38 problems

Thom's gradient conjecture. If $\gamma(t)\to x_0\in\mathcal{U}$, then $\gamma$ has a tangent at $x_0$; equivalently, the limit of secants

Let ${\mathcal P}\subset\mathbb C^n$ be a product of discs or ${\mathcal P}\subset\mathbb R^n$ a product of intervals, and let $f_1,\ldots,f_N$ be analytic functions satisfying pol…

Let $f$ be a real analytic function on an open set $U\subset\mathbb{R}^n$, and let $x(t)$ be a gradient flow line of $f$ with a limit point $x_0\in U$. Arnold–Thom conjecture. If…

Let $X$ be a closed subset of ${\mathbb R}^n$, let $p\in{\mathbb N}$, and let $f:X\to{\mathbb R}$. The paratangent bundle of order $p$ is denoted by $\tau^p(X)$, and applying the p…

Let $\mathbb{R}_{\text{an}}$ be the expansion of the real field by restricted analytic functions, let $\mathbb{R}_{\text{an}}^\mathbb{R}$ be its expansion by all real power functio…

van den Dries–Miller conjecture. There are no structures on $(\mathbb R,+,\cdot)$ lying strictly between $\mathfrak S^{an}$ and $\mathfrak S^{an,\, exp}$.

Let $M\subset\mathbb C^N$ be a real analytic generic submanifold and let $p\in M$ be a point of finite type. Let $M'\subset\mathbb C^{N'}$ be a real analytic set containing no nont…

Let $M$ and $M'$ be connected real analytic hypersurfaces through $0$ in $\mathbb C^2$, and suppose that $M$ is of finite type at some point. Let…

Let $M\subset\mathbb C^N$ be a real analytic CR submanifold. Let $p\in M$, let $M'\subset\mathbb C^N$ be a real analytic submanifold with $\dim_{\mathbb R}M=\dim_{\mathbb R}M'$, an…

Let $M\subset\mathbb C^N$ be a real analytic CR submanifold, let $p\in M$ be a point of finite type, and let $K$ be a positive integer. Let $M'\subset\mathbb C^{N'}$ be a real anal…

Baouendi–Rothschild–Zaitsev approximation conjecture. The holomorphic mapping $H$ with this approximation property should exist under these hypotheses. This would generalize the kn…

Classification conjecture. The structure theory of maximal-rank webs should yield a complete classification, up to coordinatewise reparametrization, of real-analytic diffeomorphism…

Let $\Sigma\subset\mathbb T^2$ be a real-analytic curve and let $e_\lambda$ be a toral eigenfunction. Real-analytic curve restriction conjecture. The uniform bound should hold: … U…

Let ${\mathcal P}$ be the domain of a Noetherian chain and let $P_1,\ldots,P_n:{\mathcal P}\to\mathbb C$ be Noetherian functions of format ${\mathscr F}$. Khovanskii's format-bound…

Huang–Ji–Yau conjecture. If all these syzygies are algebraic, then $M$ is locally algebraizable.

Let $X,Y\subset\mathbb{R}^n$ be two germs at the origin of irreducible real analytic subsets. Let $h\colon(\mathbb{R}^n,0)\to(\mathbb{R}^n,0)$ be a germ of a subanalytic, arc-analy…

For $s>0$, let $W_s$ be the canonical Weierstrass product for the sequence $(-n^s)_{n>0}$, namely … For $S\subseteq(0,\infty)$, write the corresponding expansion using the restrict…

Let $M$ be a real analytic manifold and let $f:M\to\mathbb R$ be a function. Consider the restrictions of $f$ to $2$-dimensional compact analytic submanifolds of $M$. Bochnak–Sicia…

Milnor vector field conjecture. If both fibrations exist under these conditions, then they are equivalent.

Kashiwara–Schapira conjecture. Every continuous subanalytic function is almost $\mathcal{C}^{\infty}$-subanalytic.

Let $f$ be an analytic function and let $x(t)$ be a gradient flow line of $f$. Suppose that $x(t)$ has a limit point $x_{\infty}$ as $t\to\infty$. Stronger Arnold-Thom conjecture.…

Factorial-growth conjecture. In the real analytic case, the Bergman kernel coefficients satisfy the sharper factorial bound above. This would improve the established estimate…

Let $M,M^*$ be real-analytic hypersurfaces in $\mathbb{C}^{N}$, with $N\geq 2$, and let $p\in M$ and $p^*\in M^*$ be reference points. Suppose that $M$ and $M^*$ are formally equiv…

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

Let the quasianalytic Ilyashenko class be the class of almost regular maps extended in this paper, and let the real field be $(\mathbb{R};+,-,\cdot,0,1)$. O-minimality conjecture.…