A workshop on tensor algebra, rewriting systems, topology and universal algebra, to investigate multiple aspects of generalized higher-order algebra. This event brings together mathematical researchers from multiple areas and to create the opportunity for intellectual cross-pollination.

Organizers:

Carlos Zapata (Wolfram, SEMF), Liubov Tupikina (Bell labs, LPI), Richard Kerner (LPTMC)

Logistics

This will be a hybrid event revolving around a 3-day on-site meeting in central Paris. A virtual forum (on Discord) will be enabled for participant discussion and collaboration prior, during and after the event. Live sessions will be held in Zoom (invites to be sent to participants who register in the link above). The live sessions will be livestreamed and archived by the Wolfram Institute.

The asynchronous discussion will take place on the Wolfram Insitute Discord server:

Live Meeting Schedule

All times are Paris local time (UTC+2).

Time | Speaker | Title of Talk |
---|---|---|

14:00 - 15:00 | Richard Kerner | Ternary and binary algebras with Z2 and Z3 grading |

15:00 - 16:00 | Andrzej Borowiec | Ternary groups and ternary algebras: a basic concept |

16:00 - 16:30 | Coffee Break | |

16:30 - 17:30 | Viktor Abramov | Biunits of ternary algebra of hypermatrices |

Tuesday 28 May

Time | Speaker | Talk |
---|---|---|

14:00 - 15:00 | Liubov Tupikina | Navigating the Hypergraph Knowledge Scape |

15:00 - 16:00 | Nicolas Behr | Algebras of Compositional Rewriting Theories |

16:00 - 16:30 | Coffee Break | |

16:30 - 17:30 | Francesco Lotito | Higher-order network analysis with HypergraphX |

Wednesday 29 May

Time | Speaker | Talk |
---|---|---|

14:00 - 15:00 | Carlos Zapata-Carratala | Plex Diagrams: Hypermatrix Algebra via Rewriting |

15:00 - 16:30 | Discussion and Coffee Break | |

16:30 - 17:30 | Josh Grochow | Weisfeiler-Leman as Bhattacharya-Mesner |

Talks

**Ternary and binary algebras with Z2 and Z3 grading**

Richard Kerner (LPTMC, Sorbonne-Université)

We expose the action of Z2 and Z3 cyclic groups on 2- and 3-forms defined on a differential manifold of low dimensions, usually 2 or 3. Then we define subspaces of forms with given symmetry properties with respect to Z2 and 23 groups. Also graded matrix algebras are investigated, with ternary generalizations of Grassmann and Clifford structures. Z3-graded differential calculus is also presented. Possible relationship with graphic representations will be mentioned at the end, too.

**Algebras of Compositional Rewriting Theories**

Nicolas Behr (CNRS, Université Paris Cité, IRIF)

Manipulations of graphs, hypergraphs, and other graph-like structures are a ubiquitous concept in many different research fields. In this talk, I will introduce compositional rewriting theory, a category-theoretical approach to formalizing operations on graph-like structures and analyzing the resulting rewriting systems mathematically and algorithmically. I will then illustrate how compositionality entails the existence of certain algebraic structures encoding combinatorial or statistical properties of sequences of rewriting operations, and provide several application examples of this theory to Markov chains and combinatorics.

**Higher-order network analysis with hypergraphx**

Francesco Lotito (University of Trento)

From social to biological systems, many real-world systems are characterized by higher-order, non-dyadic interactions. Such systems are conveniently described by hypergraphs, where hyperedges encode interactions among an arbitrary number of units. In this talk, we present an open-source python library, hypergraphx (HGX), providing a comprehensive collection of algorithms and functions for the analysis of higher-order networks. These include different ways to convert data across distinct higher-order representations, a large variety of measures of higher-order organization at the local and the mesoscale, statistical filters to sparsify higher-order data, a wide array of static and dynamic generative models, and an implementation of different dynamical processes with higherorder interactions. HGX allows to analyse hypergraphs with weighted, directed, signed, temporal and multiplex group interactions. We provide visual insights on higher-order data through a variety of different visualization tools. We accompany our code with an extended higher-order data repository, and demonstrate the ability of HGX to analyse real-world systems through a systematic analysis of a social network with higher-order interactions. The library is conceived as an evolving, community-based effort, which will further extend its functionalities over the years. The software is available at HypergraphX.

**Plex Diagrams: Hypermatrix Algebra via Rewriting**

Carlos Zapata-Carratala (Wolfram Institute)

I will introduce the notion of a plex - an abstract array on a set of indices - and the diagrammatic calculus that recovers many of the hypermatrix operations that have been identified in the literature. I will comment on recent developments in the search for associativity-like axioms.

**Navigating hypergraph knowledge scape: learning higher order structures from data**

Liubov Tupikina (Bell labs, France, LPI)

In this talk I will investigate the latent space of embedding methods using a combination of two approaches: hypergraph embeddings and dimensionality reduction techniques. Our approach aims at both developing explainable embedding methods as well as investigating the higher-order relations in data structures. We demonstrate how properties of resulting hypergraph embeddings allow to navigate knowledge maps representing high dimensional textual data. As the concrete application of our method we look at a database of scientific articles to see how the knowledge landscape can be characterized using binary and higher arity perspective.

**Ternary groups and ternary algebras: a basic concept**

Andrzej Borowiec (Institute of Theoretical Physics, University of Wroclaw)

I aim to review and discuss some earlier results on ternary groups, ternary algebras, their bi-element representations, ternary differential calculus and the concept of ternary Hopf algebras.

**Biunits of ternary algebra of hypermatrices**

Viktor Abramov (Institute of Mathematics and Statistics, University of Tartu)

We study a ternary algebra of third-order hypermatrices, where by hypermatrix we mean a complex-valued quantity with three indices. Ternary multiplications of hypermatrices have the property of generalized associativity. We introduce the concepts of q-cyclic and q-cyclic traceless hypermatrix, where q is a primitive third-order root of unity and study the structures of the corresponding subspaces. We show that q-cyclic traceless hypermatrices can be used to construct right biunits of the ternary algebra of third-order hypermatrices. We show the connection between the Clifford algebra structure induced by the quadratic invariant of q-cyclic traceless hypermatrices and ternary multiplication of hypermatrices. The motivation for studying the space of q-cyclic traceless hypermatrices is the ternary generalization of the Pauli exclusion principle.

**Weisfeiler-Leman as Bhattacharya-Mesner**

Josh Grochow (Department of Computer Science, University of Colorado)

The Bhattacharya-Mesner product arises in many places, and this talk is about a new realization that it arises in the setting of algorithms for the Graph Isomorphism problem. Specifically, the k-dimensional Weisfeiler-Leman algorithm, crucial in both practical and theoretical algorithms, corresponds to algebras of k-ary hypermatrices that contain certain elements and are closed under both Hadamard and Bhattacharya-Mesner products. For k=2 this goes back to Weisfeiler and Leman's cellular algebras and Higman's coherent configurations, and was used to great algorithmic effect by Ponomarenko in 1994. But for k>2, we are working on whether an analogous theory can be built. The talk will not assume any background in algorithms.