16 problems

Let $k$ be the ground field, and let $\Lambda_{V_d}^{S^r}$ be the exterior-algebra-like construction associated with the $d$-dimensional vector space $V_d$. Write…

Let $\mathbb{V}$ be a finite-dimensional real vector space with a euclidean or lorentzian inner product, let $p\geq 2$, and let $F\in\Lambda^p\mathbb{V}^*$. For…

Let $\mathbb K$ be a field with characteristic $\chi(\mathbb K)\ne 2$, let $n\geq 4$, and let … be a $3$-form. Its nullity is the dimension of the largest subspace on which the ass…

Let $V_n$ be an $n$-dimensional vector space, let $I_n=(f_1,\dots,f_r)$ be a generic ideal in the exterior algebra on $V_n$, and suppose that $f_i$ has degree $d_i$. Let $q_n(t)$ b…

Let $V_n$ be an $n$-dimensional vector space, let $f$ be a generic cubic form in the exterior algebra, and let $p_{n,3}(t)$ denote the Hilbert series of the quotient by $(f)$. For…

Odd-degree annihilator correction conjecture. For odd $d>3$,

Let $K$ be a field, let $V_n$ be an $n$-dimensional vector space, and let $I$ be a generic homogeneous ideal in the exterior algebra on $n$ generators, with numerical character…

Let $V_d$ be a $d$-dimensional vector space over a field $k$, and let $\Lambda_{V_d}^{S^r}$ denote the higher exterior algebra-like construction. Write $\Lambda_{V_d}^{S^r}[rd]$ fo…

Extremal self-annihilating subspace conjecture. Equality

Top-component dimension conjecture. If $\dim(V_d)=d$, then

Let $V_d$ be a vector space over $k$ of dimension $d$. For a simple tensor in $V_d^{\otimes d(2d-1)}$, written as $\otimes_{1\leq i<j\leq 2d}(v_{i,j})$, say that a function has the…

Fix vectors $v_1,\dots,v_m\in\mathbb{R}^d$ with $1\le d\le m$. For integers $k$ satisfying $2\le k\le d$, let $|v_{i_1}\wedge\cdots\wedge v_{i_k}|$ denote the Euclidean norm of the…

Moreno-Socías–Snellman conjecture. The minimal Hilbert series is

Moreno-Socías–Snellman conjecture. The Hilbert function satisfies

C.N. Yang's conjecture. The maximum is achieved at

O'Farrell's decomposability conjecture. Every $p$-form $T$ satisfying the displayed relation on a Euclidean vector space must be decomposable.