Mathematics Problem Archive

Let $D\subset\mathbb R^n$ have regular boundary for the Dirichlet problem. Prove or disprove that the following are equivalent: (PI) there is a nonzer...

A spherical quadrilateral is a disk with four marked boundary vertices, curvature-one metric, geodesic sides, and interior angles $\pi\alpha_j>0$. Cla...

Let $A$ be a set of vectors $a=(a_1,\ldots,a_n)\in\mathbb C^n$ such that every $n$ of them are linearly independent, and assign to every $a\in A$ a $\...

Let $A$ be a set of vectors $a=(a_1,\ldots,a_n)\in\mathbb C^n$ such that every $n$ are linearly independent, and let $K_a$ be plane convex compact set...

Classify entire functions $f,g$ for which $f\circ g$ is periodic. Prove that, up to the natural equivalences, the possibilities are exhausted by: $g$ ...

Does there exist an entire function whose zeros lie on the positive ray and whose $1$-points lie on two rays making angles $\pm\alpha$ with it, for so...

For $\omega=e^{2\pi i/5}$, is an entire solution of $$f(\omega z)f(\omega^{-1}z)=f(z)-1$$ unique up to rotation of the variable $z$?...

For a local inverse germ $\phi_z$ of a meromorphic function $f$ at a noncritical value $w=f(z)$, Gross's theorem gives analytic continuation along alm...

Let $F$ be entire in two variables and let a holomorphic germ $\phi$ satisfy $F(z,\phi(z))=0$ near a nonsingular point. Must $\phi$ admit analytic con...

Let $\mu$ be a positive plane measure with $\mu(\{|z|\le r\})\le cr^\alpha$ for some $0<\alpha<1/2$. Can its logarithmic potential $$u(z)=\int\log\lef...

Let subharmonic functions $u_k$ on the unit square converge uniformly to $u(x,y)=x$. If $D_k=\{u_k<0\}$ and $D_k^*$ is the component containing $-1/2$...

Let $u$ be subharmonic on $1<|z|<2$, and let pairwise disjoint open sets $D_k$ be unions of components of $\{u<0\}$. Suppose that for every $r\in(1,2)...

Let positive masses $a_k$ be placed at a discrete set $x_k\in\mathbb R^n$ and suppose $\sum_k a_k/|x_k|^{n-1}<\infty$, so that $$F(x)=\sum_k\frac{a_k(...

Let $f$ be holomorphic in the unit disk with exactly one simple zero $z_0$ and two $1$-points $z_1,z_2$, counted with multiplicity. Determine the larg...

Let $f$ be holomorphic in the unit disk, with $f(0)=0$, $f'(0)\ne0$, no other zeros, and exactly two solutions $z_1,z_2$ of $f(z)=1$, counted with mul...

Solve the two-one-point extremal problem for real holomorphic $f$: determine the minimum of $\max(|z_1|,|z_2|)$ and maximum of $|f'(0)|$ when $f(0)=0$...

Let $f$ be real and holomorphic in the unit disk, with one simple zero at $0$ and two simple $1$-points at $\pm ia$. Determine the minimum possible va...

For rational functions of each fixed degree, determine the analogues of Goldberg's constant and the unit-disk zero/one-point extremal quantities, and ...

Obtain better rigorous estimates for the Littlewood exponents $\alpha$, $\beta$, and for $\sup_c P_c$, where $P_c$ is the pressure for the hyperbolic ...

Are the iterates $p_c^n$ of hyperbolic quadratic polynomials extremal, or nearly extremal, for the Littlewood exponent $\alpha$ governing mean spheric...

For which parameters $c$ is the pressure $P_c$ of the hyperbolic quadratic polynomial $p_c(z)=z^2+c$, for the potential $|(p_c^n)'|^{-1}$, close to or...

If the definition of $\sup_E\beta_E$ is extended from connected regular compact plane sets to all regular compact sets, is the supremum attained on co...

For finitely many positive charges $a_k$ at points $x_k\in\mathbb R^3$, is the critical set of $u(x)=\sum_{k=1}^n a_k/|x-x_k|$ always finite?...

For the rectangular lattice $\Lambda=\{an+ibm:n,m\in\mathbb Z\}$ with $a^2+b^2=1$ and $a\in(0,1)$, let $f_a:\mathbb D\to\mathbb C\setminus\Lambda$ be ...

Let $u_1,u_2,u_3$ be subharmonic in the plane with $u_j(0)=0$, and let $v_1\le v_2\le v_3$ be their pointwise increasing rearrangement. Put $I(r,v)=\i...

Let $f:\mathbb C\to\mathbb P^2$ be linearly nondegenerate and let $\delta(a,f)$ be the Nevanlinna deficiency of a point $a\in\mathbb P^2$. Prove that ...

Let $f:\mathbb C\to\mathbb P^n$ be holomorphic with spherical derivative $\|f'\|(z)=O(|z|^\sigma)$ for some $\sigma>-1$, and let $a_1,\ldots,a_q$ be h...

For $p\ge3$, let $V(D)$ consist of zero-free holomorphic vectors $(f_1,\ldots,f_p)$ on $D$ with $\sum f_j=0$, and use the source's definition of a $C$...

Let $f=(f_0,\ldots,f_n)$ be a linearly nondegenerate holomorphic curve, let $T(r,f)$ have finite lower order $\lambda$, and let $N_1(r)$ be the averag...

Let $f$ be holomorphic in the unit disk with a simple zero at $0$, exactly two simple $1$-points at $a$ and $\overline a$, and no other zeros or $1$-p...

Does the three-dimensional cube admit a triangulation into tetrahedra all of whose dihedral angles are acute?...

A facet of a $d$-polytope is degenerate if it has more than $d$ vertices. Determine the maximum number of degenerate facets of an $n$-vertex $d$-polyt...

Determine the maximum total number of faces of a $d$-dimensional convex polytope with $n$ vertices and $n$ facets. In dimension four, do such fat-latt...

Given $n$ points and $m$ hyperplanes in $\mathbb R^d$ whose incidence graph contains no $K_{s,t}$, determine the maximum number of incidences. Of spec...

For an $n$-point planar set, determine the maximum number of halving lines. More generally, determine the maximum number of $k$-sets, subsets obtained...

For $n$ pseudocircles in general position, determine the maximum number of tangent pairs and the maximum number of digon cells. Determine whether the ...

Determine the worst-case complexity of the medial surface and of an offset surface of an $n$-feature polyhedron, and of the Voronoi diagram of $n$ lin...

Determine the exact Erdős–Szekeres number $f(n)$, the least number of planar points in general position forcing a convex $n$-gon. Also determine sharp...

Determine the combinatorial complexity of the visibility complex of $n$ pairwise disjoint unit spheres in three-dimensional space....

Given $n$ planar points, find a subquadratic algorithm for the minimum-area triangle or prove a quadratic lower bound in a suitable computation model....

Given $n$ points in $\mathbb C^2$, determine in quadratic time whether three lie on a complex line, or prove a quadratic lower bound; the known algori...

For fixed $d>3$, determine whether every point of an $n$-point set in $\mathbb R^d$ is a convex-hull vertex faster than the best known near-$n^{2\lflo...

Given a sorted list of $n$ real numbers, find for every $1\le k\le n$ the shortest interval containing exactly $k$ entries. Find a subquadratic algori...

Given lines in the plane and two vertices $s,t$ of their arrangement, find a subquadratic algorithm for the shortest $s$-$t$ path along arrangement ed...

Is there a near-linear-time algorithm to construct the straight skeleton of a simple polygon? Determine the optimal complexity, including for polygons...

Given motorcycles moving simultaneously along fixed rays and crashing upon reaching another track, determine the motorcycle graph in near-linear time....

Determine the optimal complexity of computing the volume of the union of axis-aligned boxes in fixed dimension at least three. In particular, is there...

Given a planar point set $P$, sample uniformly from the simple polygons with vertex set $P$ in polynomial time, or determine the complexity of countin...

Develop exact polynomial-time algorithms for the constructive forms of Aleksandrov's, Cauchy's, Minkowski's, Steinitz's, and Koebe's polytope-realizat...

Can every convex polytope be triangulated so that every vertex degree, or every edge degree, is bounded by a constant or by a polylogarithmic function...