25 problems

Zobin's conjecture. Every ideal $I^m(E)$ in $\mathcal{P}_0^m(\mathbb{R}^n)$ is equal to $I^m(V)$ for some semi-algebraic set $V$ containing the origin.

A semi-algebraic set is a subset of a real affine space defined by finitely many polynomial equalities and inequalities; its complexity is measured by the axiomatic notion introduc…

For each dimension $d$, let $\mathfrak{A}_d$ be the class of sets admitting the semigroup decomposition considered in the paper, and call a set semi-algebraic if it is defined by p…

For a fixed ReLU-network architecture, let $\mathcal{P}$ denote the parameter space obtained from the space of weights by taking into account the trivial scaling and permutation sy…

Let $\mathcal{P}$ be the parameter space obtained from the space of weights of a ReLU neural network by quotienting by the trivial scaling and permutation symmetries. An open subse…

Let $\mathcal{K}$ be a hyperplane arrangement, let $F$ be a flat, let $J$ be a set of hyperplanes, let $\iota$ be as above, and let $\omega_{\mathcal{K},\iota,F,J}$ denote the asso…

Let $X$ be a connected semi-algebraic subset of $\mathbb{R}^n$ equipped with the metric $d$ defined by the infimum of the lengths of rectifiable paths in $X$ joining two points: ……

Codimension-two conjecture. The subset ${\mathtt{G}}_{\ge 2}(\mathsf{B},m,n)$ is contained in a submanifold of codimension two in ${\mathtt{G}}(m,n)$.

Let $R$ be the real closed field under consideration, let $X$ and $Y$ be affine $R$-varieties, and let $pi:X\to Y$ be a morphism such that $pi_*\mathcal{O}_X$ is a free…

Let $X \subset \mathbb{R}^k$ be a semi-algebraic set defined by a closed formula of complexity bounded by $C$, and let $\dim X=n$. A nested semi-algebraic filtration is a sequence…

Let $S\subseteq\mathbb{R}^d$ be a semi-algebraic set of description complexity $r$, and let $a_1,\dots,a_n\in\mathbb{R}^d$. Define $f:\\{-1,1\\}^n\to\\{0,1\\}$ by letting…

For integers $k,n$ with $k>3$, let $\mathcal{A}_{k,n}$ denote the Minkowski sum of $n$ copies of the moment-curve segment $\{(t,t^2,\ldots,t^k)\mid -1\leq t\leq 1\}$. Unique bounda…

For integers $k,n$ with $k>3$, let $\mathcal{A}_{k,n}$ denote the Minkowski sum of $n$ copies of the moment-curve segment $\{(t,t^2,\ldots,t^k)\mid -1\leq t\leq 1\}$. Let…

Let $\opcat{SA}$ be the category whose objects are embedded semi-algebraic sets and whose morphisms are restrictions of polynomial mappings. Let $\cA$ consist of scalar multiplicat…

Let $d=1$ and $k=4$, so that $p=5$ and a saturated design is specified by a support $X\subset\{0,1\}^4$ with $|X|=5$. After quotienting by the hyperoctahedral symmetry, there are 1…

Algorithmic conjecture. There is an algorithm which computes $m_{i,\lambda}(S,\mathbb{Q})$ for every $\lambda\vdash k$ with $m_{i,\lambda}(S,\mathbb{Q})>0$, as well as all the Bett…

Computability conjecture. These multiplicities are computable with polynomial complexity, and consequently the Betti numbers of $S$ are also computable with polynomial complexity.

Let $K \subset \mathbb R^n$ be a definable bounded set in an o-minimal structure over $\mathbb R$, with a fixed linear order of the coordinates. A cylindrical decomposition of…

Let a symmetric semi-algebraic set be defined by real symmetric polynomials of degree bounded by $d$, in a space with $m$ non-symmetric variables, and let its equivariant cohomolog…

For every integer $k\geq 3$, there exist $d=d(k)$, $t=t(k)$, and $s=s(k)$ such that … where $c'$ depends only on $k$. Stepping-up conjecture. The preceding lower-bound result shoul…

For dimension $d$, complexity $t$, and positive integer $s$, let $R_3^{d,t}(s,n)$ denote the off-diagonal Ramsey number for $3$-ary semi-algebraic relations on points in…

For integers $k\geq 3$, dimension $d$, complexity $t$, and $s$, let $R_k^{d,t}(s,n)$ be the least integer $N$ such that every sequence of $N$ points in $\mathbb{R}^d$ and every $k$…

Let $\X$ be a complete locally convex metric space and let $f\colon\X\to\overline{\R}$ be a lower semicontinuous function. Suppose the construction used to produce a curve of near-…

Let $\mathcal{A}_{k,a}$ be the family of ordered finite lists $\mathcal{P}=(P_1,\ldots,P_s)$ of polynomials in $\mathbb{R}[X_1,\ldots,X_k]$, with total additive complexity at most…

Minimal polynomial-representation conjecture. Every $d$-dimensional polytope in ${\mathbb{R}}^d$ has a polynomial representation