24 problems

Let stable maps be maps whose qualitative behavior is unchanged under the relevant coordinate changes, and allow all homeomorphisms as coordinate changes. Thom's density conjecture…

Let $M$ be a compact manifold of dimension $m$, let $P:M\rightarrow\mathbb{R}^d$ be smooth, and let $\mathfrak{F}$ be the quasi-stratification of $M$ by the level sets…

Let $X$ be a four-manifold, let ${\cal M}_k$ be the moduli space of charge-$k$ connections, and let $\nu_p$ be the geometric representative consisting of connections for which…

Main Conjecture. The subset of points $x$ in $J(n,m,k)$ such that the map $\Psi_k^0$ is non-transversal to the stratum containing the point $\Psi_k^0(x)\in Mat^0(n,m,k)$ has codime…

Transversality conjecture for tilted Schubert cells. If $u\lesssim_{\mathbf{a}}v$, then $X^\circ_{v,\mathbf{a}}$ and $\Omega^\circ_{u,\mathbf{a}}$ intersect transversely.

Transversality conjecture. The transversality condition holds at every Misiurewicz point for all integer exponents $p,q$ with $p>q>1$ in this family.

Let $G$ be an ordered vector field, let $H$ be the fixed invariant one-dimensional set, and let $T_1,\ldots,T_k$ be the periodic orbits in $Ess(G)$. Choose tubular neighborhoods…

Let $M'$ be the configuration manifold under consideration, let $S_I$ be the specified projection, and let $Q_\gamma$ be the relevant submanifold. Let $T$ be the submanifold of…

Let $M$ be a connected manifold, let $\mathcal{R}(M)$ be the set of Riemannian metrics on $M$, and for each metric let $\Delta_g$ denote its Laplace--Beltrami operator. The strong…

Generic approximation conjecture. Any proper immersion $\beta$ admits a $C^\infty$-approximation $\tilde\beta$ such that $\tilde\beta$ is $k$-normal for all $k\in[2,n+1]$, and each…

Upward-jump representation conjecture. The representation formula by means of upward jump times of dyadic expansions has the potential to shed more light on the complex arguments u…

Let $(g,J)$ be a compatible almost Hermitian structure, and define … A biharmonic almost complex structure is a zero of $B$. Transversality conjecture. The map $B$ is transverse to…

Let $(M,\omega)$ be a symplectic manifold, and let an almost complex structure $J$ on $M$ be super-rigid if every simple $J$-holomorphic map has non-negative index, every simple in…

Let $f$ be a smooth function on a manifold whose critical set has submanifold structure, and consider situations more complicated than those covered by the preceding escape theorem…

Generic density conjecture. For any $A>0$ and $K$, ${\cal G}_{2}(3A/2,K)$ is dense in ${\cal G}_{3/2}(3A/2,K)\cap {\cal G}_{2}(A,K)$.

Let $A$ and $B$ be the two sets involved in a nonconvex alternating projections algorithm, and let subtransversality denote the regularity property imposed explicitly or implicitly…

Let $f_1:X_1\to Y_1$ and $f_2:X_2\to Y_2$ be morphisms of schemes, and let … be their product. For $i=1,2$, let $F_i\in\mathbf{T}(X_i)$, and let $F_1\boxtimes F_2$ denote their ext…

Let $(M,c omega)$ be a symplectic manifold. Denote by $c \mathscr{J}(c omega)$ the space of almost complex structures on $M$ compatible with $c omega$, and let…

Let $M$ be an open smooth $n$-manifold with a Riemannian metric $g$, and let $\alpha:N\subset M$ be a smooth compact codimension-one submanifold. An isotopy $A:N\times[0,1]\to M$ i…

Many-exponential transversality conjecture. Transversality remains true in the general case of many exponentials.

Higher-dimensional nondegenerate subspace conjecture. There does not exist a $[\frac{n-1}{2}]+1$-dimensional subspace $V\subset{\mathbb R}^n$ such that $Q_{v,w}\equiv0$ for every…

Let $(M^{2n-1},\xi)$ be a co-oriented contact manifold, so that $\xi=\ker\alpha$ for a globally defined $1$-form $\alpha$. Assume that all closed orbits of the Reeb vector field as…

Higher-jet transversality codimension conjecture. The set $\tilde A$ is a countable union of smooth submanifolds of codimension more than $n=\dim X$ in $J^{p+1}(X,Y)$.

Let $\Bbbk$ be an algebraically closed field of characteristic 0, let $X/\Bbbk$ be a projective variety, let $\sigma \in \operatorname{Aut}_{\Bbbk} X$, and let $Z \subseteq X$ be a…