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Many fundamental questions in theoretical computer science are naturally expressed as special cases of the following problem: Let G be a complex reductive group, let V be a G-module, and let be elements of V. Determine if w is in the G-orbit closure of v. I explain the computer science problems, the questions in representation theory and algebraic geometry that they give rise to, and the new perspectives on old areas such as invariant theory that have arisen in light of these questions. I focus primarily on the complexity of matrix multiplication. 

We prove that the border rank of the Kronecker square of the little Coppersmith–Winograd tensor Tcw,q is the square of its border rank for q > 2 and that the border rank of its Kronecker cube is the cube of its border rank for q > 4. This answers questions raised implicitly by Coppersmith & Winograd (1990, §11) and explicitly by Bl¨aser (2013, Problem 9.8) and rules out the possibility of proving new upper bounds on the exponent of matrix multiplication using the square or cube of a little Coppersmith–Winograd tensor in this range. In the positive direction, we enlarge the list of explicit tensors potentially useful for Strassen’s laser method, introducing a skew-symmetric version of the Coppersmith– Winograd tensor, Tskewcw,q. For q = 2, the Kronecker square of this tensor coincides with the 3 × 3 determinant polynomial, det3 ∈ C9 ⊗ C9 ⊗ C9, regarded as a tensor. We show that this tensor could potentially be used to show that the exponent of matrix multiplication is two. We determine new upper bounds for the (Waring) rank and the (Waring) border rank of det3, exhibiting a strict submultiplicative behaviour for Tskewcw,2 which is promising for the laser method. We establish general results regarding border ranks of Kronecker powers of tensors, and make a detailed study of Kronecker squares of tensors in C3 ⊗ C3 ⊗ C3. 

We make a first geometric study of three varieties inCm⊗Cm⊗Cm (for eachm), including the Zariski closure of the set of tight tensors, the tensors with continuous regular symmetry. Our motivation is to develop a geometric framework for Strassen’s asymptotic rank conjecture that the asymptotic rank of any tight tensor is minimal. In particular, we determine the dimension of the set of tight tensors. We prove that this dimension equals the dimension of the set of oblique tensors, a less restrictive class introduced by Strassen. 

We answer a question, posed implicitly in [P. Bürgisser et al., 1997] and explicitly in [M. Bläser, 2013], showing the border rank of the Kronecker square of the little Coppersmith-Winograd tensor is the square of the border rank of the tensor for all q>2, a negative result for complexity theory. We further show that when q>4, the analogous result holds for the Kronecker cube. In the positive direction, we enlarge the list of explicit tensors potentially useful for the laser method. We observe that a well-known tensor, the 3 × 3 determinant polynomial regarded as a tensor, det_3 ∈ C^9 ⊗ C^9 ⊗ C^9, could potentially be used in the laser method to prove the exponent of matrix multiplication is two. Because of this, we prove new upper bounds on its Waring rank and rank (both 18), border rank and Waring border rank (both 17), which, in addition to being promising for the laser method, are of interest in their own right. We discuss "skew" cousins of the little Coppersmith-Winograd tensor and indicate why they may be useful for the laser method. We establish general results regarding border ranks of Kronecker powers of tensors, and make a detailed study of Kronecker squares of tensors in C^3 ⊗ C^3 ⊗ C^3. 

We answer a question of K. Mulmuley. Efremenko et al. (Math. Comp., 2018) have shown that the method of shifted partial derivatives cannot be used to separate the padded permanent from the determinant. Mulmuley asked if this “no-go” result could be extended to a model without padding. We prove this is indeed the case using the iterated matrix multiplication polynomial. We also provide several examples of polynomials with maximal space of partial derivatives, including the complete symmetric polynomials. We apply Koszul flattenings to these polynomials to have the first explicit sequence of polynomials with symmetric border rank lower bounds higher than the bounds attainable via partial derivatives.