New lower bounds for matrix multiplication and the 3x3 determinant

Let $M_{⟨u,v,w⟩}\in C^{uv}\otimes C^{vw}\otimes C^{wu}$ denote the matrix multiplication tensor (and write $M_n=M_{⟨n,n,n⟩}$) and let $de{t}_3\in (C^9{)}^{\otimes 3}$ denote the determinant polynomial considered as a tensor. For a tensor $T$, let $\underset{‾}{R}(T)$ denote its border rank. We (i) give the first hand-checkable algebraic proof that $\underline R(M_2)=7$,(ii)prove$\underline R(M_{\langle 223\rangle})=10$,and$\underline R(M_{\langle 233\rangle})=14$, where previously the only nontrivial matrix multiplication tensor whose border rank had been determined was $M_2$,(iii) prove $\underset{‾}{R}(M_3)\ge 17$, (iv) prove $\underset{‾}{R}(de{t}_3)=17$, improving the previous lower bound of $12$, (v) prove $\underline R(M_{\langle 2nn\rangle})\geq n^2+1.32n$forall$n\geq 25$(previouslyonly$\underline R(M_{\langle 2nn\rangle})\geq n^2+1$wasknown)aswellaslowerboundsfor$4\leq n\leq 25$,and(vi)prove$\underline R(M_{\langle 3nn\rangle})\geq n^2+2 n+1$forall$ n\geq 21$,wherepreviouslyonly$\underline R(M_{\langle 3nn\rangle})\geq n^2+2$wasknown,aswellaslowerboundsfor$4\leq n\leq 21$. Our results utilize a new technique initiated by Buczy'{n}ska and Buczy'{n}ski, called border apolarity. The two key ingredients are: (i) the use of a multi-graded ideal associated to a border rank $r$ decomposition of any tensor, and (ii) the exploitation of the large symmetry group of $T$ to restrict to $B_T$-invariant ideals, where $B_T$ is a maximal solvable subgroup of the symmetry group of $T$.