29 problems

Projection conjecture. At most $k(n-1)$ points of $\phi_4(\Sigma)$ can be contained in a curve of degree $k$ in $\mathbb{P}^2$.

Let $D\subset\mathbb{N}^{2}$ be finite, and consider a system $\mathcal{L}_{D}(m_{1},\dots,m_{r})$. The system admits a reduction algorithm if there is a sequence of sets … that sa…

Interpolation conjecture. For every derivation $T\vdash_{\mathsf{SKS}}R$, there is a structure $P$ and derivations

Let $w_n$ be given by equation $(15)$, let ${\mathcal H}_{\mathbf w}$ be the corresponding Hilbert space, and let $Mult({\mathcal H}_{\mathbf w})$ denote its multiplier algebra. A…

Asymptotic interpolation conjecture. If $k\geqslant3$ and

Yin and Wu's size interpolation conjecture. If

$(Q_n)$ conjecture. $(Q_n)$ is true for all nonnegative integers $n$.

Let $f_n(x)$ be the Fourier interpolation basis functions. Logarithmic square-norm conjecture. For $n>1$, one has … The lower bound is already known from the result cited in the pa…

Let $f_n(x)$ be the Fourier interpolation basis functions, and let $\varepsilon,\delta>0$. Consider the region $\varepsilon n\le x\le n-\sqrt n$. Interior-region decay conjecture.…

Let $f_n(x)$ denote the basis functions of the Fourier interpolation considered in the paper, for $n\in\mathbb{Z}_{\ge0}$ and $x\ge0$. Uniform boundedness conjecture. There exists…

Assume $f obreaktext{ belongs to } C^{1,1}nu(\mathbb{R}^n)$. Let $hat{f}$ be the linear function that interpolates $f$ at any set of $n+1$ affinely independent vectors…

Let $X$ be the blowup of $\mathbb{P}^2$ at $n$ very general points, and let $D=dH-\sum_i m_iE_i$ be a divisor class with $d\geq 0$. Call $D$ special when both $h^0(\mathcal{O}(D))$…

Maximal Hilbert function conjecture. For every $s$ and $t$,

Equivalence of the optimal fractional Poincaré constants. For $t>s\geq 0$, the optimal Poincaré constant is

Let $d$, $g$, $r$, and $n$ be nonnegative integers with … A BN-curve is a stable map from a general curve of genus $g$ to $\mathbb{P}^r$ of degree $d$ corresponding to the unique c…

Let $d$ be a positive integer, and consider a uniform rectangular grid in $d$ dimensions with mesh sizes … Let the grid be shifted along any vector, and apply BFECC interpolation f…

Endpoint-maximization conjecture. The maximum of the Lebesgue function is attained at $x=-1$, namely

Alexander–Hirschowitz conjecture. Suppose that $X$ is a smooth projective variety of any dimension and $\mathcal{L}$ is ample. Then $(X,\mathcal{L})$ satisfies AH.

Let $\mathcal{S}_{even}(\mathbb{R})$ denote the space of real-valued even Schwartz functions on $\mathbb{R}$, and let $\widehat{f}$ be the Fourier transform of $f$. For a sequence…

Let $r_1,r_2,\dots$ be the positive real numbers of the form … where $j$ and $k$ are integers. Radial Schwartz functions $f\colon{\mathbb R}^2\to{\mathbb R}$ are considered through…

Fix a dimension $d\geq 3$, density $\rho=1$, and $\alpha>0$. For the potential $p(r)=e^{-\alpha r^2}$, consider the optimal linear programming bound $\widehat{f}(0)-f(0)$ from the…

Let $E$ be a quasi-Banach symmetric sequence space. For functions or sequences $f$ and $g$ with nonincreasing rearrangements $f^*$ and $g^*$, write $f\triangleright g$ when … The s…

Let $G=(\mathcal{V},E)$ be the graph underlying the neighborhood $B(v_0,K)$, let $\widetilde{\mathcal{V}}\subset\mathcal{V}$ be the interpolation set, and let $\chi(\cdot,v_0)$ and…

Fix integers $d\ge 2$ and $n>2$. Let $S\subseteq {\mathbb P}^n$ be a finite collection of $s$ points, fix integers $m_1,\dots,m_s$, and set … Let $\operatorname{w}(Z)$ be the total…

Let ${\mathcal L}={\mathcal L}_{n,d}(m^s)$ be a homogeneous linear system of degree-$d$ hypersurfaces in ${\mathbb P}^n$ through $s$ general points, each with multiplicity $m$, and…