Algebra

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.

Abstract - Dehn’s famous decision problems for finitely presented groups have been studied for over a century by combinatorial and geometric group theorists. In recent years, a variant of one of these classical problems, namely the twisted conjugacy problem, has been studied. The motivation for this problem comes from Bogopolski, Martino and Ventura who, in 2009, proved an equivalence between conjugacy in group extensions and twisted conjugacy.
In this talk I will give a brief survey of this lesser-known decision problem, and discuss some of the latest results in this area. This includes a framework which can be applied to dihedral Artin groups.

Abstract: With (almost) every finite mutation class of quivers we associate two groups: an extended affine Weyl group of type A or D, and a certain quotient of a Coxeter group which behaves nicely with respect to mutations (in most cases, the latter can also be obtained as a quotient of certain surface braid group). I will discuss a connection between these groups and a (still conjectural) characterization of mutation-finite quivers in terms of positive semidefinite symmetric matrices. The talk is based on joint works (some still in progress) with Anna Felikson, John Lawson and Michael Shapiro.

Classical knot theory associates to a combinatorial knot diagram a knot embedding in the 3-sphere. The fundamental group (and peripheral system) of the complement give rise to powerful, combinatorially computable invariants. This talk explores an analogous construction for welded knots, a diagrammatic extension of classical knot theory corresponding to ribbon knotted tori in 4-space via the tube map.

We introduce a new topological invariant, the fundamental \pi-module (built using the first and second homotopy groups), and show how it can be computed combinatorially from a diagram. We then define a natural topological generalisation of the peripheral system, using the free loop space of the complement, and again show how this admits a simple combinatorial description.