18 problems

Greedy matching conjecture. For any tangle diagram $T$, the greedy matching $M_{\operatorname{gr}}$ is a Morse matching on the delooped Khovanov complex…

Let $T$ be a merge tree, and let a path be a 1-dimensional simplicial complex with no cycles. A discrete Morse function on a path induces a merge tree by filtering the path through…

Živaljević's conjecture. Every simply connected smooth $d$-manifold admits some LC triangulations.

Let $f(n)$ denote the number of perfect Morse matchings of the $n$-simplex. The preceding lower bound gives a growth rate approximately $(1.289)^{2^n}$. Double-exponential growth c…

Let $cmathcal{M}_P(K)$ denote the pure complex of discrete Morse matchings of a simplicial complex $K$. Let $cmathrm{cDelta}^n$ be the $n$-simplex and…

Let $cmathcal{GM}(K)$ denote the simplicial complex of all matchings on the Hasse diagram of a finite simplicial complex $K$, without imposing the acyclicity condition. Let…

For fixed $m>0$, let $\mathcal{G}_m^n$, $\Delta_m^{n,r}$, and $\Gamma_n^{\alpha,r}$ be the complexes defined in the paper. Let $V(\Delta)$ denote the vertex set of an induced subco…

Symmetry conjecture. For $\ast=\boldsymbol{\mathcal B}$, the corresponding matrix elements agree under $\gamma_{\ast}^n$, namely the path sums

Optimality conjecture for $\mathcal{M}^{**}$. The augmented gradient vector field $\mathcal{M}^{**}$ is optimal; equivalently, there is a gradient vector field on $M_7$ with 22 cri…

Björner–Lovász–Vrećica–Živaljević connectivity conjecture. The connectivity bounds in this statement are sharp. The conjecture concerns the precise homotopical connectivity of matc…

Let an appropriate higher-dimensional analogue of a merge tree be given, and consider the authors' algorithm for finding cancellations of critical cells using generalized merge tre…

A discrete Morse function (dMf) is a function on a graph satisfying the discrete Morse conditions; an Ml tree and an Mo tree are the merge-tree constructions associated with the un…

Orientable pseudomanifold characterization. $K$ is an orientable pseudomanifold if and only if

Branching-tree non-realizability conjecture. If $T$ contains a vertex with degree greater than $2$, then there exists a merge tree that cannot be realized by any discrete Morse fun…

Let $K$ be a knot with bridge index $b$. Say that a $d$-complex has discrete Morse vector $(c_0,\ldots,c_d)$ if it admits a discrete Morse function with exactly $c_i$ critical face…

Heegaard–discrete Morse conjecture. Some closed $3$-manifold with Heegaard genus $g$ admits discrete Morse functions with less than $g$ critical edges.

Mazur-manifold LC triangulation conjecture. Mazur's $4$-manifold admits an LC triangulation.

Kirby's conjecture. Every simply connected closed smooth $4$-manifold is geometrically $1$-connected.