27 problems

Let $\mathcal C$ be a neural code labeled so that neuron $i$ is added as a piercing at the $i$th step, and let $T_\mathcal C$ be its toric ideal. Let $\prec$ be the monomial order…

Let $\mathbf{C}$ be a binary matrix with $k$ redundant neurons, of which $\ell \leq k$ are not simply redundant. Write $\rho(\mathbf{C})$ for the matrix obtained by removing the re…

Let $\mathcal C$ be a neural code, and let $\mathbf{P}_{\mathbf{Code}}$ denote the poset of combinatorial neural codes. A code lies below another code when it is below it in this p…

A neural code is a subset of the power set $2^{[n]}$, where $[n]=\{1,\dots,n\}$. A code is convex if it has a realization by open convex subsets of some Euclidean space. Let…

Wheel max-intersection conjecture. More generally, it is conjectured that: (1) if $(\sigma_1,\sigma_2,\sigma_3,\tau)$ is a wheel of $\mathcal{C}$, then $\tau\notin\mathcal{C}$; and…

Jeffs's conjecture. If $\mathcal{C}$ has up to four maximal codewords, then $\mathcal{C}$ is convex if and only if $\mathcal{C}$ has no local obstructions and no wheels.

Efficient degree computation conjecture. There is a polynomial-time algorithm that, given the codewords of $\mathcal C$ as binary vectors of length $n$, determines whether every ps…

Let $[n]=\{1,\ldots,n\}$, and let $\operatorname{cdim}(\mathcal C)$ denote the closed embedding dimension of a code $\mathcal C\subseteq 2^{[n]}$. Closed-dimension existence conjec…

Let $[n]=\{1,\ldots,n\}$, and for a code $\mathcal C\subseteq 2^{[n]}$ write $\operatorname{odim}(\mathcal C)$, $\operatorname{cdim}(\mathcal C)$, and…

Let $\mathcal{C}$ be a neural code, let $\Delta(\mathcal{C})$ be its simplicial complex, and let $\mathcal{W}=(\sigma_1,\sigma_2,\sigma_3,\tau)$ be a wheel of $\mathcal{C}$. A face…

Let $\mathcal{C}$ be a circulant code on $n$ neurons with support $p$. Let $\operatorname{NRH}\{\mathcal{R}_{\mathcal{C}}\}$ denote the set of neural ring homomorphisms associated…

Let $\mathcal{C}$ be a circulant code on $n$ neurons with support $2<p<n-1$. Write $n=pd+r$, where $d$ and $r$ are integers. Let $\operatorname{NRH}\{\mathcal{R}_{\mathcal{C}}\}$ d…

Let $\mathcal{C}$ be a code. Its minimal convex embedding dimension is the least dimension in which it admits a convex realization, and its minimal open convex embedding dimension…

Convex-polytope realization question. Can every convex code be realized with convex polytopes?

A code is locally good if every link of every codeword is contractible. Seven-neuron conjecture. Every locally good code on at most $7$ neurons is open convex or closed convex. The…

Let $\mathcal{C}$ be a max-intersection incomplete open convex code, where $\Delta(\mathcal{C})$ has at least two non-mandatory codewords not contained in $\mathcal C$. Suppose…

Open-convex non-closedness conjecture. Then $C$ is not a closed convex code.

Let $\mathcal C$ be a neural code on $n$ neurons, and let $T_\mathcal C$ be its toric ideal. A code is 0- or 1-inductively pierced when it belongs to the corresponding inductively…

Simplicial-complex image conjecture. Every intersection complete code lies below a simplicial complex in $d539_d53c$. More generally, every intersection complete code is the image…

Down-set conjecture. If $d53c$ has no local obstructions of the second kind, then every code below $d53c$ in $d539_d53c$ has no local obstructions of the second kind.

Dimension-two embedding conjecture. If $\mathcal{C}$ has minimal open convex embedding dimension $2$, then its minimal convex embedding dimension is $2$.

Endpoint-perturbation conjecture. The codes determined by the two realizations are equal:

Dimension-one embedding conjecture. A code $\mathcal{C}$ has minimal convex embedding dimension $1$ if and only if it has minimal open convex embedding dimension $1$.

Convex realizability conjecture. Every binary code is convex realizable.

Equivalence conjecture. For neural codes on more than $4$ neurons, the following properties are equivalent: $\mathcal C$ is convex; $\mathcal C$ is a good-cover code; and…