11 problems

Let $X$ be a normal projective variety, and let $D_j$ for $1\leq j\leq k$ be effective Weil divisors on $X$. The notation $\operatorname{Kss}(X,\sum_{j=1}^k D_j)$ denotes the K-sem…

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 $\mathcal{K}$ 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 $R$ 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 $\{(t,t^2,\ldots,t^k)\mid -1\leq t\leq 1\}$. 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…

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…

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 $k$…