A workshop on a variety of topics, such as association schemes, hypermatrices, tensor algebra or n-ary algebras, under the common theme of generalized matrix algebra. This event brings together mathematical researchers from multiple areas and to create the opportunity for intellectual cross-pollination.

Logistics

This event will take place online and will be comprised of two days of talks (Monday and Friday) and three days of free-form interactions over Discord and informal Zoom meetings (Tuesday, Wednesday and Thursday). Although exact timings may vary slightly since we will encourage informal discussions to take place organically, this is the talk schedule, times are UTC:

Monday April 24

16:00 - 17:00 "Rainbow Arrays", T. Beynon
18:00 - 19:00 "Equivalences of Hypermatrix Representations", J. Grochow
19:00 - 20:00 "Hyperdeterminants for Quantum Information", L. Oeding

Friday April 28

16:00 - 17:00 "Hypergraphs and Higher-Arity Algebras", C. Zapata-Carratalá
17:00 - 18:00 "Higher-Order Algebras", M. Rausch
18:00 - 19:00 "Wheeled PROPs", H. Derksen
19:00 - 20:00 "Complexity of Hypermatrix Equivalence", E. Gnang

Live talks, discussions and free-form interations will be coordinated on the following Discord channel (where Zoom links will be provided):

HyperMatrix Workshop Discord Channel

By accessing this link you register to attend the event. If you have issues accessing Discord or would prefer only to access the live meetings, please email Carlos Zapata (organizer) at c.zapata.carrtala@gmail.com.

Talks

On the complexity of hypermatrix equivalence
Edinah Koffi Gnang, Johns Hopkins University

Equivalence of matrices defined relative to matrix group actions are crucial to many practical numerical matrix algorithms. In this talk we discuss how equivalence classes of hypermatrices defined relative to hypermatrix actions can serve as the basis for hypermatrix algorithms. We describe insight gain by these new hypermatrix equivalence classes.

Equivalences between different kinds of objects represented by hyper-matrices
Joshua Grochow, University of Colorado

Hyper-matrices can be used to represent many kinds of mathematical objects, from hypergraphs and tensors, to polynomials, to groups and algebras. Over the past several years, with a variety of coauthors, we have shown that nonetheless nearly all these different settings are equivalent to one another under simple (affine) linear maps, computable in polynomial time. For example, there is a map that transforms any space of matrices into a related associative algebra in such a way that two spaces of matrices are equivalent up to change of bases if and only if the corresponding algebras are isomorphic (in the usual sense of isomorphisms of algebras). As another example, we show there is a similar map from k-tensors (equivalently, k-partite pure quantum states) to 3-tensors (tripartite states), that is, 3-tensors are sufficiently rich to simulate anything that can happen even with k-tensors. In this talk, we will discuss some of these results, the techniques that go into them, and some of their implications. Based on joint works with Futorny & Sergeichuk (Lin. Alg. Appl. 2019), and Y. Qiao (ITCS '21, STACS '21 also with G. Tang, CCC '21, and upcoming preprints).

Higher-order Algebras and Groups
Michel Rausch, Louis Pasteur University

Lie algebras of order $F$ (or F-Lie algebras) are possible generalisations of Lie algebras (F=1) and Lie superalgebras (F=2). An F-Lie algebra admits a Z_F-gradation, the zero-graded part being a Lie algebra. An F-fold symmetric product (playing the role of the anticommutator in the case F=2) expresses the zero graded part in terms of the non-zero graded part. This structure enables us to define various non-trivial extensions of the Poincare algebra. These extensions are studied more precisely in two different contexts. The first algebra we are considering is shown to be an (infinite dimensional) extension of the Poincare algebra in (1+2)-dimensions and turns out to induce a symmetry which connects relativistic anyons. The second extension we are studying is related to a specific finite dimensional Lie algebras of order 3 and is associated to cubic extension of the Poincare algebra. We then summarize some of the main results obtained in that context. Finally, we show that one is able to associate a group to these structures.

Wheeled Product and Permutation Categories
Harm Derksen, Northeastern University

Wheeled PROPs are certain algebraic structures related to PROPs and operads. Wheeled PROPs give a suitable framework to study classical invariant theory, representation theory, tensors and tensor networks. I will discuss some fundamental results about wheeled PROPs and their ideals, and applications to hypermatrix data.

Rainbow Arrays
Tali Beynon

Multidimensional arrays are a core data structure used in deep learning and data science. This talk will discuss how such arrays are profitably understood as functions whose inputs are the "address space" that describes the addresses of particular array cells, and whose outputs are contents of those array cells. This analogy yields unexpected correspondences: reordering function arguments ≅ array transposition; currying functions ≅ array slicing; extending functions to be "partially constant" on part of their domain ≅ array broadcasting; post-composition of functions ≅ array arithmetic. But perhaps most importantly, moving from positional to named function arguments catapults us to a cutting-edge feature of array programming called named axes, in which arrays have axes that are "colored", and array operations must be "color compatible". These "rainbow arrays" are a potential basis for a much more intuitive and flexible language of array programming.

Hyperdeterminants for Quantum Information
Luke Oeding, Auburn University

Twenty years ago Cayley's hyperdeterminant, the degree four invariant for 2x2x2 tensors, was popularized in modern physics as it separates genuine entanglement classes in the three qubit Hilbert space. I will talk about recent geometric constructions that allow us to compute for the first time the analogous hyperdeterminant for other tensor spaces that arise in quantum information. This talk is based on joint work with Frederic Holweck (UTBM, France).

Hypergraphs and Higher-Arity Algebra
Carlos Zapata-Carratalá, Wolfram Institute

One of the main results in graph theory is the equivalence between graphs and adjacency matrices which implies the close relation between topological properties of graphs and algebraic properties of matrices. A similar equivalence has not been fleshed out for hypergraphs, largely due to the lack of hypermatrix theory. We will present some basic results that identify a class of correspondences between certain hypergraph rewrite rules and higher-arity algebras.