JM Invariant at 60:
A Conference on Tensor Invariants in Geometry and Complexity Theory

May 13–17, 2024

Auburn University

Let \(f\) be a homogeneous polynomial of even degree \(d\). We study the decompositions \(f=\sum_{i=1}^r f_i^2\) where \(\deg f_i=d/2\). The minimal number of summands \(r\) is called the \(2\)-rank of \(f\), so that the polynomials having \(2\)-rank equal to \(1\) are exactly the squares. Such decompositions are never unique and they are divided into \(O(r)\)-orbits, the problem becomes counting how many different \(O(r)\)-orbits of decomposition exist.
Joint work with Ettore Turatti.

I will introduce the notion of Weddle loci, which are important in the study of configurations of points and in computer vision, and I will discuss some recent application of the loci to the description of spaces of tensors. I will mainly focus on a work in progress related the use of Weddle loci for determining the rank of linear systems of quadrics, and related partially symmetric tensors.

I call tame those tensors that are parametrized by representations appearing in cyclic gradings of simple Lie algebras; they are tame in the sense that their classification is similar to the usual Jordan classification of matrices up to conjugation. I will explain that these tensors are related with several famous chapters of classical algebraic geometry, involving elliptic curves, Kummer surfaces of Coble hypersurfaces. In fact they provide new points of view on these classical topics and allow us to add a few nice pages to the story.

A complex manifold is a contact manifold if there is a distribution in the tangent bundle which is as non-integrable as possible. I will report on recent progress in classification of projective contact manifolds focusing on the Fano case in (complex) dimensions 7 and 9. Our work implies the classification of quaternion-Kaehler manifolds of (real) dimensions 12 and 16, a famous problem from Riemannian geometry. The tools we use include representation theory and actions of (complex) reductive groups on manifolds, symplectic geometry, characteristic classes, and equivariant localisation theorems.
The talk is based on "Algebraic torus actions on contact manifolds," by Jarosław Buczyński and Jarosław A. Wiśniewski, with an appendix by Andrzej Weber arXiv:1802.05002, Journal of Differential Geometry (2022).

We propose an algebraic model of computation which formally relates symbolic listings, complexity of Boolean functions, and low depth arithmetic circuit complexity. In this model, algorithms are arithmetic formulae expressing symbolic listings of YES instances of Boolean functions, and computation is executed via partial differential operators. We consider the Chow rank of an arithmetic formula as a measure of complexity and establish the Chow rank of multilinear polynomials with totally non-overlapping monomial support. We also provide Chow rank non-decreasing transformations from sets of graphs to sets of functional graphs.
The talk is based upon joint work with Hamilton Sawczuk.

In the early 2000's Joseph M. Landsberg and Laurent Manivel wrote a series of papers using algebraic geometry to solve problems in representation theory. In one of them they investigated the geometry of the Freudenthal Magic square. Ten years later these constructions and the corresponding geometry were investigated in the quantum physics literature. It is now well understood that the orbit structure of the third row of the magic square describes entanglement patterns for three-partite quantum systems in different Hilbert spaces (bosonic qubit, qubits, fermions, fermionic fock spaces).
In this talk, I will discuss another quantum paradox, known as quantum contextuality, and explain what the second row of the Freudenthal Magic square tells us about quantum contextuality.

We investigate new lower bounds on the tensor ranks of the determinant and the permanent tensors via recursive usage of the Koszul flattening method introduced by Landsberg-Ottaviani and Hauenstein-Oeding-Ottaviani-Sommese. Our lower bounds on the ranks of determinant tensors completely separate the determinant and the permanent tensors by their tensor ranks. Furthermore, we determine the exact tensor rank of the determinant tensor of order 4 as 12 and the permanent tensor of order 4 as 8 over any field of characteristic not equal to 2.

An isotropic form is a linear form whose coefficients correspond to an isotropic point. For any natural numbers \(n\) and \(d\), the space of homogeneous harmonic polynomial of degree \(d\) in \(n\) variables is generated by the \(d\)-th powers of isotropic forms. This allows us to define the isotropic rank of a homogeneous polynomial \(h\) as the minimum natural number \(r\) such that \(h\) can be written as a linear combination of the \(d\)-th powers of \(r\) isotropic forms. Using secant varieties and Terracini's Lemma we determine the generic isotropic rank for any value of \(n\) and \(d\). This is joint work with Cristiano Bocci and Enrico Carlini.

Quantum contextuality is a phenomenon of quantum systems, less celebrated than other features such as entanglement and superposition. It shows that quantum systems can display seemingly contradictory behaviour when combined with assumptions from classical mechanics. This can be shown visually through quantum games, which provide a testing ground for experimentally ruling out such classical assumptions. In this talk we will cover the most basic and celebrated quantum game - the Mermin Game - and talk through how to play it on a quantum computer. We will also examine the geometry underpinning this game and a Cayley invariant of degree 3 connecting related geometries, their games, and quantum contextuality.

Loosely speaking, \(GL\)-varieties are affine schemes whose coordinate rings are polynomial representations. Their closed points can be obtained as the inverse limits of \(\textrm{Vec}\)-varieties, functors from the category of finite dimensional vector spaces to affine varieties.
In a joint work with Christopher Chiu and Jan Draisma, we proved that the inverse limit of the singular loci of a Vec-variety defines a closed \(\textrm{Vec}\)-subvariety of “singular points”. In a forthcoming paper with Andrew Snowden, we show that the \(GL\)-variety arising from these "singular points" is the locus of singular points, in the classical sense, of the initial \(GL\)-variety.
In this talk, I will present the necessary background and the aforementioned results.

We introduce a broad lemma, one consequence of which is the higher order singular value decomposition (HOSVD) of tensors defined by DeLathauwer, DeMoor and Vandewalle (2000). By an analogous application of the lemma, we find a complex orthogonal version of the HOSVD. Kraus' (2010) algorithm used the HOSVD to compute normal forms of almost all \(n\)-qubit pure states under the action of the local unitary group. Taking advantage of the double cover \(\operatorname{SL}_2(\mathbb{C})\times \operatorname{SL}_2(\mathbb{C})\to\operatorname{SO}_4(\mathbb{C})\), we produce similar algorithms (distinguished by the parity of \(n\)) that compute normal forms for almost all \(n\)-qubit pure states under the action of the SLOCC group.

Secant varieties are among the main protagonists in tensor decomposition, whose study involves both pure and applied mathematical areas. Any generalised Grassmannian \(G/P\) is a set of rank-1 "tensors" in the corresponding (minimal) irreducible representation, like Grassmannians for skew-symmetric tensors. Despite the geometry of secant varieties to \(G/P\) is in general not completely understood, several contributions appear in the literature, among which many works by J.M. Landsberg and L. Manivel on the geometry of the Freudenthal magic square. In this talk we discuss the secant variety of lines to a generalised Grassmannian, and we give results on the identifiability and singularity in the case of Grassmannians. Some results are from a joint work with Reynaldo Staffolani.

It is known that for \(r \leq 13\) and any natural numbers \(n, d\), the \(r\)-th cactus variety of \(v_d(\mathbb{P}^n)\) is equal to the \(r\)-th secant variety. In the first extremal case, we know that \(\kappa_{14}(v_d(\mathbb{P}^n))\) has two irreducible components for \(d\geq 5, n\geq 6\). We look at the Grassmann secant variety of lines \(\kappa_{r,2}(v_d(\mathbb{P}^n))\) and investigate what is the least \(r\) such that this variety is reducible for almost all \(n\). In order to do this, we analyze the locus of socle dimension at most two in the Hilbert scheme of \(r\) points on \(\mathbb{P}^n\) for some small \(r\).

Tensor rank, identifiability, and the number of orbits has been studied extensively in many different contexts. In the case of \(2\times \cdots \times 2\) tensors, the maximum rank over \(\mathbb{F}_2\) is known[SL2020]. We survey results on the distribution of ranks and identifiability for \(2\times 2\times 2\) and \(2\times 2\times 2 \times 2\) tensors over \(\mathbb{F}_2\). We investigate the distribution of tensor ranks, identifiability, and orbits for symmetric and partially symmetric tensors over \(\mathbb{F}_2\) and conclude with future directions.