Mathematics Problem Archive

For an infinite $d$-regular Ramanujan graph, does random-walk neighborhood sampling converge to the $d$-regular tree?...

For each $d\geq3$, does the independence ratio of a uniformly random $d$-regular graph converge in probability as the number of vertices tends to infi...

Let $G$ be an infinite Cayley graph of a group that is not virtually cyclic. Prove that there exists $p<1$ for which Bernoulli $p$-edge percolation on...

Does every infinite connected Cayley graph admit an invariant random perfect matching?...

For a nonamenable group $\Gamma$, does the Bernoulli shift $\{0,1\}^{\Gamma}$ factor onto $\{0,1,2\}^{\Gamma}$?...

Can every $d$-regular graphing without multiple edges be properly edge-colored by $d+1$ colors?...

If $A$ and $B$ are countably infinite groups, does $A\times B$ have fixed price $1$?...

Let an amenable group $\Gamma$ act ergodically and essentially faithfully on $X$. Is the groupoid cost of the action $1$? If $\Gamma$ is finitely pres...

In a symmetric space, is every Killing tensor a sum of symmetric products of Killing vectors? Equivalently, is it true that the algebra of all polynom...

Assume that the exterior of an outer billiard table is foliated by invariant curves. Prove that the table is an ellipse....

{\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...

The Bieberbach conjecture Is it true that $|a_n|\leq n$ for $f$ in $S$ with equality only for $f(z)\equiv f_\theta(z)$? The result is known to be true...

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)...

Is the weak order on S_(n) (the "inversion" poset) Sperner?...

If the critical set of $u(x)=\sum_{k=1}^n a_k/|x-x_k|$ for positive point charges in $\mathbb R^3$ is finite, how many points can it contain? In parti...

For a monic polynomial $p$ of degree $d$, determine the maximum length of the lemniscate $E(p)=\{z:|p(z)|=1\}$. Is the extremal asymptotically $p(z)=z...

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....

Given a polygonal room with perfectly reflecting sides and a point light source, characterize when every point of the room is illuminated. In particul...

Does there exist a family of bounded-degree expander graphs, with spectral gap bounded away from zero and diameter tending to infinity, whose coarse R...

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...

Let $S\subset\mathbb R$ be a unital subring and let $A$ be a square matrix over $S$ whose nonzero spectrum satisfies the spectral-conjecture condition...

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