15 problems

Let $W=(W_1,W_2,\ldots)$ be a sequence of representations, with $W_i$ an $\operatorname{SL}_{n_i}$-representation of dimension $N_i$. An arithmetic circuit succinctly encodes the g…

Let $G=SL_{n^2}(\mathbb{C})$ and let $H=SL_n(\mathbb{C})\times SL_n(\mathbb{C})$ have its natural embedding in $SL(\mathbb{C}^n\otimes\mathbb{C}^n)$. A subgroup is strongly $G$-sep…

Let $g=\operatorname{det}(Y)$, $h=\operatorname{perm}(X)$, and $f=\phi(h)$. A strong obstruction is a representation-theoretic obstruction of the type defined in the paper for the…

Let $v$ be the determinant function and let $\Delta_V[v]$ be the class variety associated with $NC$, namely the orbit closure of the determinant. Let $X(Pi_v)$ be the variety and sc…

Let $f$ be the form under consideration, let $f_0$ be its leading form, let $L(\lambda)$ be the Levi subgroup associated with the special one-parameter subgroup $\lambda$, and let…

Let $g(\lambda,n^d,n^d)$ be a rectangular Kronecker coefficient, and let $\gamma_{\lambda,d,n,m}$ be the multiplicity of the irreducible representation indexed by $\lambda$ in the…

Let $\mathrm{dc}(\mathrm{per}_m)$ be the determinantal complexity of the $m\times m$ permanent. For integers $d,n,m$, let $\gamma_{\lambda,d,n,m}$ and $\delta_{\lambda,d,n}$ denote…

Let $\ell$ be a linear coordinate on $\mathbb C^1$, and define the projective orbit closures … … Let $n(m)$ be a polynomial. Mulmuley–Sohoni's orbit-closure conjecture. For all suf…

The complexity classes ${\mathsf{VP}}_{ws}$ and ${\mathsf{VNP}}$ are considered through representation-theoretic multiplicities associated with them. Mulmuley–Sohoni multiplicity c…

For $n>m$, let $Z_{n,m}$ be the orbit closure of the padded permanent $X_{11}^{n-m}\mathrm{per}_m$ in $\mathsf{Sym}^n\mathbb{C}^{n^2}$, and let ${{\mathcal{D}}et}_n$ be the orbit c…

For $n>m$, let $X_{11}^{n-m}\mathrm{per}_m$ be the padded permanent, viewed as an element of $\mathsf{Sym}^n\mathbb{C}^{n^2}$, and let ${{\mathcal{D}}et}_n$ be the Euclidean (equiv…

Let $W$ be an explicit variety, and distinguish explicit close-to-defining equations from explicit defining equations as in the source. Explicit equations conjecture. (a) Any expli…

Let $W\subseteq K^m$ be an explicit variety. A strict separating e.s.o.p. is a strict separating homogeneous system of parameters for the coordinate ring $K[W]$; for a weakly expli…

Let $K$ be an algebraically closed field of characteristic zero, let $G$ be a connected reductive algebraic group of the form $cprod G_i$, with each factor a torus or a classical s…

Mulmuley–Sohoni's strengthened conjecture. For each $c>0$ and infinitely many $n$, $\mathcal{P}erm_n^{n^c}\not\subseteq\mathcal{D}et_{n^c}$. This is a central geometric-complexity-…