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**11 - 18**of**18**### ABELIAN TENSORS

"... Abstract. We analyze tensors in Cm⊗Cm⊗Cm satisfying Strassen’s equations for border rank m. Results include: two purely geometric characterizations of the Coppersmith-Winograd ten-sor, a reduction to the study of symmetric tensors under a mild genericity hypothesis, and numerous additional equations ..."

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Abstract. We analyze tensors in Cm⊗Cm⊗Cm satisfying Strassen’s equations for border rank m. Results include: two purely geometric characterizations of the Coppersmith-Winograd ten-sor, a reduction to the study of symmetric tensors under a mild genericity hypothesis, and numerous additional equations and examples. This study is closely connected to the study of the variety of m-dimensional abelian subspaces of End(Cm) and the subvariety consisting of the Zariski closure of the variety of maximal tori, called the variety of reductions. 1.

### Non-Commutative Formulas and Frege Lower Bounds: a New Characterization of Propositional Proofs

"... Does every Boolean tautology have a short propositional-calculus proof? Here, a propositional calculus (i.e. Frege) proof is a proof starting from a set of axioms and deriving new Boolean formulas using a set of fixed sound derivation rules. Establishing any super-polynomial size lower bound on Freg ..."

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Does every Boolean tautology have a short propositional-calculus proof? Here, a propositional calculus (i.e. Frege) proof is a proof starting from a set of axioms and deriving new Boolean formulas using a set of fixed sound derivation rules. Establishing any super-polynomial size lower bound on Frege proofs (in terms of the size of the formula proved) is a major open problem in proof complexity, and among a handful of fundamental hardness questions in complexity theory by and large. Non-commutative arithmetic formulas, on the other hand, constitute a quite weak computational model, for which exponential-size lower bounds were shown already back in 1991 by Nisan [Nis91] who used a particularly transparent argument. In this work we show that Frege lower bounds in fact follow from correspond-ing size lower bounds on non-commutative formulas computing certain polynomi-als (and that such lower bounds on non-commutative formulas must exist, unless NP=coNP). More precisely, we demonstrate a natural association between tau-tologies T to non-commutative polynomials p, such that:

### UNIFYING AND GENERALIZING KNOWN LOWER BOUNDS VIA GEOMETRIC COMPLEXITY THEORY

"... Abstract. We show that most arithmetic 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 Smolens ..."

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Abstract. We show that most arithmetic 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 de-gree bound (Strassen, Baur–Strassen), the connected components technique (Ben-Or), depth 3 arithmetic circuit lower bounds over finite fields (Grigoriev–Karpinski), lower bounds on perma-nent versus determinant (Mignon–Ressayre, Landsberg–Manivel–Ressayre), lower bounds on ma-trix multiplication (Bürgisser–Ikenmeyer) (these last two were already known to fit into GCT), the chasms at depth 3 and 4 (Gupta–Kayal–Kamath–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 and broad generalization of known results. It also shows that the framework of GCT is at least as powerful as known 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. 1.