28 problems

Conjecture 1.3 (Maxwell; [11], see also Section 4). The total number of points of equilibrium, all assumed nondegenerate, of any configuration with $l$ charges in $\mathbb{R}^{3}$…

Let a Morse spacetime be a singular Lorentzian spacetime built from a Morse function, whose critical points have an index and a coindex. Borde–Sorkin conjecture. A Morse spacetime…

Legendre-transform equivalence conjecture. The natural map induced by the Legendre transform,

Let $V_{\alpha}(\bar x)$ be the potential of a generic configuration of point charges of the same sign, and let $a_{\alpha}^j$ denote the number of its critical points of index $j$…

Let $X$ be a closed $3$-manifold with $b_1(X)>0$, let $o$ be a homology orientation, and let $\operatorname{Spin}^c(X)$ denote the set of spin-c structures on $X$. Let…

Let $f_1$ and $f_2$ be functions compatible with the same input vector field, and fix the orders of simplices within each Morse set. Connection-matrix invariance conjecture. The co…

Gabrielov–Novikov–Shapiro conjecture. For $n$ unit point charges in generic position, and all $p\geq 1$, the number of index-$j$ equilibria of $V_p$ is at most $\#_j$.

Let $Y^+$ be a cylindrical-end $3$-manifold with harmonic Morse function $f^+$, let $Z$ be the set of its index-$1$ and index-$2$ critical points, and let a t-orbit be a weighted c…

Let $G,G',H,H'$ be compact Lie monoids, let $X,X'$ be closed smooth manifolds acted on by $G\times H$ and $G'\times H'$ respectively, let $\varphi\colon G\to G'$ and…

Let $C$ be a convex body in $\mathbb{R}^n$ with smooth boundary. An inward normal to $\partial C$ is a line segment or ray normal to the boundary and directed inward. The concurren…

Correspondence conjecture. For any rigid isotopy class, the isotopy classes of strictly Morse polynomials contained in it are in a natural one-to-one correspondence with the extens…

Let $X^{00}$ be the relevant open space, let $\pi^*\mathcal{O}_{an}$ be the pulled-back analytic-function sheaf, and let $C^*(X^{00},\pi^*\mathcal{O}_{an})$ be its Morse complex. F…

Let $(\mathcal{M},h,f,\zeta)$ be a Morse geometry of dimension $n$, and let $(M,g)$ be the corresponding Morse spacetime. Assume that $f$ has a single critical point for each criti…

Let $\mathcal{M}$ be a compact cobordism of dimension $n$, $h$ a Riemannian metric, $\zeta>1$ a constant, and $f$ a Morse function. Let…

The infinity-categorical structure conjecture. The simplicial sets $\mathrm{HOM}_{\mathsf{\Omega B As\text{-}Alg}}(A,B)_\bullet$ are $\infty$-categories.

Let $x_1,\dots,x_p$ with $p\geq 2$ be distinct vectors in $\mathbb R^m$. Define the symmetric matrix $A=(a_{ij})_{1\leq i,j\leq p}$ by … For $k=1,\dots,p$, let…

Let $V$ be a real-valued periodic potential, let $H_0=-\Delta+V$ be the associated self-adjoint operator on $\ell^2(\mathbb{Z}^d)$, and let $\lambda_m(k)$, $k\in\mathbb{R}^d$, be i…

Highest abelian subgroup conjecture. Any highest abelian subgroup in $G$ is a Morse quasiflat.

Let $f$ be the function for the equal-mass planar $N$-body problem, and let $M(t)$ denote its Morse polynomial. Newtonian Morse-polynomial conjecture. For $A=3$, the Morse polynomi…

Consider the equal-mass planar six-body problem, with exponent $A$, function $f$, and Morse polynomial $M(t)$; let $P(t)$ be the Poincaré polynomial of the quotient configuration s…

Let $f$ be the function for the equal-mass planar five-body problem, let $\mathcal{C}_5$ be the quotient configuration space, and let $P(t)$ be its Poincaré polynomial. Five-body M…

Let $f$ be the function whose critical points represent central configurations in the four-body problem, and let $\mathcal{C}_4$ be the quotient configuration space with Poincaré p…

Let $N>4$, let $A$ be the exponent of the homogeneous potential, and let $f$ be the function whose Morse index is considered on the quotient configuration space $\mathcal{C}_N$. Fo…

Higher-dimensional barcode-functional conjecture. There exists a constant $\kappa_g>0$ such that, for any $\lambda>0$,

Let $f:M\to\mathbb R$ be a Morse function on a closed manifold $M$ of dimension $n+1$, and let $v$ be its gradient-like field satisfying the Morse–Smale transversality condition. F…