Program

Bell-Rogalski algebras are a family of algebras that further extends generalized Weyl algebras (GWAs). These are all defined over a fixed base ring R with an automorphism σ. In previous work with J. Hartwig, we showed that the tensor product over R of two weight modules for two GWAs is a weight module over another GWA, which allows us to define a ring structure on the Grothendieck group of the category of all such modules. In this work we extend this construction to BR algebras and draw some connections to quiver representations. This is a preliminary report of joint work with Jason Gaddis. 

Quantum SL(2) plays an important role in mathematical physics and quantum topology. In many settings, the central object of study is the braided tensor category of representations of the small quantum group. Lusztig constructed the small quantum group as a Hopf subalgebra of the Lusztig quantum group. However, at even roots of unity, this Hopf subalgebra is not quasitriangular, so its category of representations does not carry a braiding. Motivated by physics, Creutzig-Gainutdinov-Runkel and Gainutdinov-Lentner-Tobias-Ohrmann constructed braided tensor categories of small quantum group representations at even roots of unity as representation categories of quasi-Hopf algebras.  

In this talk, following work of Negron, I will explain a way to define the category of small quantum group representations uniformly at all roots of unity, recovering the previous definitions at relevant parameters. We will then study fiber functors on these categories, making use of a classification result by Ostrik. This recovers the known classifications at odd roots of unity, while at even roots it shows that the braided tensor category cannot be realized as the representation category of a Hopf algebra.

The anyonwiki is a repository for data on anyon models and fusion categories. In this talk I will present its main features and show how to to finw and download computable data on fusion categories. I will also show how anyone can help expand the wiki by writing pages and adding references.

The classification of quantum phases of matter is a fundamental topic in the study of quantum many-body systems. A central question in this classification is whether or not there is a nonvanishing spectral gap above the ground state energy and, if so, whether or not this gap is stable under sufficiently short-range perturbations. While its importance is well known, determining the existence of a gap is notoriously difficult, especially for multidimensional models. In their seminal work, Affleck, Kennedy Lieb and Tasaki (AKLT) conjectured the existence of a spectral gap for the model they defined on the hexagonal lattice. Significant evidence supporting this long-standing claim was only recently achieved, and naturally leads to the question of whether or not this gap is stable. In this talk, we review some recent progress related to the classification of gapped ground state phases, and then discuss a soon-to-be-appearing work where cluster expansion techniques are used to show that the finite volume ground states of the AKLT model on the hexagonal lattice satisfy an indistinguishability condition known as Local Topological Quantum Order, the key property for establishing gap stability via the Bravyi-Hastings-Michakalis method. This talk is based on joint work with Thomas Andrew Jackson and Bruno Nachtergaele.

Tambara-Yamagami (TY) categories form one of the simplest families of non-pointed fusion categories. Despite their elementary description, these categories play a central role in mathematical physics. Studying generalizations of TY fusion rings can then provide a controlled framework to explore the new phenomena that arise when one moves beyond the classical TY setting. In this talk, I will focus on the generalization of the TY fusion ring, proposed by Jordan-Larson. One of the most useful tools to study fusion rings is the classification of their non-negative integer matrix (NIM-) representations. NIM-reps can be used as an effective method to detect algebra objects in the fusion category underlying the fusion ring. In this talk, we will compute and classify the irreducible NIM-reps of both proposed extensions as well as detect candidate algebra objects associated to these NIM-reps. This work is joint with Agustina Czenky, Emily McGovern, Monique Müller, and Ana Ros Camacho. 

Atiyah, Drinfeld, Hitchen, and Manin provided a correspondence between solutions to Yang-Mills equations on R^4 and complex linear algebraic data. This correspondence was built using the Ward correspondence which links instantons to holomorphic vector bundles on an associated twistor space. Later, Nakajima built an expanded correspondence between framed torsion-free sheaves on CP^2 (as an algebraic compactification of R^4) and stable representations of the doubled framed Jordan quiver. This allowed solutions to a non-linear PDE (YM equations) to be studied using linear algebra. 

In this talk we will present a new result that extends this correspondence where the base space is not CP^2 but Bl(CP^2) the blow up of CP^2 at a point, in an effort to understand how resolving singularities affects the correspondence. Along the way we build new tools to accomplish this, since all of the holomorphic symplectic tools that Nakajima utilized no longer apply. In particular, Nakajima’s quiver doubling is no longer useful and the standard Nakajima quiver variety construction, which is a symplectic reduction and GIT quotient, can no longer be leveraged to create a nice moduli space that is a variety.

Recent work suggests a manifestly unitary construction of higher Hilbert spaces. A key desideratum for higher Hilbert spaces is that they are the local data of fully extended unitary (reflection positive) TQFTs. "Disk-like" higher categories in the sense of Morrison–Walker are defined to describe the local data of a fully extended TQFT in the absence of unitarity.  In this talk, I will introduce a notion of unitary disk-like $n$-category meant to capture the local data of a fully extended unitary TQFT, briefly review 2- and 3-Hilbert spaces, and discuss the equivalence between unitary disk-like $n$-categories and $(n+1)$-Hilbert spaces in the cases $n=1$ and $n=2$.

The algebra of the infrared provides an algebraic framework for encoding structures related to Fukaya–Seidel categories. In this talk, I will discuss a possible extension of this picture from complex-valued potentials to curve-valued potentials. The emphasis will be on the algebraic side, namely the appearance of 𝐴∞- and 𝐿∞-structures arising from geometric and combinatorial data. I will explain the basic idea of this generalization and describe some expected features. 

In this talk, we will recall the quantum toroidal superalgebra of type $A$ and then present two realizations of it. First, we will construct a quantum group associated to a quiver $\widetilde{Q}$ and show that there is a choice for which this quantum group is isomorphic to the quantum toroidal superalgebra. We will also present a shuffle realization of quantum toroidal superalgebra. Further, we will show that there is a factorization of this superalgebra into quantum affine superalgebras. 

Let G be a complex, connected, reductive algebraic group and N be the nilpotent cone of G.

In 1985-1986, Lusztig proved a remarkable property of cuspidal perverse sheaves on N. He proved when the sheaf coefficient has characteristic 0 then the cuspidal perverse sheaves are “clean,” meaning that their stalks vanish outside a single orbit. This property is crucial in making

character sheaves computable by an algorithm, and it plays an important role in “block decompositions” of generalized Springer correspondence.

About 10 years ago, Mautner conjectured that cuspidal perverse sheaves remain clean when the sheaf coefficient has positive characteristic

(with some exceptions for small p). In this talk, I will discuss the history and context of the cleanness

phenomenon, along with recent progress on Mautner’s conjecture. This is joint work with P. N. Achar.

Ordinary vector spaces admit a scalar multiplication from their field. For higher vector spaces, the analogue of scalar multiplication is understood to be a module category structure over (n-1)Vec. However, as higher categories generally possess more structure, this is not the only possible notion of scalar multiplication. We develop and discuss various forms of relative scalar multiplication for 2- and 3-vector and Hilbert spaces. These additional structures are useful for formally proving results about adjoints, modules, and higher Hilbert spaces. This work is joint with Chumeng Di, Giovanni Ferrer, David Penneys, and Greyson Wesley. 

In 2023, Sözer and Virelizier developed monoidal categories graded by crossed modules, aka strict 2-groups, to build invariants of 3-manifolds with a map from their fundamental crossed module to a given crossed module. Following standard reconstruction theory, they defined Hopf crossed-module coalgebras which can be used to define Kuperberg-like invariants of manifolds. In this expository talk, we will explore the structure of the fundamental crossed module, discuss strategies for computing it from a trisection diagram for a smooth 4-manifold X, and, if there is time, give preliminary results defining a Kuperberg-like invariant of the 2-skeleton of X from a semisimple Hopf crossed-module coalgebra. 

There has been much recent progress on the preparation of non-Abelian topological order through measurement, both theoretically and experimentally. Notably, the topological order associated with the quantum double of the dihedral group D4 (of order 8) has been simulated in quantum processors. Motivated by this, in ongoing work, we study the computational power of the anyons in the D4 and D8 quantum double models. It is expected that the anyons in these models are not universal for quantum computing, even when allowing for measurement as an additional resource (resource assisted quantum computation). Although we do not find a universal gate set, we compute all braid group representations on four strands, and describe supplemental gates obtained via anyon fusion. In addition to anyons, we also explore the computational power of the Z2 symmetry defects in D4. We do so by two complementary methods: directly from D4 using the framework developed by Barkeshli et al., and via anyon condensation from D8. With the latter, we explicitly compute the data of the Z2 symmetry defects of D4 (6j-symbols, R-symbols, etc), thus relating the D4 and D8 topological orders.

Down-up algebras, originally defined by Benkart and Roby, have drawn significant interest in representation theory, noncommutative algebraic geometry, and invariant theory. In this talk, I will introduce a generalization of down-up algebras to quivers on many vertices. These algebras arise through various constructions including Ore extensions, generalized Weyl algebras, and normal extensions, which allow us to consider various ring-theoretic properties. This is joint work with Dennis Keeler. 

Cobordism categories describe the algebraic gluing structure of manifolds and play a central role in the functorial formulation of topological quantum field theories (TQFTs). We introduce a “nested” variation of a cobordism category in which manifolds are equipped with embedded submanifolds, and cobordisms with corresponding sub-cobordisms. A motivating example is the category of cylinders with lines. In this talk, I will describe current work on these nested cobordism categories and the algebraic structures associated with them.

The Jordan and super Jordan planes are examples of infinite dimensional Nichols algebras for which suitable Quantum Doubles have been constructed and studied. In this talk, we will discuss some properties of the super Jordan plane over an algebraically closed field with characteristic zero. In particular, we will discuss the classification of its finite dimensional irreducible representations and compare these results to that of the Jordan and restricted super Jordan planes.  

We study the relation subspace $O_n^\circ(V)$ that appears in the definition of the level-$n$ Zhu algebra $A_n(V)=V/O_n(V)$, where $O_n(V)=O^L(V)+O_n^\circ(V)$ and $O^L(V)=(L(-1)+L(0))V$. Using residue calculus, we introduce operators $R^n_{m,k}$ that encode the circle products $u\circ_n v$ and prove explicit change-of-generators formulas between the standard generators $u_{-m}\mathbf{1}\circ_n v$ and the residue generators $u\cdot R^n_{m,0}v$, together with a binomial inversion. These identities provide a practical framework for computing $O_n^\circ(V)$, especially in strongly generated VOAs. As progress toward understanding the overlap $O_n^\circ(V)\cap O^L(V)$, we use that the $k=0$ tower collapses modulo the $k\ge1$ subspace and derive the induced action of $L(-1)+L(0)$ on the resulting circle-layer quotient. This leads to a conjecture describing $O_n^\circ(u,v)\cap O^L(V)$ as $(L(-1)+L(0)),O_n^\circ(u,v)$.

The level-n Zhu algebra A_n(V) serves as a vital bridge between vertex operator algebras and their representation theory, yet obtaining explicit presentations remains a significant challenge due to the complexity of the defect subspace O_n(V). In this talk, we present a new framework designed to handle the circle-type component O_n^\circ(V), which governs the most difficult relations in higher-level constructions (n \ge 1).

We show that O_n^\circ(V) forces a deep-mode recursion: modes beyond a fixed negative cutoff can be replaced by a finite band of less negative modes, provided they appear in the leftmost position. To operationalize this, we introduce the left-truncated field Y_L(u,x), which explicitly records these replacements. We further develop an axiomatic theory of left level-n truncations, providing a suite of Jacobi, commutator, and normal-ordering identities that are natively compatible with this replacement regime.

 Finally, we discuss the computational implementation of this framework. We define an ascending chain of subspaces O_n^\circ(N,S,V) based on word length and formulate a finiteness conjecture concerning the dimensionality of the resulting operator quotients. These tools provide a systematic, computation-friendly path toward finding explicit generators and relations for A_n(V) in a wide range of examples.  We verify the finiteness conjecture for N=2.