Last updated: April  07, 2026, 22:25 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, TBA

Abstract: TBA

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.

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. 

Justin Desrochers, Homological invariants and resolution by spread modules.

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. Instead, we consider incomplete invariants, which can distinguish some modules.

Relative projective resolutions are an excellent source of invariants, via Betti numbers. This presentation discusses invariants given by Betti numbers.  I will introduce resolutions and approximations, with a focus on spread-resolutions over a grid poset. I will discuss literature of these invariants and their stability. I provide an upper bound on the spread global dimension of any finite poset. This also gives an upper bound on the spread global dimension of certain infinite posets.

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

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, TBA 

Abstract: TBA.

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.

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.

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: TBA

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.

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, TBA.

Abstract: TBA

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

Abstract: TBA.

Peter Webb, TBA

Abstract: TBA

Yunmeng Wu, Quantum Laurent phenomenon algebras.

Abstract: TBA

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.