz. An important problem is the classification of rings that admit only finitely many Cohen-Macaulay modules (finite Cohen-Macaulay type).

The rings we will encounter in this talk are non-commutative Iwanaga-Gorenstein k-algebras of Gorenstein dimension one. The stable category of Cohen-Macaulay modules is then a triangulated category whose inverse shift is given by the syzygy functor. An important family  of such algebras is given by 2-Calabi-Yau tilted algebras, for example the Jacobian algebra of a quiver with potential which are essential in the theory of cluster algebras. In this setting, the stabe syzygy category has the additional property of being 3-Calabi-Yau. We will discuss examples of algebras of finite Cohen-Macaulay type. 

Markus Schmidmeier, Gorenstein-projective modules over the ring of dual integers.

Abstract: We consider a cube which has as vertices eight categories, as edges functors and as faces commutative diagrams.  Since the presentation is 10 minutes, I can only discuss one walk from the top of the cube to the bottom:  Start from the complexes of free abelian groups, modulo the shift arrive at the Gorenstein-projective modules over the ring of dual integers, up to projectives get to the orbit category of the derived category of the integers, and finally, modulo the ideal generated by the stalk objects, arrive at the category of finite abelian groups located at the bottom of the cube.  This is a report about a paper with Xiuhua Luo from Nantong in China.

Khrystyna Serhiyenko, Skein relations in the derived category of gentle algebras.

Abstract: Derived categories of gentle algebras can be described via the geometry of surface dissections, where complexes of projectives correspond to arcs in the surface.  Furthermore, morphism are encoded by crossings between the arcs and their cones are given by resolving the crossings.  We associate a Laurent polynomial to every (finite) complex and show that the skein relations hold whenever the corresponding arcs cross in the interior.  Furthermore, for each complex their Lauren polynomial specializes to the corresponding element of the Grothendieck ring. This is joint work in progress with Esther Banaian, Azzurra Ciliberti, Ilaria Di Dedda, Yadira Valdivieso-Diaz and Kayla Wright. 

Emre Sen, Cluster and higher preprojective categories.

Abstract: We show that the category of coherent functors on the fundamental domain of the cluster category has a 4-cluster-tilting subcategory whose objects are projective and injective functors, and that it has global dimension 4. Furthermore, there is a bijection between projective non-injective and injective non-projective functors via the 4-Auslander-Reiten translation, which mimics the description of 4-representation-finite algebras. This enables us to prove that representable functors on the cluster category decompose as coherent functors on the fundamental domain of the cluster category—a process we call the 5-preprojective decomposition, mimicking 5-preprojective algebras. Moreover, we prove that 5-cluster-tilting subcategories can be mutated and have periodicity 5. This is joint work with O. Iyama and G. Todorov.

Shashank Singh, Cyclic quivers, Taft algebras, and the classification of bimodules.

Abstract: This talk explores how quivers can be used not only to study module categories over algebras, but also the representation theory of Hopf algebras, with a focus on the Taft algebras. I will discuss how cyclic quivers naturally encode symmetries appearing in Taft algebra representations and how this perspective helps organize their structure. I will conclude with classification results for bimodules over these algebras.

Hugh Thomas, Representation theory of lattices and rowmotion.

Abstract: In this talk, I will discuss Auslander regularity for lattices. It was shown by Iyama and Marczinzik that the incidence algebra of a lattice L is Auslander regular if and only if L is distributive. I will explain that, for L a lattice, Auslander regularity can also be detected in terms of linear algebraic properties of the Coxeter matrix: the lattice is Auslander regular if and only if the Coxeter matrix, with respect to a linear extension of L, can be factorized as a permutation matrix times an upper-triangular matrix. The permutation matrix so obtained turns out to encode both the Auslander--Reiten bijection between indecomposable injectives and indecomposable projectives for Auslander regular algebras, and also the rowmotion permutation of the distributive lattice, a permutation which has received a lot of study over the past fifteen years within dynamical algebraic combinatorics. This talk will mainly draw on joint work with Viktória Klász and René Marczinzik, arXiv:2501.09447.

Helene Tyler, Cohomology for gentle algebras.

Abstract: This is joint work with Thomas Brüstle and David Pauksztello. We describe a geometric approach to computing cohomology in the derived category of a gentle algebra. Using surface models, indecomposable objects correspond to graded curves, while hearts correspond to graded dissections. We show how to read off the cohomology of an object directly from its curve by cutting it at points where a simple local condition fails. The resulting pieces are 0-graded zigzag curves, which represent the cohomology objects, and their degrees are determined by the grading. This provides a combinatorial and visual method for understanding cohomology with respect to arbitrary length hearts.

Jose Velez-Marulanda, Euler characteristic curves and 1-Wasserstein distances for triangulated categories.

Abstract: TBA.

Peter Webb, The structure of biset functors.

Abstract:  The biset category has as its objects finite categories (or, more traditionally, finite groups) and morphisms are formal linear combinations of bisets, also known as profunctors or distributors. Biset functors are linear functors from this category to abelian groups. They originally arose in contexts having to do with classifying spaces and cohomology and are closely related to Mackey functors. Examples include representation rings, the Burnside ring, Hochschild cohomology and cohomology of the nerve of the category (with suitable modification of the biset category) and one reason for doing all this is to create a setting that will provide insight into these examples. It turns out that the biset category is rigid symmetric monoidal, giving a symmetric monoidal structure on the biset functors. The biset functors in special circumstances are a highest weight category. In other special circumstances, in work with Andrew Snowden, they are not only locally Noetherian but are locally finite length despite being non-Noetherian in general. We will give an overview of such structural properties.

Lauren Williams, Signed cells and G-gradings on cluster algebras.

Abstract: This presentation is based on recent work by Benjamin Blanchette, Eric Hanson, Luis Scoccola, and me.

Yunmeng Wu, Quantum Laurent phenomenon algebras.

Abstract: Laurent Phenomenon Algebras (LPAs), introduced by Lam and Pylyavskyy, generalize cluster algebras by allowing arbitrary exchange polynomials while preserving the Laurent Phenomenon property under mutation. Although LPAs provide a flexible framework for commutative mutation systems, yet a corresponding quantum theory has not been developed. In this talk, I outline ongoing work that defined Quantum Laurent Phenomenon Algebras (qLPAs)—a new class of noncommutative algebras that extend the structural ideas of LPAs to the quantum setting. I will describe how quasi-commutation relations and a generalized compatibility condition can be used to formulate mutation rules and establish a quantum Laurent phenomenon. This short talk aims to illustrate how the qLPA provides a unifying framework for certain import quantum algebras that are not quantum cluster algebras, including quantized Weyl algebra at roots of unity, allowing their representation theory to be treated analogously to quantum cluster algebras and LPAs. This is based on joint work with Charles Barth and Milen Yakimov.

Jie Xiao, Root categories and Lie groups. 

Abstract: D. Happel introduced the root category as a two-periodic orbit triangulated category R of the derived category of Dynkin quiver. The Gabriel Theorem can be stated with the Auslander-Reiten quiver of R, not only for the positive roots Φ^+  but also the whole root system Φ. We introduce here a process to build up semi-simple Lie algebras and Chevalley groups via Hall algebra approach. The construction can be applied to a realization of compact real form and maximal compact subgroups from the root category R, and obtain the Peter-Weyl Theorem and the Plancherel Theorem for compact groups. This is a joint work with Buyan Li.

Shijie Zhu, Reflection functors on stable monomorphism categories.

Abstract: Let A be an artin algebra. We say three subcategories (X,Y,Z) in A-mod form a Frobenius cotorsion triple, if both (X,Y) and (Y,Z) are complete hereditary cotorsion pairs and both X and Z are Frobenius subcategories of A-mod. Let Q be a finite acyclic quiver and Q’ the quiver obtained by reversing all the arrows at a sink vertex of Q.  

For a Frobenius cotorsion triple (X,Y,Z) in A-mod, we show that the BGP-reflection functor F: rep(Q,A) --> rep(Q’,A) between the quiver representations over A-modules induces an equivalence smon(Q,X) -->smon(Q’,X) between the stable categories of separated monic representations. In particular, it shows an equivalence between stable categories of Gorenstein-projective modules.