19 problems

Let $G$ be a graph, let $\mu(G)$ denote the Colin de Verdière parameter of $G$, and let $k$ be a positive integer. Let $\Delta(G)$ denote either the symmetric or exterior algebraic…

Characteristic-zero exterior shifting conjecture.

Barycentric subdivision conjecture. If $K$ is a surface triangulation, then $\Delta^{ex}(\operatorname{sd}(K))$ is a homology lex-segment.

Uniform surface conjecture. For every fixed genus $g$, a.a.s.,

Lex-segment prevalence conjecture. As $n$ tends to infinity,

Lexicographic edge conjecture. If $\sigma<_{\mathrm{lex}}\sigma'$, then

Let $d\ge 3$, and let $K$ be a triangulation of the $(d-1)$-sphere. Removing a single $(d-1)$-simplex means deleting that facet and its interior. Higher-dimensional volume-rigidity…

A triangulation of a compact connected surface without boundary is a simplicial complex representing the surface; removing a single triangle means deleting that triangular face and…

Kalai–Sarkaria conjecture. If $K$ admits an embedding into $S^{r}$, then

Let $[n]=\{1,\ldots,n\}$, and let $C(n)$ be the family … For a $3$-uniform hypergraph $H$, say that $H$ dominates $C(n)$ when the corresponding rows and columns of the third compou…

Algebraic and symmetric shifting conjecture. For all $i,j$,

Let $K$ be a $d$-dimensional balanced simplicial complex, and let $<$ be a $(3,\ldots,3)$-admissible order. Let $K^{b,<}$ denote the corresponding balanced shift. Balanced linkless…

Let $K$ be a $d$-dimensional balanced complex, and let $<$ be a $(2,\ldots,2)$-admissible order. Let $K^{b,<}$ denote the corresponding balanced shift. Balanced PL-embeddability co…

Let $K$ be a $d$-dimensional balanced complex, and let $<$ be a $(2,2,\ldots,2)$-admissible order, meaning that the least two vertices of each color class form an initial segment o…

Let $K$ be a $d$-dimensional balanced simplicial complex, meaning that its vertices are colored with $d+1$ colors so that every edge has vertices of different colors. Let $K^b$ den…

Let $K$ be a $2$-dimensional simplicial complex, and let $K^s$ denote its symmetric algebraic shifting. Symmetric-shifting conjecture. If $K$ is embeddable in $\mathbb{R}^4$, then…

Let $t\leq r$ be two natural numbers. A family of faces is $t$-intersecting if every pair of faces in it has intersection of cardinality at least $t$. Let $\Delta$ be a simplicial…

Toric $g$-vector conjecture. The toric $g$-vector of a non-simplicial polytope is an $M$-sequence, i.e. it satisfies Macaulay inequalities.

Symmetric shifting obstruction conjecture. If