Mathematics Problem Archive

{\rm (A. Glutsyuk, S. Tabachnikov )} Given two nested closed convex hypersurfaces, assume that the billiard transformations on the set of oriented lin...

Construct real analytic examples of Nijenhuis operators on closed two-dimensional surfaces whose eigenvalues are real and generically distinct....

[ Maxwell, seems bad, no tools] For any system of $N$ isolated fixed point charges in $\mathbb{R}^3$, the number of points of equilibrium (assumed fin...

[B. Sh., seems good] For any real polynomial $p(x)$ of degree $k$ with simple real zeros, $$ \#_{r}\left[(k-1)(p'(x))^2-kp(x)p''(x)\right] \le \#_{nr}...

[B. Sh] For any real polynomial $p(x)$ of even degree, $$ \#_{r}\left[(k-1)(p'(x))^2-kp(x)p''(x)\right] + \#_{r}p(x)>0, $$...

If $f(z)$ is meromorphic of finite order $\rho$ and $\sum\delta(a,f)=2$, it is conjectured that $\rho=n/2$, where $n$ is an integer and $n\geq 2$, and...

(Edrei's spread conjecture) If $f(z)$ is meromorphic in the plane and of lower order $\lambda$, and if $\delta=\delta(a,f)>0$, is it true that, for a ...

Let $f$ denote a rational or entire function of a complex variable, and $f^n, n=1, 2, \ldots$, the $n$-th iterate of $f$, so that $f^1=f, f^{n+1}=f\ci...

Define $A_n=\sup_{f\in S}|a_n|$. It is shown by Hayman that \[\frac{A_n}{n}\to K_0,\hspace{1cm}\text{ as }n\to\infty.\] Is it true that $K_0=1$? The b...

(Schoenberg's conjecture) If $f(z)=\sum^\infty_{n=1}a_nz^n$ and $g(z)=\sum^\infty_{n=1}b_nz^n$ are convex, and $f$, $g$ belong to $S$, is it true that...

Suppose $f$ in $S$ and define $h(z) = \{f(z^2)\}^{\frac{1}{2}} = z + c_3z^3 + c_5z^5 +\ldots$ Robertson's conjecture (see Sheil-Small ) asserts that \...

If $f$ in $S$, write \[\log[f(z)/z]=2\sum^\infty_{k=1}\gamma_kz^k.\] If $f$ is star-like then $|\gamma_k| \leq1/k$; this is false in general, even in ...

It was proved by Boyarski\u\i\, that the partial derivatives of a plane $K$-quasiconformal mapping are locally $L$-integrable for $2\leq p < 2+c$, whe...

This problem is equivalent to Problem 7.9 due to Gehring and Reich about best bounds for area distribution under quasiconformal mapping. Let $E$ denot...

(Subadditivity problem for analytic capacity) Prove or disprove the existence of a constant $M$ such that \[\gamma(K_1\cup K_2)\leq M\{\gamma(K_1)+\ga...

Let $\Gamma$ be a Fuchsian group in $\mathbb{D}$, and let $i(z) \equiv z$. Is it true that \[\sum_{\gamma\in\Gamma}|\gamma'(0)|\geq\prod_{\gamma\in\Ga...

Characterise the Hankel operators on the Hardy space $H^2$ on the circle that are of trace class. (F. Holland)...

Let $A_1,\dots,A_n\in \mathbb{R}^{d\times d}$ be symmetric matrices and $g_1,\dots,g_n\sim\mathcal{N}(0,1)$ i.i.d.. Does the following hold? $\mathbb{...

Let $X\in\mathbb{R}^{d\times d}$ be a symmetric matrix with (otherwise) independent gaussian entries. Prove (or disprove): $\mathbb{E} \|X\| \lesssim\...

Prove or disprove the following conjecture by Feige : Given $n$ independent random variables $X_1,\dots,X_n$ s.t., for all $i$ , $X_i \geq 0$ and $\ma...

Let $N = 2M$ . For which $s$ is there a polynomial time algorithm that is guaranteed to, with high probability, certify that a gaussian matrix $A$ is ...

What is the partial recovery threshold for the Stochastic Block Model on $k\geq 3$ communities....

What is the exact recovery threshold for the Stochastic Block Model with a logarithm number of communities? Both computational and information theoret...

Does the projected power method converge (with high probability) to the optimal solution of the angular synchronization problem with (small enough) ga...

Let G=(V,E) be an Eulerian graph with minimum degree at least 4, and let W be a closed Eulerian walk of G. Is it true that G has a cycle decomposition...

Beck: Show that the discrepancy of any hypergraph H is at most c|E(H)|^(1/2)...

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

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....

Does there exist a Hamiltonian system with a smooth invariant torus on which the induced dynamics is mixing? No such mixing can occur on a two-dimensi...

If positive square matrices $A,B$ are shift equivalent over $\mathbb Q_+$, prove that they are strong shift equivalent over $\mathbb Q_+$....

Let $S\subset\mathbb R$ be a unital subring. Prove that a tuple $\Lambda=(\lambda_1,\ldots,\lambda_k)$ is the nonzero spectrum of a primitive matrix o...

Prove that every expansive automorphism of a one-sided full shift is topologically conjugate to a two-sided full shift....

If $(a(n))$ is good for $\ell$-recurrence of powers, is then $(a(n)^k)$ good for $1$-recurrence for $k=1,\ldots, \ell$?...

Suppose that the polynomials $p_1,\ldots, p_\ell\in \mathbb Z[t]$ are pairwise independent. Show that there exists $d\in \mathbb N$ such that the fact...

Suppose that the polynomials $p_1,\ldots, p_\ell\in \mathbb Z[t]$ are rationally independent. Show that $\mathcal{K}_{rat}(T_1),\ldots, \mathcal{K}_{r...

Is it true that one always has a decomposition $$ \int f_0 \cdot T_1^n f_1 \cdot \ldots \cdot T_\ell^n f_\ell \ d\mu= \psi(n)+e(n) $$ where $(\psi(n))...

Here $\mathcal F=\{a_1,\ldots,a_\ell\}$ is a family of functions of polynomial growth in one Hardy field, and $\operatorname{span}^*(\mathcal F)$ deno...

Here $\mathcal F=\{a_1,\ldots,a_\ell\}$ is a family of functions of polynomial growth in one Hardy field, and $\operatorname{span}^*(\mathcal F)$ deno...

Let $c$ be a positive non-integer. Show that the sequence $([p_n^c])$ is good for multiple recurrence and convergence of powers....

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Can a Siegel disk have a boundary that is not a Jordan curve?...

If a rational map $f$ has $J(f)=\widehat{\mathbb C}$, is $\omega(z)=\widehat{\mathbb C}$ for almost every $z$, and is $f$ conservative with respect to...

Given a sufficiently dense sample of points on a surface (technically, an $\epsilon$-sample), reconstruct a surface homeomorphic to the original....

An optimization problem that naturally arises in the study of ``swarm robotics'' is to wake up a set of ``asleep'' robots, starting with only one ``aw...

Does every $n$-vertex planar graph have a 3D grid drawing with $O(n)$ volume? A 3D grid drawing of a graph is a placement of the vertices at distinct ...

Does every planar graph have $O(1)$ queue-number? A queue layout of a graph consists of a linear order of the vertices and a partition of the edges in...

There almost surely exists a horofunction h such that lim n→∞ − 1 n h(xn) = A, where A = lim n→∞ 1 n d(x0, xn). A theorem of Karlsson states that ∀ϵ >...

Prove that the pictured standard triple soap bubble is the least-perimeter way to enclose and separate three given volumes in $\mathbb{R}^3$....