Friedl and Löh (2021, Confl. Math.) prove that testing whether or not there is an epimorphism from a finitely presented group to a virtually cyclic group, or to the direct product of an abelian and a finite group, is decidable. Here we prove that these problems are $\mathsf{NP}$-complete. We also show that testing epimorphism is $\mathsf{NP}$-complete when the target is a restricted type of semi-direct product of a finitely generated free abelian group and a finite group, thus extending the class of virtually abelian target groups for which decidability of epimorphism is known. Lastly, we consider epimorphism onto a fixed finite group. We show the problem is $\mathsf{NP}$-complete when the target is a dihedral groups of order that is not a power of 2, complementing the work on Kuperberg and Samperton (2018, Geom. Topol.) who showed the same result when the target is non-abelian finite simple.

The Ideal Proof System (IPS) of Grochow & Pitassi (FOCS 2014, J. ACM, 2018) is an algebraic proof system that uses algebraic circuits to refute the solvability of unsatisfiable systems of polynomial equations. One potential drawback of IPS is that verifying an IPS proof is only known to be doable using Polynomial Identity Testing (PIT), which is solvable by a randomized algorithm, but whose derandomization, even into NSUBEXP, is equivalent to strong lower bounds. However, the circuits that are used in IPS proofs are not arbitrary, and it is conceivable that one could get around general PIT by leveraging some structure in these circuits. This proposal may be even more tempting when IPS is used as a proof system for Boolean Unsatisfiability, where the equations themselves have additional structure.

The Cohn-Umans (FOCS '03) group-theoretic framework for matrix multiplication produces fast matrix multiplication algorithms from three subsets of a finite group G satisfying a simple combinatorial condition (the Triple Product Property). The complexity of such an algorithm then depends on the representation theory of G. In this paper we extend the group-theoretic framework to the setting of infinite groups. In particular, this allows us to obtain constructions in Lie groups, with favorable parameters, that are provably impossible in finite groups of Lie type (Blasiak, Cohn, Grochow, Pratt, and Umans, ITCS '23). Previously the Lie group setting was investigated purely as an analogue of the finite group case; a key contribution in this paper is a fully developed framework for obtaining bona fide matrix multiplication algorithms directly from Lie group constructions. As part of this framework, we introduce "separating functions" as a necessary new design component, and show that when the underlying group is G = GL_n, these functions are polynomials with their degree being the key parameter. In particular, we show that a construction with "half-dimensional" subgroups and optimal degree would imply ω = 2. We then build up machinery that reduces the problem of constructing optimal-degree separating polynomials to the problem of constructing a single polynomial (and a corresponding set of group elements) in a ring of invariant polynomials determined by two out of the three subgroups that satisfy the Triple Product Property. This machinery combines border rank with the Lie algebras associated with the Lie subgroups in a critical way. We give several constructions illustrating the main components of the new framework, culminating in a construction in a special unitary group that achieves separating polynomials of optimal degree, meeting one of the key challenges. The subgroups in this construction have dimension approaching half the ambient dimension, but (just barely) too slowly. We argue that features of the classical Lie groups make it unlikely that constructions in these particular groups could produce nontrivial bounds on ω unless they prove ω = 2. One way to get ω = 2 via our new framework would be to lift our existing construction from the special unitary group to GL_n, and improve the dimension of the subgroups from (dim G)/2 - Θ(n) to (dim G)/2 - o(n).

In this paper, we explore the descriptive complexity theory of finite groups by examining the power of the second Ehrenfeucht--Fraïssé bijective pebble game in Hella's (Ann. Pure Appl. Log., 1989) hierarchy. This is a Spoiler--Duplicator game in which Spoiler can place up to two pebbles each round. While it trivially solves graph isomorphism, it may be nontrivial for finite groups, and other ternary relational structures. We first provide a novel generalization of Weisfeiler--Leman (WL) coloring, which we call 2-ary WL. We then show that 2-ary WL is equivalent to the second Ehrenfeucht--Fraïssé bijective pebble game in Hella's hierarchy. Our main result is that, in the pebble game characterization, only $O(1)$ pebbles and $O(1)$ rounds are sufficient to identify all groups without Abelian normal subgroups (a class of groups for which isomorphism testing is known to be in $\mathsf{P}$; Babai, Codenotti, & Qiao, ICALP 2012). We actually show that $7$ pebbles and $7$ rounds suffice. In particular, we show that within the first few rounds, Spoiler can force Duplicator to select an isomorphism between two such groups at each subsequent round. By Hella's results (ibid.), this is equivalent to saying that these groups are identified by formulas in first-order logic with generalized 2-ary quantifiers, using only $7$ variables and $7$ quantifier depth.

We study the complexity of isomorphism problems for d-way arrays, or tensors, under natural actions by classical groups such as orthogonal, unitary, and symplectic groups. Such problems arise naturally in statistical data analysis and quantum information. We study two types of complexity-theoretic questions. First, for a fixed action type (isomorphism, conjugacy, etc.), we relate the complexity of the isomorphism problem over a classical group to that over the general linear group. Second, for a fixed group type (orthogonal, unitary, or symplectic), we compare the complexity of the decision problems for different actions. Our main results are as follows. First, for orthogonal and symplectic groups acting on 3-way arrays, the isomorphism problems reduce to the corresponding problem over the general linear group. Second, for orthogonal and unitary groups, the isomorphism problems of five natural actions on 3-way arrays are polynomial-time equivalent, and the d-tensor isomorphism problem reduces to the 3-tensor isomorphism problem for any fixed d>3. For unitary groups, the preceding result implies that LOCC classification of tripartite quantum states is at least as difficult as LOCC classification of d-partite quantum states for any d. Lastly, we also show that the graph isomorphism problem reduces to the tensor isomorphism problem over orthogonal and unitary groups.

We investigate the constant-depth circuit complexity of the Isomorphism Problem, Minimum Generating Set Problem (MGS), and Sub(quasi)group Membership Problem (Membership) for groups and quasigroups (=Latin squares), given as input in terms of their multiplication (Cayley) tables. Despite decades of research on these problems, lower bounds for these problems even against depth-2 <?TeX $\mathsf {AC}$?> Math 1 circuits remain unknown. Perhaps surprisingly, Chattopadhyay, Torán, and Wagner (FSTTCS 2010; ACM Trans. Comput. Theory, 2013) showed that Quasigroup Isomorphism could be solved by <?TeX $\mathsf {AC}$?> Math 2 circuits of depth O(log log n) using O(log 2n) nondeterministic bits, a class we denote <?TeX $\exists ^{\log ^{2}n}\mathsf {FOLL}$?> Math 3 . We narrow this gap by improving the upper bound for these problems to <?TeX $\mathsf {quasiAC^0}$?> Math 4 , thus decreasing the depth to constant.

In this paper we combine many of the standard and more recent algebraic techniques for testing isomorphism of finite groups (GpI) with combinatorial techniques that have typically been applied to Graph Isomorphism. In particular, we show how to combine several state-of-the-art GpI algorithms for specific group classes into an algorithm for general GpI, namely: composition series isomorphism (Rosenbaum-Wagner, Theoret. Comp. Sci., 2015; Luks, 2015), recursively-refineable filters (Wilson, J. Group Theory, 2013), and low-genus GpI (Brooksbank-Maglione-Wilson, J. Algebra, 2017). Recursively-refineable filters -- a generalization of subgroup series -- form the skeleton of this framework, and we refine our filter by building a hypergraph encoding low-genus quotients, to which we then apply a hypergraph variant of the k-dimensional Weisfeiler-Leman technique. Our technique is flexible enough to readily incorporate additional hypergraph invariants or additional characteristic subgroups.

Abstract There is no single canonical polynomial-time version of the Axiom of Choice (AC); several statements of AC that are equivalent in Zermelo-Fraenkel (ZF) set theory are already inequivalent from a constructive point of view, and are similarly inequivalent from a complexity-theoretic point of view. In this paper we show that many classical formulations of AC, when restricted to polynomial time in natural ways, are equivalent to standard complexity-theoretic hypotheses, including several that were of interest to Selman. This provides a unified view of these hypotheses, and we hope provides additional motivation for studying some of the lesser-known hypotheses that appear here. Additionally, because several classical forms of AC are formulated in terms of cardinals, we develop a theory of polynomial-time cardinality. Nerode & Remmel (Contemp. Math.106, 1990 and Springer Lec. Notes Math. 1432, 1990) developed a related theory, but restricted to unary sets. Downey (Math. Reviews MR1071525) suggested that such a theory over larger alphabets could have interesting connections to more standard complexity questions, and we illustrate some of those connections here. The connections between AC, cardinality, and complexity questions also allow us to highlight some of Selman’s work. We hope this paper is more of a beginning than an end, introducing new concepts and raising many new questions, ripe for further research.

Abstract. We study the complexity of isomorphism problems for tensors, groups, and polynomials. These problems have been studied in multivariate cryptography, machine learning, quantum information, and computational group theory. We show that these problems are all polynomial-time equivalent, creating bridges between problems traditionally studied in myriad research areas. This prompts us to define the complexity class [Formula: see text], namely problems that reduce to the tensor isomorphism problem in polynomial time. Our main technical result is a polynomial-time reduction from [Formula: see text]-tensor isomorphism to 3-tensor isomorphism. In the context of quantum information, this result gives a multipartite-to-tripartite entanglement transformation procedure that preserves equivalence under stochastic local operations and classical communication.

We report our experiences implementing standards-based grading at scale in an Algorithms course, which serves as the terminal required CS Theory course in our department's undergraduate curriculum. The course had 200-400 students, taught by two instructors, eight graduate teaching assistants, and supported by two additional graders and several undergraduate course assistants. We highlight the role of standards-based grading (SBG) in supporting our students during the COVID-19 pandemic. We conclude by detailing the successes and adjustments we would make to the course structure.

In this paper 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. These problems can all be cast as orbit problems on multi-way arrays under different group actions. Our first two main results are: 1. All the aforementioned isomorphism problems are equivalent under polynomial-time reductions, in conjunction with the recent results of Futorny-Grochow-Sergeichuk (Lin. Alg. Appl., 2019). 2. Isomorphism of $d$-tensors reduces to isomorphism of 3-tensors, for any $d \geq 3$. Our results suggest that these isomorphism problems form a rich and robust equivalence class, which we call Tensor Isomorphism-complete, or TI-complete. We then leverage the techniques used in the above results to prove two first-of-their-kind results for Group Isomorphism (GpI): 3. We give a reduction from GpI for $p$-groups of exponent $p$ and small class ($c < p$) to GpI for $p$-groups of exponent $p$ and class 2. The latter are widely believed to be the hardest cases of GpI, but as far as we know, this is the first reduction from any more general class of groups to this class. 4. We give a search-to-decision reduction for isomorphism of $p$-groups of exponent $p$ and class 2 in time $|G|^{O(\log \log |G|)}$. While search-to-decision reductions for Graph Isomorphism (GI) have been known for more than 40 years, as far as we know this is the first non-trivial search-to-decision reduction in the context of GpI. Our main technique for (1), (3), and (4) is a linear-algebraic analogue of the classical graph coloring gadget, which was used to obtain the search-to-decision reduction for GI. This gadget construction may be of independent interest and utility. The technique for (2) gives a method for encoding an arbitrary tensor into an algebra.

We study the complexity of isomorphism problems for tensors, groups, and polynomials. These problems have been studied in multivariate cryptography, machine learning, quantum information, and computational group theory. We show that these problems are all polynomial-time equivalent, creating bridges between problems traditionally studied in myriad research areas. This prompts us to define the complexity class TI, namely problems that reduce to the Tensor Isomorphism (TI) problem in polynomial time. Our main technical result is a polynomial-time reduction from d-tensor isomorphism to 3-tensor isomorphism. In the context of quantum information, this result gives multipartite-to-tripartite entanglement transformation procedure, that preserves equivalence under stochastic local operations and classical communication (SLOCC).

We study the problems of testing isomorphism of polynomials, algebras, and multilinear forms. Our first main results are average-case algorithms for these problems. For example, we develop an algorithm that takes two cubic forms $f, g\in \mathbb{F}_q[x_1,\dots, x_n]$, and decides whether $f$ and $g$ are isomorphic in time $q^{O(n)}$ for most $f$. This average-case setting has direct practical implications, having been studied in multivariate cryptography since the 1990s. Our second result concerns the complexity of testing equivalence of alternating trilinear forms. This problem is of interest in both mathematics and cryptography. We show that this problem is polynomial-time equivalent to testing equivalence of symmetric trilinear forms, by showing that they are both Tensor Isomorphism-complete (Grochow-Qiao, ITCS, 2021), therefore is equivalent to testing isomorphism of cubic forms over most fields.

Covering spaces of graphs have long been useful for studying expanders (as "graph lifts") and unique games (as the "label-extended graph"). In this paper we advocate for the thesis that there is a much deeper relationship between computational topology and the Unique Games Conjecture. Our starting point is Linial's 2005 observation that the only known problems whose inapproximability is equivalent to the Unique Games Conjecture - Unique Games and Max-2Lin - are instances of Maximum Section of a Covering Space on graphs. We then observe that the reduction between these two problems (Khot-Kindler-Mossel-O'Donnell, FOCS '04; SICOMP '07) gives a well-defined map of covering spaces. We further prove that inapproximability for Maximum Section of a Covering Space on (cell decompositions of) closed 2-manifolds is also equivalent to the Unique Games Conjecture. This gives the first new "Unique Games-complete" problem in over a decade. Our results partially settle an open question of Chen and Freedman (SODA, 2010; Disc. Comput. Geom., 2011) from computational topology, by showing that their question is almost equivalent to the Unique Games Conjecture. (The main difference is that they ask for inapproximability over Z_2, and we show Unique Games-completeness over Z_k for large k.) This equivalence comes from the fact that when the structure group G of the covering space is Abelian - or more generally for principal G-bundles - Maximum Section of a G-Covering Space is the same as the well-studied problem of 1-Homology Localization. Although our most technically demanding result is an application of Unique Games to computational topology, we hope that our observations on the topological nature of the Unique Games Conjecture will lead to applications of algebraic topology to the Unique Games Conjecture in the future.

We study the computational power of deciding whether a given truth table can be described by a circuit of a given size (the minimum circuit size problem, or MCSP for short) and of the variant denoted as MKTP, where circuit size is replaced by a polynomially related Kolmogorov measure. Prior to our work, all reductions from supposedly intractable problems to MCSP/MKTP hinged on the power of MCSP/MKTP to distinguish random distributions from distributions produced by hardness-based pseudorandom generator constructions. We develop a fundamentally different approach inspired by the well-known interactive proof system for the complement of graph isomorphism (GI). It yields a randomized reduction with zero-sided error from GI to MKTP. We generalize the result and show that GI can be replaced by any isomorphism problem for which the underlying group satisfies some elementary properties. Instantiations include linear code equivalence, permutation group conjugacy, and matrix subspace conjugacy. Along the way we develop encodings of isomorphism classes that are efficiently decodable and achieve compression that is at or near the information-theoretic optimum; those encodings may be of independent interest.

The isomorphism problem for finite groups of order $n$ (GpI) has long been known to be solvable in $n^{\log n+O(1)}$ time, but only recently were polynomial-time algorithms designed for several interesting group classes. Inspired by recent progress, we revisit the strategy for GpI via the extension theory of groups. The extension theory describes how a normal subgroup $N$ is related to $G/N$ via $G$, and this naturally leads to a divide-and-conquer strategy that “splits” GpI into two subproblems: one regarding group actions on other groups, and one regarding group cohomology. When the normal subgroup $N$ is abelian, this strategy is well known. Our first contribution is to extend this strategy to handle the case when $N$ is not necessarily abelian. This allows us to provide a unified explanation of all recent polynomial-time algorithms for special group classes. Guided by this strategy, to make further progress on GpI, we consider central-radical groups, proposed in Babai et al. [Code equivalence and group isomorphism, in Proceedings of the 22nd Annual ACM--SIAM Symposium on Discrete Algorithms (SODA'11), SIAM, Philadelphia, 2011, ACM, New York, pp. 1395--1408]: the class of groups such that $G$ modulo its center has no abelian normal subgroups. This class is a natural extension of the group class considered by Babai et al. [Polynomial-time isomorphism test for groups with no abelian normal subgroups (extended abstract), in International Colloquium on Automata, Languages, and Programming (ICALP), 2012, pp. 51--62], namely those groups with no abelian normal subgroups. Following the above strategy, we solve GpI in $n^{O(\log \log n)}$ time for central-radical groups, and in polynomial time for several prominent subclasses of central-radical groups. We also solve GpI in $n^{O(\log\log n)}$ time for groups whose solvable normal subgroups are elementary abelian but not necessarily central. As far as we are aware, this is the first time there have been worst-case guarantees on an $n^{o(\log n)}$-time algorithm that tackles both aspects of GpI---actions and cohomology---simultaneously. Prior to this work, the best proven upper bounds on algorithms for groups with central radicals were $n^{O(\log n)}$, even for groups with a central radical of constant size, such as ${Rad}(G) = Z(G)=\mathbb{Z}_2$. To develop our new algorithms we utilize several mathematical results on the detailed structure of cohomology classes, as well as algorithmic results for code equivalence, coset intersection, and cyclicity testing of modules over finite-dimensional associative algebras. We also suggest several promising directions for future work.

One fundamental question in the context of the geometric complexity theory approach to the VP vs. VNP conjecture is whether VP = !VP, where VP is the class of families of polynomials that can be computed by arithmetic circuits of polynomial degree and size, and VP is the class of families of polynomials that can be approximated infinitesimally closely by arithmetic circuits of polynomial degree and size. The goal of this article is to study the conjecture in (Mulmuley, FOCS 2012) that !VP is not contained in VP. Towards that end, we introduce three degenerations of VP (i.e., sets of points in VP), namely the stable degeneration Stable-VP, the Newton degeneration Newton-VP, and the p-definable one-parameter degeneration VP*. We also introduce analogous degenerations of VNP. We show that Stable-VP subseteq Newton-VP subseteq VP* subseteq VNP, and Stable-VNP = Newton-VNP = VNP* = VNP. The three notions of degenerations and the proof of this result shed light on the problem of separating VP from VP. Although we do not yet construct explicit candidates for the polynomial families in !VP\VP, we prove results which tell us where not to look for such families. Specifically, we demonstrate that the families in Newton-VP \VP based on semi-invariants of quivers would have to be nongeneric by showing that, for many finite quivers (including some wild ones), Newton degeneration of any generic semi-invariant can be computed by a circuit of polynomial size. We also show that the Newton degenerations of perfect matching Pfaffians, monotone arithmetic circuits over the reals, and Schur polynomials have polynomial-size circuits.

We show that most algebraic circuit lower bounds and relations between lower bounds naturally fit into the representation-theoretic framework suggested by geometric complexity theory (GCT), including: the partial derivatives technique (Nisan–Wigderson), the results of Razborov and Smolensky on AC 0[p], multilinear formula and circuit size lower bounds (Raz et al.), the degree bound (Strassen, Baur–Strassen), the connected components technique (Ben-Or, Steele–Yao), depth 3 algebraic circuit lower bounds over finite fields (Grigoriev–Karpinski), lower bounds on permanent versus determinant (Mignon–Ressayre, Landsberg–Manivel–Ressayre), lower bounds on matrix multiplication (Bürgisser–Ikenmeyer, Landsberg–Ottaviani) (these last two were already known to fit into GCT), the chasms at depth 3 and 4 (Gupta–Kayal–Kamath–Saptharishi, Agrawal–Vinay, Koiran, Tavenas), matrix rigidity (Valiant) and others. That is, the original proofs, with what is often just a little extra work, already provide representation-theoretic obstructions in the sense of GCT for their respective lower bounds. This enables us to expose a new viewpoint on GCT, whereby it is a natural unification of known results and broad generalization of known techniques. It also shows that the framework of GCT is at least as powerful as previous methods, and gives many new proofs-of-concept that GCT can indeed provide significant asymptotic lower bounds. This new viewpoint also opens up the possibility of fruitful two-way interactions between previous results and the new methods of GCT; we provide several concrete suggestions of such interactions. For example, the representation-theoretic viewpoint of GCT naturally provides new properties to consider in the search for new lower bounds.

We show that most algebraic circuit lower bounds and relations between lower bounds naturally fit into the representation-theoretic framework suggested by geometric complexity theory (GCT), including: the partial derivatives technique (Nisan-Wigderson), the results of Razborov and Smolensky on AC0[p], multilinear formula and circuit size lower bounds (Raz et al.), the degree bound (Strassen, Baur-Strassen), the connected components technique (Ben-Or, Steele-Yao), depth 3 algebraic circuit lower bounds over finite fields (Grigoriev-Karpinski), lower bounds on permanent versus determinant (Mignon-Ressayre, Landsberg-Manivel-Ressayre), lower bounds on matrix multiplication (Burgisser-Ikenmeyer, Landsberg-Ottaviani) (these last two were already known to fit into GCT), the chasms at depth 3 and 4 (Gupta-Kamath-Kayal-Saptharishi, Agrawal-Vinay, Koiran), matrix rigidity (Valiant) and others. That is, the original proofs, with what is often just a little extra work, already provide representation-theoretic obstructions in the sense of GCT for their respective lower bounds. This enables us to expose a new viewpoint on GCT, whereby it is a natural unification of known results and broad generalization of known techniques. It also shows that the framework of GCT is at least as powerful as previous methods, and gives many new proofs-of-concept that GCT can indeed provide significant asymptotic lower bounds. This new viewpoint also opens up the possibility of fruitful two-way interactions between previous results and the new methods of GCT, we provide several concrete suggestions of such interactions. For example, the representation-theoretic viewpoint of GCT naturally provides new properties to consider in the search for new lower bounds.

The isomorphism problem for groups given by their multiplication tables (GPI) has long been known to be solvable in nO(log n)time, but only recently has there been significant progress towards polynomial time. For example, Babai et al. (ICALP 2012) gave a polynomial-time algorithm for groups with no abelian normal subgroups. Thus, at present it is crucial to understand groups with abelian normal subgroups to develop no(log n)-time algorithms. Towards this goal we advocate a strategy via the extension theory of groups, which describes how a normal subgroup N is related to G/N via G. This strategy "splits" GPI into two sub problems: one regarding group actions on other groups, and one regarding group cohomology. The solution of these problems is essentially necessary and sufficient to solve GPI. Most previous works naturally align with this strategy, and it thus helps explain in a unified way the recent polynomial-time algorithms for other group classes. In particular, most prior results in the multiplication table model focus on the group action aspect, despite the general necessity of cohomology, for example for p-groups of class 2-believed to be the hardest case of GPI. To make progress on the group cohomology aspect of GPI, we consider central-radical groups, proposed in Babai et al. (SODA 2011): the class of groups such that G mod its center has no abelian normal subgroups. Recall that Babai et al. (ICALP 2012) consider the class of groups G such that G itself has no abelian normal subgroups. Following the above strategy, we solve GPI in nO(log log n)time for central-radical groups, and in polynomial time for several prominent subclasses of central-radical groups. We also solve GPI in nO(log log n)-time for groups whose solvable normal subgroups are elementary abelian but not necessarily central. As far as we are aware, this is the first time that a nontrivial algorithm with worst-case guarantees has tackled both aspects of GPI-actions and cohomology-simultaneously. Prior to this work, only nO(log n)-time algorithms were known, even for groups with a central radical of constant size, such as Z(G) = Z_2. To develop these algorithms we utilize several mathematical results on the detailed structure of cohomology classes, as well as algorithmic results for code equivalence, coset intersection and cyclicity testing of modules over finite-dimensional associative algebras. We also suggest several promising directions for future work.

We introduce a new and natural algebraic proof system, whose complexity measure is essentially the algebraic circuit size of Nullstellensatz certificates. This enables us to exhibit close connections between effective Nullstellensatzë, proof complexity, and (algebraic) circuit complexity. In particular, we show that any super-polynomial lower bound on any Boolean tautology in our proof system implies that the permanent does not have polynomial-size algebraic circuits (VNP ≠ VP). We also show that super-polynomial lower bounds on the number of lines in Polynomial Calculus proofs imply the Permanent versus Determinant Conjecture. Note that there was no proof system prior to ours for which lower bounds on an arbitrary tautology implied any complexity class lower bound. Our proof system helps clarify the relationships between previous algebraic proof systems. In doing so, we highlight the importance of polynomial identity testing (PIT) in proof complexity. In particular, we use PIT to illuminate AC 0 [ p ]-Frege lower bounds, which have been open for nearly 30 years, with no satisfactory explanation as to their apparent difficulty. Finally, we explain the obstacles that must be overcome in any attempt to extend techniques from algebraic circuit complexity to prove lower bounds in proof complexity. Using the algebraic structure of our proof system, we propose a novel route to such lower bounds. Although such lower bounds remain elusive, this proposal should be contrasted with the difficulty of extending AC 0 [ p ] circuit lower bounds to AC 0 [ p ]-Frege lower bounds.

We study the problem of matrix isomorphism of matrix Lie algebras (MatIsoLie). Lie algebras arise centrally in areas as diverse as differential equations, particle physics, group theory, and the Mulmuley -- Sohoni Geometric Complexity Theory program. A matrix Lie algebra is a set L of matrices that is closed under linear combinations and the operation [A, B] = AB - BA. Two matrix Lie algebras L, L' are matrix isomorphic if there is an invertible matrix M such that conjugating every matrix in L by M yields the set L'. We show that certain cases of MatIsoLie -- for the wide and widely studied classes of semi simple and abelian Lie algebras -- are equivalent to graph isomorphism and linear code equivalence, respectively. On the other hand, we give polynomial-time algorithms for other cases of MatIsoLie, which allow us to mostly derandomize a recent result of Kayal on affine equivalence of polynomials.

The isomorphism problem for groups given by their multiplication tables has long been known to be solvable in time nlog n+O(1). The decades-old quest for a polynomial-time algorithm has focused on the very difficult case of class-2 nilpotent groups (groups whose quotient by their center is abelian), with little success. In this paper we consider the opposite end of the spectrum and initiate a more hopeful program to find a polynomial-time algorithm for semisimple groups, defined as groups without abelian normal subgroups. First we prove that the isomorphism problem for this class can be solved in time nO(log log n). We then identify certain bottlenecks to polynomial-time solvability and give a polynomial-time solution to a rich subclass, namely the semisimple groups where each minimal normal subgroup has a bounded number of simple factors. We relate the results to the filtration of groups introduced by Babai and Beals (1999). One of our tools is an algorithm for equivalence of (not necessarily linear) codes in simply-exponential time in the length of the code, obtained by modifying Luks's algorithm for hypergraph isomorphism in simply-exponential time in the number of vertices (FOCS 1999). We comment on the complexity of the closely related problem of permutational isomorphism of permutation groups.