Invited talks
Jonathan Beardsley – University of Nevada, Reno
Title: Classifying Spaces, Projective Geometries and Quantum Logics
Abstract: So Nakamura and I recently constructed a fully faithful embedding from the category of axiomatic projective geometries (and collineations) into the category of Connes-Consani 𝔽₁-modules. This construction is a version of the usual classifying space construction for Abelian groups, but has been extended to apply to partially defined and multi-valued “additions” (one way of encoding the structure of a projective geometry). In this talk, I will describe the most elementary possible computation with this machinery; I will apply the classifying space functor to the trivial geometry with one point and no lines. This can also be interpreted as computing the bar construction of the Krasner hyperfield. The result is an 𝔽₁-module which, in degree n, encodes the set of so-called quantum logic structures (a generalization of σ-algebra structure) on the set with n+1 elements. This is a “toy example” associated to a longer-term project of constructing classifying stacks in Arakelov geometry.
Ken Brown – University of Glasgow (slides)
Title: Conjectures and results on pointed Hopf algebras, especially group algebras
Abstract: This talk is in two parts. In the first part, for a pointed noetherian Hopf k-algebra H, I propose a relation between the GK-dimension of H, the nature of its group-likes G(H), and the Dixmier-Moeglin Equivalence. In the second part, I discuss when a group algebra kG is noetherian. This is joint work with J. Bell and J. T. Stafford.
Ken Goodearl – University of California, Santa Barbara
Title: Catenarity for Prime Ideals
Abstract: The classical property of catenarity is the condition, on a partially ordered set, that all saturated chains between any two fixed elements have the same length. This property holds for the poset of prime ideals in the coordinate ring of any affine algebraic variety, and it has been established for the prime spectra of many noncommutative algebras, including many quantized coordinate rings. We will survey catenarity for prime ideals, illustrate its widespread appearance, and propose that it should be more widespread still.
Hongdi Huang – Shanghai University
Title: Relative Cancellation
Abstract: In this talk, we introduce and study a relative cancellation property for associative algebras. We also prove a characterization result for polynomial rings which partially answers a question of Kraft. This work is joint with Zahra Nazemian, Yanhua Wang, and James J. Zhang.
Colin Ingalls – Carleton University
Title: Artin-Schelter regular algebras
Abstract: We review some well-known properties of AS-regular algebras, which are non-commutative analogues of polynomial rings. This will be an expository talk.
Ellen Kirkman – Wake Forest University
Title: The Invariant Theory of Artin-Schelter Regular Algebras: The McKay Correspondence
Abstract: The classical McKay Correspondence concerns certain categories of modules over ℂ[x₁,…,xₙ]^G, representations of “small” finite groups G, and resolutions of singularities. Important ingredients include a theorem of Maurice Auslander. Aspects of the correspondence have been generalized in various directions, including work by Buchweitz, Faber, and Ingalls for reflection groups. I will discuss work related to the McKay Correspondence in the setting of an action of a semisimple Hopf algebra on an Artin-Schelter regular algebra.
Frank Moore – Wake Forest University
Title: Observations on Normal Packets
Abstract: Let A be a Koszul AS-regular algebra. Shelton and Tingey have shown that if f_1,…,f_c is a normal regular sequence of quadratic elements, then A/(f_1,…,f_c) is again Koszul and its Koszul dual is again Koszul AS-regular. Part of their work involves passing the notion of normality and 1-regularity from A to its Koszul dual A^!. In joint work with Goetz, we provide examples of this phenomenon, extend some results from elements to packets, and give partial results regarding Koszulness of the quotient A/(f_1,…,f_c).
Izuru Mori – Shizuoka University
Title: A noncommutative ℙ¹ × ℙ¹ and a noncommutative 𝔽₀
Abstract: It is known that ℙ¹ × ℙ¹ is isomorphic to 𝔽₀ = ℙ_{ℙ¹}(𝒪_{ℙ¹} ⊕ 𝒪_{ℙ¹}), the Hirzebruch surface of degree 0. A noncommutative ℙ¹ × ℙ¹ is the noncommutative projective scheme Proj_nc A associated to a 3-dimensional cubic AS-regular ℤ-algebra A, and a noncommutative 𝔽₀ is the noncommutative ℙ¹-bundle ℙ_{ℙ¹}(ℰ) associated to a locally free sheaf bimodule E of rank 2 over ℙ¹ such that 𝒪_{ℙ¹}(i) ⊗ ℰ = 𝒪_{ℙ¹}(i) ⊕ 𝒪_{ℙ¹}(i) for every i ∈ ℤ. Relationships between these noncommutative objects are discussed via semi-orthogonal decompositions.
References:
[1] M. Van den Bergh, Noncommutative quadrics, Int. Math. Res. Not. IMRN 2011, no. 17, 3983--4026.
[2] M. Van den Bergh, Non-commutative ${\mathbb P}^1$-bundles over commutative schemes, Trans. Amer. Math. Soc. vol. 364 (2012), 6279--6313.
Daniel K. Nakano – University of Georgia
Title: A Tale of Three Spectra
Abstract: Let T be the stable module category for a finite tensor category. I will introduce three important topological spaces that govern the geometry associated with T: (i) homological spectra, (ii) Balmer spectra, (iii) cohomological spectra (via the categorical center). After presenting background material, I will state an open conjecture relating these spaces. Examples include representations of finite-dimensional Hopf algebras, including small quantum groups, and finite group schemes. This is joint work with Kent Vashaw and Milen Yakimov.
Dan Rogalski – University of California, San Diego
Title: Division algebras in noncommutative geometry
Abstract: We survey the birational part of noncommutative geometry. In particular, we review Artin's conjecture on the function fields of noncommutative surfaces and related recent work on division rings. Compared to a recent talk at SLMath, this talk will have a more geometric flavor.
Xin Tang – Fayetteville State University
Title: Log-unimodularity for Poisson Algebras
Abstract: Poisson homology and cohomology are important invariants for Poisson algebras. Usually, there is a Poincaré-type duality between Poisson homology and cohomology facilitated by the modular derivation. In this talk, we observe that, in many cases, the modular derivation can be realized by a Poisson normal element and investigate some implications.
Chelsea Walton – Rice University
Title: Representations of braided categories
Abstract: I will discuss recent work with Monique Müller on quasitriangular comodule algebras. This is based on ArXiv:2508.19845.
Xingting Wang – Louisiana State University
Title: Valuation method for Nambu-Poisson algebras
Abstract: We discuss automorphism, isomorphism, and embedding problems for a family of Nambu-Poisson algebras (or n-Lie Poisson algebras) using Poisson valuations. This is joint work with Hongdi Huang, Xin Tang, and James Zhang.
Sarah Witherspoon – Texas A&M University
Title: Support for modules and bimodules
Abstract: We discuss two types of topological spaces associated to modules and bimodules for some types of rings, namely the Balmer support and cohomological support. These spaces are important tools for understanding the structure of categories of modules and bimodules, best understood in classical settings such as finite group representations. Recent results for the bimodule setting will be presented. This is joint work with Kent Vashaw and Oeyvind Solberg.
Quanshui Wu – Fudan University
Title: Irreducible representations of Hopf algebras with large centers
Abstract: Any affine Hopf algebra admitting a large central Hopf subalgebra can be endowed with a Cayley-Hamilton Hopf algebra structure. The irreducible representations of such an algebra H are the disjoint union of the irreducible representations over its fiber algebras. The category of finite-dimensional modules over any fiber algebra of H is an indecomposable exact module category over the tensor category of finite-dimensional modules over the identity fiber algebra. The talk is based on recent work with Huang, Mi, and Qi.
Short talks
Yasmeen Baki
Title: Obstructions towards defining nonabelian group-graded twisted Calabi–Yau algebras
Abstract: In this talk, we motivate the study of nonabelian group-graded twisted Calabi–Yau (tCY) algebras by examining a result of Crawford which gives an example of a 2-dimensional AS regular algebra graded by a finite nonabelian group. While historically the study of graded tCY algebras has meant the study of such algebras under N-gradings, recent results have given a suitable definition of G-graded tCY algebras for G an arbitrary abelian group. A natural next question is if such a suitable definition can be given if we are to drop the abelian assumption on G. We discuss the major obstructions and possible remedies towards providing a definition of graded tCY algebras in the nonabelian case.
Lucas Buzaglo
Title: Hopf coactions on quantum polynomial rings
Abstract: Hopf coactions on two-dimensional AS-regular algebras have been widely studied. For example, Crawford proved that if H = kG is a noncommutative group algebra which coacts inner faithfully on a two-dimensional AS-regular algebra A, then A = k<x,y>/(xy + yx) and G is a quotient of the group <a,b | a^2 = b^2>. On the other hand, little is known about the situation in three or more dimensions. I will talk about some of my recent work with Daniel Rogalski about Hopf coactions on three-variable quantum polynomial rings.
Sarafina Ford
Title: Stanley’s theorem for gentle algebras
Abstract: Stanley’s theorem is a result from commutative algebra which allows us to determine if an algebra is Gorenstein from its Hilbert series. For gentle algebras, a well-studied class of bound quiver algebra, there is an analogue of Stanley’s theorem that allows us to determine whether a gentle algebra is AS Gorenstein from symmetries of its matrix Hilbert series.
Peter Goetz
Title: Noncommutative Matrix Factorizations from Quadratic Normal Elements
Abstract: Motivated by recent work with Kirkman-Moore-Vashaw, I will explain how to construct noncommutative matrix factorizations from quadratic normal elements in a Koszul algebra. This builds upon work of Smith-van den Bergh, Mori-Ueyama and He-Ye.
Mengwei Hu
Title: On certain Lagrangian subvarieties in minimal resolutions of Kleinian singularities
Abstract: Kleinian singularities are quotients of C^2 by finite subgroups of SL_2(C). They are in bijection with the ADE Dynkin diagrams via the McKay correspondence. In this talk, I will introduce certain singular Lagrangian subvarieties in the minimal resolutions of Kleinian singularities that are related to the geometric classification of certain unipotent Harish-Chandra (g,K)-modules. The irreducible components of these singular Lagrangian subvarieties are P^1's and A^1's. I will describe how they intersect with each other through the realization of Kleinian singularities as Nakajima quiver varieties. Time permitting, we will also discuss the deformations of these singular Lagrangian subvarieties.
Thomas Lamkin
Title: Point Modules of Non-Connected Graded Algebras
Abstract: One of the earliest successes of noncommutative projective geometry was the use of (truncated) point modules---or more precisely, their (sequence of) fine moduli spaces---by Artin-Tate-Van den Bergh to study 3-dimensional Artin-Schelter regular algebras. Since then, there has been much interest in studying point modules of connected graded algebras. In this short talk, we discuss what we believe is the right notion of a (truncated) point module for a path algebra with relations, show they are parameterized by a quotient stack, and show the resulting sequence of stacks stabilizes for a family of 3-dimensional twisted graded Calabi-Yau algebras.
Ian Martin
Title: Bicharacter twists of quantum groups
Abstract: It has long been known that multiparameter quantum groups can be obtained from their one-parameter counterparts by twisting the algebra structure by a Hopf 2-cocycle. However, such a description is often not suitable for explicit computations, and it has further limitations for affine quantum groups, since there is no explicit description of the coproduct in the new Drinfeld realization. We shall indicate how a different method -- twisting a bigraded Hopf algebra by a skew-bicharacter -- applies to quantum groups, and show that this procedure allows one to prove fundamental results on multiparameter quantum groups by reducing to the corresponding one-parameter quantum groups. In particular, we will see how to establish isomorphisms between the Drinfeld-Jimbo, new Drinfeld, and R-matrix realizations of two-parameter quantum affine algebras in all classical types. This is based on joint work with Alexander Tsymbaliuk.
Shashank Singh
Title: A Quiver-Theoretic Description of Module Categories over Taft Algebras
Abstract: The Taft algebras $T_\ell(q)$ are a fundamental class of non-commutative, non-cocommutative finite-dimensional Hopf algebras. While the classification of exact module categories over $\text{Rep}(T_\ell(q))$ is established, explicit combinatorial realizations of these categories provide deeper structural insight. In this talk, I present a quiver-theoretic description of these module categories. We demonstrate how the action of the tensor category can be encoded into the path algebra of a directed quiver. This approach translates the abstract categorical data into concrete graph-theoretic properties, offering a clear visualization of the indecomposable module categories associated with Taft algebras.
Padmini Veerapen
Title: A potpourri of twists
Abstract: In this talk we explore twists of algebras and bialgebras/Hopf algebras and discuss how they can be explicitly computed.
Trung Vu
Title: On De Concini-Kac forms of quantum groups
Abstract: A quantum group over complex numbers is a Hopf algebra depending on a semisimple Lie algebra and a complex parameter q. When q is a root of unity, there are several forms that people consider. The De Concini-Kac form is the easiest to define, and it behaves like the enveloping algebra in positive characteristics; in particular, it is finitely generated over the center. When the order of q is odd, there is a nice description of the center, and its representation theory has been extensively studied in the works of De Concini-Kac-Procesi and others. In contrast, when the order of q is even, the center is more complicated, and its representation theory is less understood. I will discuss a modification to the De Concini-Kac form that places the cases of odd and even order roots of unity on equal footing.
Yunmeng Wu
Title: 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.