50 problems

Does every polynomial map F: ℂⁿ → ℂⁿ with constant nonzero Jacobian determinant have a polynomial inverse?

How many pairwise non-overlapping infinite unit cylinders can simultaneously touch a unit ball?

Does the centered simplex maximize volume among convex bodies whose barycenter is their unique interior lattice point?

Do the logarithmic-vector-field conditions proposed by da Silva Machado and Seade characterize weighted homogeneous isolated hypersurface singularities?

Does a general pencil of plane cubics over $\mathbb C$ have exactly $12$ common flex lines?

Let $g_d(n)$ be the fewest points in $\mathbb R^d$ that always determine at least $n$ distances. Estimate $g_d(n)$; in particular, does $g_d(n)/d^{n-1}$ have a limit as…

For planar point sets with no $k$ collinear points, how many ordinary lines force an $r$-point subset whose every connecting line is ordinary? Is the threshold $o(n^2)$, or even…

Can $n$ planar points determine only $O(n/\sqrt{\log n})$ distances while every four-point subset determines at least three distinct distances?

If every $n$-point subset of an infinite planar set contains at least $\epsilon n$ points with no three collinear, must the whole set be a finite union of sets with no three collin…

If u(n) is the maximum number of unit-distance pairs among n planar points, is u(n) = n^(1+o(1))?

Is every finite locally free group scheme of order n killed by n, without assuming that the group scheme is commutative?

Let $X\to B$ be a semistable one-parameter family of complex projective varieties, let $X_t$ be a smooth nearby fiber, and let $T$ act on $H^1(X_t,\mathbb{Z})$. Is the natural map…

For every amphicheiral knot $K$, does there exist $f(z)\in\mathbb{Z}[z]$ such that …

How many unit set-distances can occur among $n$ pairwise disjoint translates of one planar compact convex set?

What is the largest possible measure of a subset of a radius-$R$ disk in $\mathbb R^2$ containing no pair of points at a positive integer distance?

Are total Cartier indices of rational singularities uniformly bounded in the bounded-family setting of Han--Jiang Problem 4.5?

Is the index of a numerically trivial log-canonical foliated log Calabi--Yau triple uniformly bounded in dimension three?

What is the optimal universal bend-and-break constant for rational curves tangent to a rank-$r$ foliation?

Is every infinite-dimensional Lie group modelled on a complete locally convex space regular?

What is the smallest universal order of lattice-covering density needed for an arbitrary convex body in $\mathbb R^n$?

Must every projective variety with Gorenstein canonical singularities and polynomial stringy $E$-function have nonnegative stringy Hodge numbers?

For an infinite planar set in strong general position, how large can the chromatic and clique numbers of its positive-integer-distance graph be; in particular, can the chromatic nu…

If $f(n)$ is the maximum total side length of $n$ interior-disjoint squares packed in the unit square, is $f(k^2+1)=k$?

What is the exact shortest route guaranteed to escape the golden-gnomon triangle from an unknown starting point and heading?

Can the braid-group action on full exceptional collections fail to be transitive for $D^b(X)$ when $X$ is a smooth projective variety?