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, J. Groups, Complexity, Cryptology, 2022 also with G. Tang, CCC '21, and upcoming preprints).

Many versions of the Axiom of Choice (AC), though equivalent in ZF set theory, are inequivalent from the computational point of view. When we consider polynomial-time analogues of AC, many of these different versions can be shown to be equivalent to other more standard questions about the relationship between complexity classes. We will use some of these formulations of AC to motivate several complexity questions that might otherwise seem a bit bespoke and unrelated from one another.

Next, as many versions of AC are about cardinals, in the second half of the talk we introduce a polynomial-time version of cardinality, in the spirit of polynomial-time model theory. As this is a new theory, we will discuss some of the foundational properties of polynomial-time cardinality, some of which may be surprising when contrasted with their set-theoretic counterparts. The talk will contain many open questions, and the paper contains even more! Based on arXiv:2301.07123 [cs.CC].

Part I: Motivation, history, and introduction to algebraic proof complexity (Toniann Pitassi)

This first talk will give a quick primer on proof complexity, especially as it relates to algebraic proof systems. It will cover some of the history and motivation that led to the study of algebraic proof systems, as well as introduce some well-studied algebraic proof systems and the relations between them.

Part II: Circuit complexity, ideals and proof systems: connections and recent results (Joshua A. Grochow)

The second talk will cover connections between the Ideal Proof System (IPS), circuit complexity, and more generally complexity in ideals. In particular, it will cover the relation between IPS and VP vs VNP, as well as the recent result of Andrews & Forbes that extends the constant-depth algebraic circuit lower bound of Limaye-Srinivasan-Tavenas to a lower bound on refuting certain polynomial equations in IPS. The talk will highlight the connection between algebraic circuit complexity, algebraic proof complexity, and even Boolean circuit complexity to the complexity not of individual (families of) polynomials, but of arbitrary (families of) polynomials inside an ideal or coset of an ideal.

We overview some relationships between codes (and various code-like objects) and constructing algorithms for multiplying matrices, ranging from concrete results to intriguing observations of things that look similar if you squint just right.

This includes a connection between codes and bilinear complexity from Lempel & Winograd '77, symmetric algorithms for matrix multiplication (studied by many authors), and a construction of Strassen's 1969 matrix multiplication algorithm from a unitary 2-design (G. & Moore, arXiv:1708.09398). And more connections we won't have time for, but promise at least one slide about.

We consider the problems of testing isomorphism of tensors, p-groups, cubic forms, algebras, and more, which arise from a variety of areas, including machine learning, group theory, and cryptography. Despite a perhaps seeming similarity with Graph Isomorphism, the current-best algorithms for these problems (when given by bases) are still exponential - for most of them, even qⁿ² over GF(q). Similarly, while efficient practical software exists for Graph Isomorphism, for these problems even the best current software can only handle very small instances (e.g., 10 x 10 x 10 over GF(13)). This raises the question of finding new algorithmic techniques for these problems, and/or of proving hardness results.

We show that all of these problems are equivalent under polynomial-time reductions, giving rise to a class of problems we call Tensor Isomorphism-complete (TI-complete). We further show that testing isomorphism of d-tensors for any fixed d (at least 3) is equivalent to testing isomorphism of 3-tensors. Using the same techniques, we show two first-of-their-kind results for Group Isomorphism (GpI): (a) a reduction from isomorphism of p-groups of exponent p and class c < p, to isomorphism of p-groups of exponent p and class 2, and (b) a search-to-decision reduction for the latter class of groups in time |G|^{O(log log|G|)}. We note that while p-groups of class 2 have long been believed to be the hardest cases of GpI, as far as we are aware this is the first reduction from any larger class to this class of groups. Finally, we discuss a way to apply combinatorial methods from Graph Isomorphism (namely, Weisfeiler-Leman) to Group and Tensor Isomorphism.

Based on joint works with Vyacheslav V. Futorny and Vladimir V. Sergeichuk (Lin. Alg. Appl., 2019; arXiv:1810.09219), with Peter A. Brooksbank, Yinan Li, Youming Qiao, and James B. Wilson (arXiv:1905.02518), and with Youming Qiao (arXiv:1907.00309).

The first lecture will be about what computational complexity can tell us about tensors. We will review notions of reductions between problems, how we can use these notions of reduction to learn about computational difficulty. We will then cover what is known about the complexity of a variety of problems on tensors, including a first look at some very recent results (only going up on the arXiv this week!).

This second lecture will be about what tensors can tell us about computational complexity. We will see several ways - both direct and indirect - in which problems on tensors lie at the heart of lower bounds in complexity theory.