14 problems

A semialgebraic set $\EuScript T\subset\mathbb R^n$ is locally connected by analytic paths if, for every $x\in\EuScript T$ and every open semialgebraic neighborhood…

Let $U \in \mathcal{U}(n,m)$, and let … be its unique partition into irreducible sets. Let $\gamma \in C_{\operatorname{cv}}(U) \cap \mathbb{R}_{>0}^m$, and define … Here…

Equitable regularity conjecture. The equitable semialgebraic regularity lemma holds with partitions into

Shiota's conjecture. ${\EuScript S}$ is a Nash image of ${\mathbb R}^d$ if and only if ${\EuScript S}$ is pure dimensional and there exists an analytic path

Fix $m\ge 0$. Let $(A_{ij}(x))_{1\le i\le N,1\le j\le M}$ be a matrix of semialgebraic functions on $\mathbb{R}^n$, and let the system … L{mu,nu}f(x)=sum{i=1}^Nsum{|alpha|le bar m}…

Adaptive-radius homology computation conjecture. One can compute the homology of

Link and double-point image criterion. The map germs $f$ and $g$ are bi-Lipschitz equivalent if and only if $link(f)$ and $link(g)$ are bi-Lipschitz equivalent and the images…

Let $(S_0,0)$ and $(S_1,0)$ be two normally embedded real semialgebraic surface germs. They are normally embedded when their inner and outer metrics are bi-Lipschitz equivalent; fo…

Let $X$ be an $\mathscr{A{\!}R}$-closed set in $\mathbb{R}^n$. Denote by $\mathscr{A}_a(X)$ the ring of arc-analytic semialgebraic functions on $X$, by…

Rank-one factor abundance conjecture. Every such matrix has a psd factorization in which either at least $k$ matrices $A_i$ and $k-1$ matrices $B_j$ are rank one, or at least $k-1$…

Let a spectrahedral shadow be the affine projection of a spectrahedron, that is, a set of the form … where $F_k \in {\mathbb S}^m$ for $k=0,1,\ldots,n$ and $G_l \in {\mathbb S}^m$…

Semialgebraic universal-cover classification conjecture. The following are equivalent:

Let $\mathcal{C}_{\mathcal{G}}$ denote the cone associated with a graph $\mathcal{G}$. A basic open semialgebraic set is an open semialgebraic set defined by finitely many strict p…

Let $G$ be a graph and let $k$ be a positive integer. The $(G,k)$-stratum is the set of frameworks $G(P)$ for which the dimension of the fiber $W(G,P)$ is at least $k$. Such a stra…