Photo by Fractal Hassan on Unsplash
Talk Location: Student Innovation Center (SICTR), room 2221.
Saturday, November 8, 2025
8:00-9:00
Registration
9:00-10:00
Speaker: N. Rozhkovskaya
Talk title: Action of $W_{1+\infty}$ and $W^B_{1+\infty}$ on generating functions of symmetric functions
Abstract: Central extension $W_{1+\infty}$ of the Lie algebra of differential operators on the circle appeared in mathematical physics literature in nineties, in the relation to various models of two-dimensional quantum field theory and integrable systems. A recent rise of interest in the structure and representations of this infinite-dimensional Lie algebra is due to its role in the interpretation of generating functions of certain intersection numbers on moduli spaces of stable curves as tau-functions of soliton integrable hierarchies. This in turn led to the studies of the action of $W_{1+\infty}$ on particular families of symmetric functions.
In this talk we describe the action of $W_{1+\infty}$ operators and their B-type analogues on Schur and Schur Q-functions respectively through the techniques of formal distributions. We observe an interesting self-duality property possessed by these compact formulas.
10:00-10:30
Speaker: S. Cui
Talk title: Quantum invariants from Hopf algebras and tensor categories beyond semisimplicity
Abstract: Topological quantum field theories (TQFTs) provide quantum invariants that are useful to distinguish manifolds. In dimension 4, it is known that semisimple TQFTs cannot detect smooth structures of manifolds. We develop a framework to construct quantum invariants of
4-manifolds that can potentially extend to the nonsemisimple setting, based on nonsemisimple Hopf algebras or certain ribbon categories. The resulting invariants reduce to the Crane-Yetter invariants in special cases. We also provide extensions to include cohomology data of
4-manifolds in the construction.
10:30-10:45
Speaker: N. Bridges
Talk title: On Invariants of 3-Manifolds with Embedded Links
10:45-11:00
Coffee/Tea
11:00-11:15
Speaker: A. Watkins
Talk title: Exploring Modular Tensor Categories with Egyptian Fractions
Abstract: Solutions $[d_1,...,d_R]$ to the equation $\sum_{i=1}^R \frac{1}{d_i}$ for a fixed $R \in \Z$ have been studied for millennia and are called Egyptian fraction representations of 1 of rank $R$. For an integral modular tensor category (MTC) with simples ${X_1, …, X_R}$, the numbers $d_i = dim(\C)/dim(X_i)$ form an Egyptian fractions representation of 1. The largest number $d_i$ in this solution is of particular interest because it is equal to $dim(\C)$. It has been shown that the sequence [d_1,...,d_R] is bounded by a function which is doubly exponential with respect to $R$. This gives a bound on the dimension of an integral MTC with $R$-many simple objects. In recent work with Agustina Czenky, Emily McGovern, Eric Rowell, and Julia Plavnik, we study Egyptian fractions representations of 1 with a limited number of prime divisors in hopes of getting improved bounds on the terms $[d_1,...,d_R]$.We explore this question and the question of how many Egyptian fractions representations of 1 there are in a given rank $R$ via an experimental approach.
11:15-11:45
Speaker: M. Molander
Talk title: A Well-Defined Jellyfish Algorithm for Affine E Planar Algebras
Abstract: A planar algebra is an infinite collection of vector spaces equipped with an action of planar tangle diagrams. A first example is when the vector spaces are chosen to be the Temperley-Lieb algebras. Vaughan Jones first introduced planar algebras in the study of subfactors. Bigelow, Morrison, Peters, and Snyder introduced a powerful diagrammatic evaluation method on planar algebras called the jellyfish algorithm.
In quantum topology, evaluation algorithms typically reduce some measure of complexity in each step. For instance, the Kauffman bracket reduces the crossing number at every step. The jellyfish algorithm stands out in that it may initially increase complexity.
In this talk, I will give a generators-and-relations presentation for planar algebras corresponding to subfactors with principal graph affine E7. Then I will define a jellyfish algorithm on this planar algebra and show it gives rise to a well-defined map onto the complex numbers.
11:45-12:15
Speaker: D. Rosso
Talk title: Weight modules for mirabolic quantum sl(n)
Abstract: Mirabolic quantum sl(n) is an associative algebra defined by a convolution product on the space of triples of two partial flags and a vector. We show that it has the structure of a comodule algebra for the quantum enveloping algebra U_q(sl(n)), and use this to classify all of its finite dimensional representations. We also give an explicit description of the correspondence between these representations and the ones for the mirabolic Hecke algebra, which is given by a mirabolic quantum Schur-Weyl duality. This is joint work with Pallav Goyal (UC Riverside).
12:15-1:45
Lunch Break
1:45-1:50
Conference Photo
2:30-3:00
Speaker: J. Gaddis
Talk title: Actions of Taft algebras on noetherian down-up algebras
Abstract: When $A$ is a polynomial ring and $G$ a finite group, the Shephard--Todd--Chevalley Theorem gives conditions for the invariant ring $A^G$ to be a polynomial ring. Similarly, Watanabe's Theorem gives conditions for $A^G$ to be Gorenstein. There are extensions to both theorems when $A$ is Artin--Schelter regular and $G$ is replaced by a semisimple Hopf algebra. However, few results exist for the case when $G$ is replaced by a nonsemisimple Hopf algebra. In this talk, I will discuss actions of Taft algebras on noetherian graded down-up algebras. Precise conditions will be given for the invariant ring to be commutative and, more generally, to be Artin--Schelter regular. This is joint work with Simon Crawford and Robert Won.
3:00-3:30
Speaker: P. Veerapen
Talk title: Hopf Algebras Through the Lens of Various Twists
Abstract: In this talk, we first explore how a twist of an algebra's multiplicative structure by either a automorphism or by a set of bijective linear maps, commonly known as Zhang twists, can be extended to a twist of the associated Manin's universal quantum group of the algebra. Moreover, we determine when a Zhang-twisted Hopf algebra coincides with its 2-cocycle-twisted one. Second, we discuss applications of the above-mentioned results to Koszul regular algebras.
3:30-3:45
Speaker: J. Lim
Talk title: Planar algebras associated to cocommuting squares
Abstract: The ‘generalized symmetries’ of subfactors are encoded by their planar algebras. A natural foundational question is “What minimal structure can be universally expected from these symmetries?” For arbitrary subfactors, Jones showed that the associated planar algebra contains the Temperley-Lieb-Jones algebra. When an intermediate subfactor is present, the planar algebra contains the Fuss-Catalan-Bisch-Jones algebra, introduced by Bisch and Jones. However, the situation with two intermediate subfactors remains an open problem. As a natural special case, we study cocommuting squares of factors with ‘group-like’ properties. We exhibit the skein relations of their associated planar algebras and show that these algebras extend the partition algebras. This is based on joint work with Dietmar Bisch.
3:45-4:00
Speaker: D. Wallick
Talk title: $G$-crossed braided categories from quantum spin systems
Abstract: Naaijkens’ superselection theory provides an operator-algebraic description of anyonic excitations for (2+1)-dimensional topologically ordered quantum spin systems. This approach has proven to be a useful method to describe the physics of these systems mathematically rigorously in the infinite volume. Notably, these anyonic excitations form a braided tensor category. Our work generalizes Naaijkens’ methods to include symmetry-enriched topological orders. These are topologically ordered spin systems that also have a symmetry action by a finite group $G$. Given an onsite $G$-action for a quantum spin system and a $G$-invariant state $\omega_0$, we define a notion of $G$-defect sector that extends Naaijkens’ definition of anyon superselection sector. Under suitable assumptions, we show that the $G$-defect sectors form a $G$-crossed braided tensor category, as expected from the physics literature. We compute this category for some short-range and long-range entangled models. This is joint work with Kyle Kawagoe and Siddharth Vadnerkar.
4:00-4:20
Coffee/Tea
4:20-4:35
Speaker: S. P. Singh
Talk title: Module categories over representations of Taft algebras
Abstract: Gabriel’s theorem classifies finite-dimensional basic associative algebras using bound quivers, viewing them as algebra objects in the tensor category of vector spaces. This talk explores a generalization of this theorem by replacing vector spaces with other tensor categories. I will present my work on this problem for the specific case of the representation categories of Taft algebras.
4:35-5:05
Speaker: D. Green
Talk title: Some problems in the structural theory of Hopf monoidal categories
Abstract: Hopf monoidal categories are a straightforward vertical categorification of the well known Hopf algebra axioms. I will provide a short intro to the definitions, a brief survey of known structural results (categorifications of well known theorems for Hopf algebras), and an example (computed jointly with Brett Hungar and Sean Sanford). The remainder of the time will be spent describing a few open problems, some more well formed than others.
5:05-5:35
Speaker: G. Faurot
Talk title: Homotopy 3-type of Abelian C*-Algebras
Abstract: C*-categories have a rich history in the study of C*-algebras, going back to work of Rieffel in the 1970s. Recent work of Ferrer provides a framework for C*-3-categories, a higher category theoretic analogue of a C*-category. Given an object $C$ of a higher category, one may compute the homotopy groups of $C$ in this category. In the C*-3-category $\mathbf{AbC^*Alg}$, we decompose the first higher homotopy group of $C(T)$ as an extension of the homeomorphism group of $T$ by a subgroup of its third cohomology group. This work is joint with Giovanni Ferrer.
6:00-8:00
Conference Dinner
Sunday, November 9, 2025
8:00-8:30
Coffee/Tea
8:30-9:00
Speaker: H. M. Peña Pollastri
Talk title: The relationship of bicrossed products of fusion categories and extensions
Abstract: In this talk, we review the notions of exact factorizations of fusion categories and extensions of them. We show that some bicrossed product of fusion categories, which are exact factorizations, are related to the notion of crossed extension of fusion categories introduced by Natale. The relation is under the correspondence of exact factorizations and extensions proven by Basak and Gelaki. As a result of this correspondence, we completely characterize the exact factorizations where one of the factors is the category of G-graded vector spaces showing they are all bicrossed products.
9:00-10:00
Speaker: R. Kaufmann
Talk title: Classical and quantum aspects of actions, twists and quotients.
Abstract: There is well known theory of group actions, invariants, and quotients. Starting from this, we add in more structures that have been considered in the past decades, many of them motivated by physics, notably orbifold CFTs and mirror symmetry. Twists are a natural occurrence in many branches of mathematics. Informally one twists with a one—dimensional objects or cycles, or more precisely by invertible objects. We will examine the effect of such twists which are comparable to zesting. Group quotients naturally come equipped with a torsor of possible twists. The type of twist was found algebraically by Drinfel’d when considering certain types of Hopf algebras.
We will then discuss mirror type operations in the framework of equivariantization and de-equivariantization. Finally, and this is work in progress with A. Davydov, we show how these relate to the classical constructions using Hochschild homology.
10:00-10:15
Speaker: J. Van Grinsven
Talk title: Equivariant Morita Equivalences of Loewy-Graded Comodule Algebras
Abstract: We discuss questions related to preferred representatives of equivariant Morita equivalence classes of Loewy-graded comodule algebras. In particular, if A is a Loewy-graded H-comodule algebra and H(0)-equivariant equivalence A(0)~X, then there exists a Loewy-graded comodule algebra B, that is equivariant equivalent to A and isomorphic to X in degree zero.
10:15-10:30
Speaker: A. Bhardwaj
Talk title: Superselection sectors for posets of von Neumann algebras
Abstract: Starting with a poset that satisfies some simple geometric axioms, we study a collection of commutant closed von Neumann algebras indexed by this poset. We then construct a braided W^* tensor category of superselection sectors analogous to the construction by Gabbiani and Frohlich for conformal nets on S^1.
10:30-11:00
Speaker: Y. Liu
Talk title: The Homotopy Theory of Categorified Relative Hopf Modules
Abstract: Differential graded (dg) categories and dg-functors can be viewed as models for noncommutative spaces and their representations.
The homotopy theory of dg-categories and dg-modules fits naturally into the framework of model structures on enriched categories and enriched functors over chain complexes, providing a powerful foundation for derived and homotopical algebraic geometry.
Pareigis observed that the category of chain complexes is equivalent to the category of comodules over a Hopf algebra $B$ (a particular coFrobenius Hopf algebra), allowing this theory to be reinterpreted in terms of categorified comodule algebras and relative Hopf modules_.
In joint work with Julia Plavnik and Samarpita Ray, we extend this framework to arbitrary coFrobenius Hopf algebras $H$.
We show that the category of $H$-comodules admits a combinatorial, non-symmetric monoidal model structure, and that the category of categorified relative Hopf modules --defined in terms of enriched functors -- also carries a combinatorial model category structure.
These results generalize Ohara’s construction of the model structure on relative Hopf modules over a comodule algebra and set the foundation for derived Morita theory in a more general setting.
11:00-11:30
Speaker: D. Addabbo
Talk title: Modularity of Vertex Operator Algebra Correlators with Zero Modes
Abstract: We will discuss modular transformation properties of vertex operator algebra correlators that have zero modes inserted. In particular, we will discuss recursion relations for these correlators and describe how to use these recursions to establish the modular transformation properties (based on joint work with Christoph A. Keller).
11:30-12:30
Speaker: Z. Lin
Talk title: Homotopy Lie algebras and support varieties for representations of vertex algebras
Abstract: Given a vertex algebra, one can attach a Poisson variety, called the associated variety, defined by the $C_2$-algebra. In fact, each representation of the vertex algebra will give rise to a sheaf of Poisson modules on the associated variety. The support of this sheaf is a Poisson subvariety. Now one can use the geometric tools to stratify the representation category of the vertex algebra in terms of the closed Poisson subvarieties of the associated variety. However, when the vertex algebra is rational and $C_2$-cofinite, the associated variety becomes a single point. In this talk, we define a different collection of varieties as a cohomological dual, by the maximal commutative quotient of the Yeneda algebra at each closed subvariety of the associated variety. It turns out that the Yoneda algebra is the universal enveloping algebra of a graded Lie algebra, called homotopy Lie algebra. Now there is a second approach to define the associated variety as closed subvariety of the dual of the graded Lie algebra (which is automatically a graded Poisson variety) if the module is finitely generated in the same way as associated varieties for modules of Lie algebras. This is a joint work with Antione Caradot.