Algebra

Abstract: Let Hn be a Hecke algebra over an arbitrary field, and let e be its quantum characteristic. A central open problem in the representation theory of Hn is describing composition factors of Specht modules (which are irreducible if e=0) for e>1. We provide new elementary formulas for many entries of decomposition matrices of Hn - matrices that give these composition factors. The new formulas provide at least one column in every block and are independent of the underlying field, making them highly applicable and insightful. The talk is based on results proved with David Hemmer and with Bim Gustavsson, Stacey Law, Lorenzo Putignano and Liron Speyer.

Abstract: The Schröder problems are a set of 4 combinatorial questions relating to counting different kinds of bracketings. In this talk, we introduce the basic idea of combinatorial species, demonstrate a common species approach which can solve the four of these problems, and look at some of the objects produced by extending this approach.

Abstract: In this talk we give a snapshot of developments in the local-global representation theory of finite groups in recent years. We will begin by describing some of the context and motivating concepts behind counting conjectures and open problems, and present some recent progress in the case of symmetric and alternating groups.

Abstract - The discriminant of a finite reflection group is an early example of a Saito free divisor which is not normal crossing. Consequently, the vector fields which are tangent to the discriminant define a Lie algebroid. In this talk, I will discuss a family of algebraic Lie groupoids (and categories) which integrate these Lie algebroids.

Abstract - This talk focuses on the recently introduced notion of bounded cohomology for quandles. We establish sufficient criteria ensuring that the second bounded cohomology of a quandle is infinite-dimensional. As a topological application, we prove that the second bounded cohomology of the fundamental quandles of most links is infinite-dimensional. If time permits, we will further explore this cohomology for families of quandles arising on surfaces.

Abstract - Symmetry and its anomaly constrain a system's dynamics and provide a universal characterization of its behaviors. It serves as a powerful tool to understand exotic phases of matter, especially symmetry-protected topological (SPT) orders. The notion of generalized symmetries requires extending our previous understanding of topological phases to those enriched by fusion category symmetries, yet new mathematical formalism is needed to implement unitary fusion category symmetries in a quantum many-body system. In this talk, I will introduce a general fixed-point lattice construction of (1+1)d SPTs with unitary fusion category symmetries, realized in a tensor-product Hilbert space with an “onsite” matrix-product-operator (MPO) version of the Hopf C*-algebra symmetry operators. Within this construction, I will address that the UV description of an anomaly-free fusion category symmetry must include the fiber functor, giving rise to a local symmetry action, a charge category and a trivial phase, and discuss an alternative characterization of SPT phases using the Q-system in the charge category. As an example, I will provide an explicit microscopic realization of all three 𝖱𝖾𝗉(D8) SPT phases, including a trivial phase, and further demonstrate the S3-duality among these three SPT phases.

Abstract - A fundamental notion in quantum topology is that of topological quantum field theory (TQFT) formulated by Witten and Atiyah. This notion originates in ideas from quantum physics and constitutes a framework that organizes certain topological invariants of manifolds, called quantum invariants, which are defined by means of quantum groups.
Homotopy quantum field theories (HQFTs) are a generalization of TQFTs. The idea is to use TQFT techniques to study principal bundles over manifolds and, more generally, homotopy classes of maps from manifolds to a (fixed) topological space called the target. The resulting invariants are called quantum homotopy invariants.

Turaev and Virelizier have constructed quantum homotopy invariants of 3-manifolds (by state sum) when the target space is a 1-type, and Sözer and Virelizier have recently constructed quantum homotopy invariants of 3-manifolds when the target space is a 2-type. Using state sum techniques, Douglas and Reutter have constructed quantum invariants of 4-manifolds using fusion 2-categories. In this talk, we combine both of these approaches: we construct quantum homotopy invariants of 4-manifolds with a 3-type target from 3-group extensions of fusion 2-categories.

Abstract - A fundamental notion in quantum topology is that of topological quantum field theory (TQFT) formulated by Witten and Atiyah. This notion originates in ideas from quantum physics and constitutes a framework that organizes certain topological invariants of manifolds, called quantum invariants, which are defined by means of quantum groups.
Homotopy quantum field theories (HQFTs) are a generalization of TQFTs. The idea is to use TQFT techniques to study principal bundles over manifolds and, more generally, homotopy classes of maps from manifolds to a (fixed) topological space called the target. The resulting invariants are called quantum homotopy invariants.

Turaev and Virelizier have constructed quantum homotopy invariants of 3-manifolds (by state sum) when the target space is a 1-type, and Sözer and Virelizier have recently constructed quantum homotopy invariants of 3-manifolds when the target space is a 2-type. Using state sum techniques, Douglas and Reutter have constructed quantum invariants of 4-manifolds using fusion 2-categories. In this talk, we combine both of these approaches: we construct quantum homotopy invariants of 4-manifolds with a 3-type target from 3-group extensions of fusion 2-categories.

Abstract - An associahedron is a polytope arising from combinatorics of Catalan-type objects (for example, from a collection of all triangulations of a given polygon). Fomin and Zelevinsky found a way to construct the same combinatorial structure from considering the Coxeter group of type A_n. This allowed them to define a generalized associahedron for every finite reflection group. For generalized associahedra arising from crystallographic reflection groups, it was also shown that they can be realized as polytopes. We use the folding technique to construct polytopal realisations of generalized associahedra for all non-simply-laced root systems, including non-crystallographic ones. This is a joint work with Pavel Tumarkin and Emine Yildirim.

Abstract - The Graph Reconstruction Conjecture is a long-standing problem in Graph Theory formulated by Kelly (1957) and Ulam (1960). The conjecture states that every graph with at least three vertices can be uniquely reconstructed (up to isomorphism) from their deck one-vertex deleted subgraphs. It is well known that the graph isomorphism problem can be worded using Invariant Theory, although this is not particularly interesting in practice as the computations get quickly out of hand. In this talk, we explore how invariant theory can be used to approach the Graph Reconstruction conjecture. This naturally brings the focus to K-weighted graphs. We focus on the attempt by Thiéry (2000), which led to a disproof of a stronger statement using a computational argument. This also turns out not to be particularly practical, but we'll see how it still brings valuable insight. This talk is based on a survey paper joint with Gabriela Jerónimo, Jenny Kenkel, Haydee Lindo and Nelly Villamizar.