Invited Speakers

• Weronika Buczyńska (U. Warsaw)

• Jarosław Buczyński (IMPAN)

• Harm Derksen (Northeastern U.)

• Jan Draisma (U. Bern)

• Katsuhisa Furukawa (Josai U.)

• Christian Ikenmeyer (U. Warwick)

• Petteri Kaski (Aalto U.)

• Hanieh Keneshlou (U. Würzburg)

• Yeongrak Kim (PNU)

• Khazhgali Kozhasov (U. Côte d'Azur)

• Joseph M. Landsberg (Texas A&M)

• Mateusz Michalek (U. Konstanz)

• Giorgio Ottaviani (U. Florence)

• Kristian Ranestad (U. Oslo)

• Tim Seynnaeve (IMPAN)

• Jeroen Zuiddam (U. Amsterdam)

Title and Abstract

Talk information will be updated.

• Weronika Buczyńska (U. Warsaw)
  Title :  Apolarity for border cactus decomposition
  
  Abstract : A couple of years ago with Jarek Buczyński we introduced a border apolarity method that helps to either compute or bound the border rank of a tensor or polynomial or solve similar problems. Our initial motivation was to compute the border rank of monomials, which remains only partially solved. However, the method we created allowed several researchers to compute or bound border ranks of some tensors or find a shorter proof for known results. Already on the first official presentation of this theory, I was asked whether we have considered other components of the Hilbert scheme. The result came a couple of years later – this part of the story is a bit more technical and algebraic in nature. The answer is more interesting: I will show examples of what properties can the map from a component of the multigraded Hilbert scheme to the usual one have depending on the chosen component. One example discussed in Tomasz Mandziuk PhD thesis is 4 collinear points on the projective plane – the multigraded Hilbert scheme has two components: one birational to Hilbert scheme of 4 points, the other collapses to a lower dimensional object. In this talk I will present the apolarity method adapted for cactus varieties.
  
  

• Jarosław Buczyński (IMPAN)
  Title :  Apolarity you have never seen before
  
  Abstract : Apolarity, multigraded apolarity, border apolarity, non-abelian apolarity, and Schur apolarity are notions and statements familiar to numerous experts in tensor decompositions, secant varieties, and finite algebras/schemes. Both their efficiency and barriers to applicability are well known and exploited. A long-standing quest is to find a uniform theory that encompasses various versions of apolarity. The talk will consist of four or five parts. First, I will briefly describe the notions of divisorial algebra, Cox ring, and Mori Dream space. Then, in the second part, called "apolarity you have probably already seen", I will outline the various known versions of apolarity, focusing on multigraded apolarity (following Gałązka, Teitler, Gellet-Ranestad-Villamizar), non-abelian apolarity (aka vector bundle method by Landsberg-Ottaviani), and Schur apolarity (based on Arrondo-Bernardi-Marques-Mourrain, Staffolani, Isoldi). In part three (called "apolarity you might have seen before"), I will bring your attention to a situation where the simplest version of apolarity action also appears in a seemingly unrelated algebro-geometric context. In part four "Apolarity you have never seen before..." [fine print: "...unless I have already told you about that"], I will present the above situation in a more general setting. Finally, in the optional fifth part (time permitting), I will explain how to use this new method to partially unify Schur and multigraded apolarity in an elegant theory that also admits the border analogue (including the Fixed Ideal Theorem). Parts 1, 2, and perhaps 3 can be considered general knowledge (with some tiny tweaks by the speaker); Part 4 is a joint work in progress with Tiago Duerte-Guerreiro, while Part 5 is an initial stage of a research project with Maciej Gałązka.
  
  

• Harm Derksen (Northeastern U.)
  Title :  Invariant Tensors and Wheeled PROPs
  
  Abstract : For a representation V of an algebraic group G we consider the invariant tensors in a tensor product of several copies of V and its dual. For G=GL(V) these are described by the Fundamental Theorems of Invariant Theory. If we consider all invariant tensors for a fixed G we get an algebraic structure known as a wheeled PROP. This gives us first and second fundamental theorems for any reductive group. We will characterize which wheeled PROPs appear as invariant tensors for some representation V of some reductive group G. In this framework invariant tensors are represented by diagrams. This is joint work with Visu Makam.
  
  

• Jan Draisma (U. Bern)
  Title :  Tensor subrank: symmetric, border, and more
  
  Abstract : The (border) subrank of a tensor is the largest unit tensor to which it can be restricted (respectively, degenerated). I will discuss recent work on this topic, both in the case of ordinary tensors and in the setting of symmetric tensors.
  
  

• Katsuhisa Furukawa (Josai U.)
  Title :  Kronecker products and iterated matrix multiplication
  
  Abstract : The $k$-secant variety $\sigma_k(X)$ of a projective variety $X \subset \mathbb{P}^N$ is the closure of the union of $(k-1)$-planes spanned by $k$ points on $X$. Since the singular locus of $\sigma_k(X)$ naturally contains $\sigma_{k-1}(X)$, we investigate when non-trivial singularities appear outside this subvariety.  In this talk, we focus on cases where $X$ is a basic variety, such as a Veronese or Segre variety. For the $d$-uple Veronese variety $v_d(\mathbb{P}^n) \subset \mathbb{P}^N$ with $N=\binom{n+d}{d}-1$, we characterize the (non-)singularity of $\sigma_k(v_d(\mathbb{P}^m))$ for any $m$-plane $\mathbb{P}^m \subset \mathbb{P}^n$ and arbitrary $k$ by investigating the geometry of moving tangents, secant defectivity, and the identifiability of symmetric tensors. Our results reveal an interesting trichotomy for these singularities. We also examine $\sigma_4(v_3(\mathbb{P}^3)) \subset \mathbb{P}^{19}$, which is a remarkable exception to this trichotomy; it lacks non-trivial singularities and is, in fact, a "del Pezzo 4-secant variety" in the sense of J. Choe and S. Kwak.
  
  

• Christian Ikenmeyer (U. Warwick)
  Title :  Singular loci of higher secant varieties and equations on the space of tensors
  
  Abstract : We observe that the Kronecker product of tensors is the operation that converts the determinant polynomial into Cayley's first hyperdeterminant. We apply the Kronecker product to iterated matrix multiplication, which results in the hypercomputant, a VNP-complete and VW[1]-complete polynomial whose hardness we prove via the equivariance of the Kronecker product. The construction works over arbitrary commutative semirings and also for the tensor algebra and the exterior algebra. For the tensor algebra this gives a version of "noncommutative VNP", and for polynomials over the nonnegative real numbers this gives a version of "monotone VNP", each with the hypercomputant as the complete object. We take a parameterized complexity viewpoint and compare the noncommutative setting and the monotone setting. Using standard techniques we obtain optimal algebraic branching program width lower bounds in both settings, and these are notably not always the same. We also prove the polystability of the hypercomputant and that its isotypic components are characterized by their stabilizer.
  
  Our approach has several implications:
  (1) A generalization of the Cayley–Hamilton theorem, new record ABP construction for the determinant (Mahajan-Vinay 1997), and an answer to question from Mahajan-Vinay (1999) about combinatorial algorithms for the determinant.
  (2) Gurvits' (2004, 2005) VNP-hardness results for the mixed discriminant and Cayley's first hyperdeterminant hold over every commutative ring.
  (3) Christandl-Fawzi-Ta-Zuiddam (2022) about the asymptotic spectrum of symmetric tensors can be translated to homogeneous polynomials over arbitrary characteristic.
  
  

• Hanieh Keneshlou (U. Würzburg)
  Title :  Varieties of tensors and stratification of tensor spaces
  
  Abstract : A linear space of matrices gives rise to a family of varieties,  unirational varieties parametrized by size k minors, or obtained by compactification in a projective variety. In this talk, we address the study of the some of these varieties starting from the initial case of pencil of matrices. This approach naturally leads to new stratification of tensor spaces. We present some of these stratifications, and recovery of interesting parameter spaces of tensors and their invariants in terms of our strata and the geometric invariants of these varieties.  This is based on joint works with Fulvio Gesmundo and Vincenzo Galgano.
  
  

• Yeongrak Kim (PNU)
  Title :  Determinant vs. Permanent
  
  Abstract : The determinant of a matrix is one of the most fundamental concepts in mathematics, and thus it appears and applies almost everywhere. We usually define this notion using Leibniz's formula, which expresses the determinant as a sum of signed products of matrix entries running over all the permutations. On the other hand, if we take a sum of unsigned products of those matrix entries, then the resulting invariant is called the permanent of a matrix. It may not be familar as the determinant but it is also very useful in several areas including combinatorics. One of the central problems in complexity theory is to compare their computational complexities, often called a "Determinant vs. Permanent" problem. This problem and its variants also have connections to the famous P vs. NP problem.  In this talk, I will begin with a quick survey on complexity notions motivated by a determinantal representation of a polynomial and by the rank of a matrix, and compare the determinant and the permanent with respect to these complexities. And then, by considering them as n-linear maps (tensors), we discuss the tensor rank behaviors for the corresponding determinant and permanent tensors. In particular, we will see that there is a huge difference between the tensor rank of the determinant and permanent tensors. If time permits, I will also talk about the exact ranks when n=4, and a few questions beyond. The talk is based on joint works with Jong In Han, Jeong-Hoon Ju, and Taehyeong Kim.
  
  

• Khazhgali Kozhasov (U. Côte d'Azur)
  Title :  Geometry of Euclidean distance minimization to tensor varieties and their duals
  
  Abstract : For a symmetric tensor F, critical points of the Bombieri–Weyl distance from F to the Veronese variety V of rank-1 symmetric tensors are well understood: they are given by the fixed points of the polar map x ↦ ∇f(x), where f is the homogeneous form associated with F. The number of points x in complex projective space satisfying this condition for a generic F was first computed by Fornæss and Sibony (1992). Later, the same count was obtained in the context of the spectral theory of symmetric tensors (Cartwright and Sturmfels, 2013), and it also equals the Euclidean distance degree (ED degree) of the Veronese variety (Draisma, Horobet, Ottaviani, Sturmfels, and Thomas, 2016). An analogous problem - counting the complex critical points of the Euclidean distance to the Segre variety S of general rank-1 tensors - was solved by Friedland and Ottaviani (2014).  While the problem of computing the ED degree of secant varieties of V or S remains largely open, we obtain a closed formula for the ED degree of the tangential variety of V. As I will explain, this count appears to be relevant to the problem of distance minimization to the discriminant (namely the variety of singular homogeneous forms of a given degree) that is projectively dual to V. The talk is based on joint work with B. Mourrain and A. Parusiński.
  
  

• Joseph M. Landsberg (Texas A&M)
  Title :  Matrix multiplication tensors and centroids
  
  Abstract : A fundamental problem in algebraic complexity theory is to classify tensors of minimal border rank. The problem is now understood to be intractable so one must be content with a qualitative understanding of these objects. For tensors satisfying a mild genericity condition the problem may be translated to a classical problem in algebraic geometry but for other tensors there had been no structure to exploit until recently. I will discuss the new structure (centroids), minimal border rank tensors including a class of such with remarkable centroids, and their connections to matrix multiplication and its complexity. This is joint work with Martin Kassabov, Victor Souza, and Philip Speegle.
  
  

• Mateusz Michalek (U. Konstanz)
  Title : Beyond Linear Flattenings of Tensors
  
  Abstract : Equations of secant varieties play an important role in several branches of pure and applied mathematics. Landsberg, Manivel and Ottaviani were among the pioneers in studying such equations using modern representation theory and algebraic geometry. Among the most useful equations are Koszul and Young flattenings, developed by Landsberg and Ottaviani, with ideas going back to Strassen. However, it was known that this approach in fact always produces equations for a larger cactus variety.
  
  I will report on recent joint work with Dolezalek in which we provide determinantal equations that distinguish secant varieties from cactus varieties. As an application, we obtain a simple, computer-free proof that the border rank of the 2×2 matrix multiplication tensor is seven.
  
  

• Giorgio Ottaviani (U. Florence)
  Title :  Kronecker product of polynomials and the exponent of matrix multiplication
  
  Abstract : We show a slight improvement on the upper bound of the exponent of matrix multiplication, by a border rank computation.
  
  

• Kristian Ranestad (U. Oslo)
  Title :  Betti stratifications of higher secant varieties of Veronese varieties
  
  Abstract : The Betti table of a the apolar ideal of a homogeneous form is a discrete invariant. Together with the variety of sum of powers of the form it conveyes the forms position in the higher secant variety of the corresponding Veronese variety. In examples I will show how they may define a stratification of these higher secant varieties. I willl survey joint work with G. Kapustka, M. Kapustka, H. Schenck, M. Stillman, B. Yuan and work of A. Brugård.
  
  

• Tim Seynnaeve (IMPAN)
  Title :  Chisel-based tensor decomposition
  
  Abstract : I will report on two ongoing projects concerning chiseling, a general algorithm for tensor sparsification recently introduced by Brooksbank, Kassabov and Wilson. Briefly: given an order d tensor T and a matrix C (the "chisel") with d columns, the chiseling algorithm attempts to compute a basis in which the tensor T is block sparse, where the found sparsity pattern depends on the chisel C.  The first project, joint with Nathaniel Collins, is about the combinatorics of these sparsity patterns. We call two chisels equivalent if they detect the same sparsity patterns. For tensors of order 3, it is easy to see that there are only finitely many equivalence classes, determined by the matroid associated to the chisel C. We show that for tensors of order 4 or higher the combinatorics become much more wild. In particular, there are now uncountably many equivalence classes of chisels.  In the second project, joint with Daniele Taufer and Nick Vannieuwenhoven, we leverage and extend the chiseling algorithm to compute additive tensor decompositions. It applies to a variety of elementary tensors and their corresponding decompositions, such as tensor rank, subspace, Chow, Grassmann, and Waring decompositions. The essential steps are as follows: First, the high-order input tensor is put into the appropriate third-order mold by reshaping. Second, the appropriate number of matrix slices are cut off from this remolded third-order tensor by random linear projections. Third, chiseling is applied to the matrix slices to extract the projected elementary tensor components. We show that for a fairly general class of varieties X and a generic tensor of sufficiently low X-rank, this algorithm indeed recovers the decomposition, and provide a numerical implementation.
  
  

• Jeroen Zuiddam (U. Amsterdam)
  Title :  On stabiliser dimension, additive cancellation and concise restriction
  
  Abstract : Tensor restriction is a fundamental preorder on tensors, relevant in many applications. It is still surprisingly poorly understood. We will discuss several recent results in this theme, in particular regarding stabiliser dimension (a fundamental tensor invariant and monotone), additive cancellation properties and the interplay between restriction and conciseness.

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