28 problems

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…

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…

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…

Let $G$ be a compact Lie group and let $P$ be a principal $G$-bundle over a Riemann surface $\Sigma$. Write $\mathcal A(P)$ for the space of connections on $P$, $\mathcal G_0(P)$ f…