75 problems

Let $f:M\to S^1$ be a Morse map and let $v$ be a generic $f$-gradient satisfying the Transversality Condition. The associated Novikov complex $\mathcal C_*(v)$ is a free based chai…

Flow-closedness conjecture. If $\mu_c$ is a complex moment map associated to a linear action by a compact group $G$ on a vector space, then $|\mu_c|^2$ is flow-closed.

Let $M$ be a locally symmetric space associated to a group $G$ without discrete series, equivalently of the form $M=K \backslash G/\Gamma$ with Euler characteristic zero. A finite…

Spanier–Whitehead duality conjecture. Reversing the ordering of the critical sets of an arbitrary framed flow category, while preserving the stable normal framing, results in takin…

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…

The Yang–Mills functional is defined on the infinite-dimensional space of connections over a compact Riemann surface, with the gauge group acting on this space. An equivariantly pe…

Let $M$ be a closed smooth manifold and let $\omega$ be a closed $1$-form on $M$. Let $\operatorname{Nov}$ denote the Novikov ring associated with the cohomology class represented…

Let $M$ be a closed connected manifold of dimension at least $6$ with fundamental group $G=\pi_1(M)\cong \mathbf{Z}^{n+1}$. Let $\mathbb{G}=\{G_k\}$ be a nested tower of finite-ind…

Let $X$ be a compact $3$-fold with nonempty boundary. Let $gc(X)$ denote its gradient complexity and $c(X)$ its Matveev complexity, namely the minimum number of $Q$-singularities a…

Let $X$ be a compact $3$-manifold, and let $gc(X)$ denote its gradient complexity, as introduced in the paper. A nonsingular gradient flow on $X$ is a gradient flow without singula…

Let $M$ be an infinite-dimensional isoparametric submanifold in a Hilbert space, or one of its parallel manifolds, and let the setting and hypotheses of Theorem 1 apply, including…

Monotonicity conjecture. The coefficients satisfy

Let $(L,h)\to(M,\omega_h)$ be a positive holomorphic line bundle over a compact Kähler manifold of complex dimension $m$, and let $\beta_{2q}(m)$ be the universal coefficient of th…

For a link $L$, let $R$ be a Seifert surface and let $h(R)$ denote its handle number. Define … Let $R_1\natural R_2$ denote the plumbing of Seifert surfaces. Przytycki–Rudolph–Wada…

Let $\{(f_t,g_t)\mid t\in[0,1]\}$ be a family of Morse data, and let $\Phi'$ be the bifurcation chain map and $\Phi$ the continuation map defined by counting flow lines of…

Fibre-wise Morse function conjecture. The theorem holds without the assumption that such a function exists.

Let $M\to B$ be a family of compact manifolds, and let $(F,\nabla^F)\to M$ be a flat complex vector bundle with a parallel Hermitian metric such that $(H,\nabla^H)$ also admits a p…

Let $(E,\pi)$ be an exact Morse fibration over $D$ with relative Maslov map $\delta_{E/D}$, and let $(N,p)$ be a good pseudo-real submanifold compatible with $\delta_{E/D}$. Let…

Let $(E,\pi)$ be an exact Morse fibration over $D$ with a good and complete real structure $\iota$ compatible with a relative Maslov map $\delta_{E/D}$. Choose admissible paths…

Flow-up generator conjecture. There exists a set of independent generators

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 $\omega$ be a Morse form on a closed manifold, let $v$ be an $\omega$-gradient, and let $p:M'\to M$ be a free abelian covering such that $p^*\omega=df$ for a function…

The lifted complex Morse model conjecture. The induced deformation $R_{\mathfrak C_S,\gamma_S}$ is $A_\infty$-quasi-isomorphic to the directed total algebra $R_S$:

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…