30 problems

Let $D$ be a completely nonholonomic rank $2$ distribution on an $n$-dimensional manifold, and let its class be the constant integer $m$ with $2\leq m\leq n-4$. Nonexistence conjec…

The conjecture. If $u\in\mathcal G(\mathbb R^d)$ is weakly homogeneous of degree $\alpha$, then there is a representative $(u_\varepsilon)_\varepsilon$ of $u$ such that, for every…

Conjecture A. If

Let $f$ be a holomorphic cusp form of weight $2$ for a fixed cofinite discrete subgroup $\Gamma$ of $\operatorname{SL}_2(\mathbb R)$, and let $\Gamma_{\!\infty}$ be the stabilizer…

Let $P^\varepsilon(x,y)$ be the regularized fermionic projector and define the regularized product … The proposed procedure is to choose the regularization tails so that the singul…

Let ${\mathcal{M}}[A^\varepsilon_{xy}]$ denote the distributional expression associated with the regularized kernel $A^\varepsilon_{xy}$, and let … denote the scalar and vector con…

Connes–Estrada conjecture. Hadamard's finite part theory is in principle enough to deal with quantum field theory divergences.

Let $x\in R^n$ and $t\in R^r$, let $\Omega\subseteq R^r$ be open, and consider distribution solutions of the functional equation (1.1) satisfying assumptions A-1 through A-5. Let…

Let $D$ be a bracket-generating distribution with $5$-dimensional cube on a manifold $M$. For a point $q_0\in M$, an abnormal extremal trajectory is a horizontal curve starting at…

Let $\pi:S[\{0,1\}]\to\mathbb{Z}_{\geq0}$ interpret a binary word as an integer, and let $u_1<u_2<\ldots<u_{k_n}$ be the ordered elements of $\pi(\mathscr{U}_n)$. Define … Let…

Let $S$ be a finite set of primes, let $b\geq 2$ satisfy $\#(S\cup F_b)>1$, and let $M_S$ denote the set of nodes associated with $S$. If a node $m\in M_S$ has a $b$-ary expansion…

Marino's conjecture. Singularities and symmetries pair down the test function space, expanding the space of distributions to accommodate the variety of quantum fields observed in n…

Let $W_n$ be the topological vector space of Wilsonian renormalization group flows, let $\mathcal{D}^{\times}_{n}{}'$ be the space of ordinary distributional correlators, and let…

Let $\Omega\subset\mathbb{R}^n$ be an open subset, let $f$ be a real analytic function on $\Omega$, and let $T$ be a distribution on $\Omega$. Schwartz's conjecture. There exists a…

Consider the minimum $\ell_1$-norm interpolator $\widehat{\boldsymbol{\theta}}^{\mathsf{Int}}$ under an i.i.d. Gaussian design, and consider the finite-sample distribution of its c…

Let $f(x,y)$ be a bivariate polynomial, and let the Newton polytope of $f$ have $k$ vertices. For $a\in\mathbb{C}\setminus\{0\}$, let…

Let $f$ be a polynomial of degree $n>1$, let $\sigma$ be a permutation with $Pr(f,\sigma)>0$, and let $D=\overline{D^\circ}\subset[0,1)^n$ be the proposed limiting region for the n…

Let $n\ge 1$ be an integer and let $G=\operatorname{SL}_n(\mathbb{F}_p)$. Consider the solutions $(g,h)\in G\times G$ of … for a fixed $x\in G$. Hu–Li's conjecture. The solutions s…

Let $\mathcal{E}'(\mathbb{R})$ be the commutative unital convolution algebra of compactly supported distributions on $\mathbb{R}$. Its Bass stable rank…

Massless-field kinetic-tensor conjecture. The renormalized IN-region kinetic tensor should have $tt$-component

Let $f(x)$ be a polynomial of degree $n$. For a prime $p$ in $Spl(f)$, number the roots of $f(x)$ modulo $p$ as … For a subset $S\subseteq\{1,2,\dots,n\}$, write $Pr(f,S)$ for the…

Let $A$ be the abelian part of a semidirect-product symmetry group $\mathcal B=A\rtimes_{T}\mathcal G$, let $A^{\prime}$ be the topological dual of $A$, and consider the…

Redundancy conjecture. The convergence of $a^m(x,D)u^m$ to $a(x,D)u$ in $\mathcal{S}'$ should be redundant; it should suffice to assume that $a(x,D)u\in\mathcal{S}'$.

The distributional approach yields interaction distributions concentrated at the origin, and regularizations using $\delta'$-converging sequences must converge in the sense of dist…

Schwartz's division conjecture. There exists a distribution $S$ such that $fS=T$. Lojasiewicz proved this conjecture using his inequality, while Hörmander independently proved the…