Mathematics Problem Archive

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

If a quadratic polynomial $f_c$ is infinitely renormalizable of bounded type, must $J(f_c)$ be locally connected? In particular, is the Julia set of t...

Can a nowhere-dense Julia set of a rational map have positive Lebesgue measure?...

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...

Consider minimal immersions $S^n\to S^N(1)$ with total area at most a fixed constant $A$, identifying immersions which differ by a motion of the ambie...

Let $B$ be a set of $n+2$ hyperplanes in general position in complex projective space $\mathbb{P}^n(\mathbb{C})$. Prove that there is no nondegenerate...

Let $\Delta$ be a ball in $\mathbb{C}^n$, $n\geq2$, and let $F$ be a complete Hermitian manifold. Does every holomorphic map from $\Delta\setminus\{0\...

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 >...

Determine all single sided (resp. double sided) sequences that are pattern Sturmian....

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

Is every Laplacian algebra of polynomials maximal?...

Does every curve bounding a surface of positive curvature in 3-space have (at least) four points where the torsion vanishes?...

Does connectedness of the shadows imply that f(M) is convex?...

Are there any complete surfaces of negative curvature in Euclidean 3-space whose principal curvatures are bounded away from zero?...

Deligne's conjecture on Hochschild cohomology about the operadic structure on Hochschild cochain complex....

Does every convex polyhedron have Rupert's property?...

The asymptotic behavior of a variety of combinatorial problems has now been analyzed in great detail. Here we have in mind Ulam's problem for the leng...

For a full-rank integer lattice, is finding a nonzero vector of minimum Euclidean norm NP-hard?...

Under the promise that $b^2-4ac$ is not a square, is deciding whether $ax^2+bxy+cy^2+dx+ey+f=0$ has an integral solution NP-hard?...

Is the same binary quadratic Diophantine solvability problem NP-hard under randomized reductions?...

Develop a fast algorithm to compute the Tate–Lichtenbaum pairing $T_n$....

Factor $N=pqr$ from fewer known bits of its prime factors....

Given integers $n$ and a prime power $q$, construct, when possible, an elliptic curve $E/\mathbb{F}_q$ with $\#E(\mathbb{F}_q)=n$....

Speed up the computation of cryptographic pairings....

Give the cited pairing-based signature constructions and their security proofs without relying on distortion maps....

Construct efficient pseudorandom generators from ideal-lattice problems....

Construct efficient digital-signature schemes from ideal-lattice problems....

Base cryptographic constructions directly on worst-case hardness assumptions from algebraic number theory....

Construct an efficient ideal-lattice version of Regev's quantum-SVP-based cryptosystem....

For the classical-group pairs and generic local $L$-parameters in the Gan–Gross–Prasad setting, is there exactly one relevant representation $\pi$ in ...

Can every integer be written as a sum of four perfect cubes?...

Does every unimodular random countable locally finite rooted graph arise as a local weak limit of finite graphs?...

Let $S_n=S_0+X_1+\cdots+X_n$ be a renewal process whose i.i.d. strictly positive recurrence times have infinite mean and a non-lattice distribution. I...

Let $S$ and $S'$ be random walks on $\mathbb{R}$ with the same i.i.d. step-length distribution, starting at $0$ and $x$. Suppose the step lengths are ...

Prove that every bounded-degree transient hyperbolic graph contains a transient subtree....

For simple random walk $(X_n)$ on the uniform infinite planar triangulation, prove that the graph distance from the starting point has exponent $1/4$,...

Take a uniform quadrangulation with $N$ faces conditioned to have genus $CN$, where $0<C<1/4$. Prove that its rooted local limit is the stochastic hyp...

Find a 3-dimensional topological interpretation of the Jon es polynomial of links....

Understand Khovanov’s categorification of the Jones polyno - mial....

(H. Murakami) For a torus knot K, calculate CS(S3− K) (giving an appropriate definition of it) and calculate lim log JN (K) N (fixing an appropriate c...

[154] Every Vassiliev invariant of classical knots can be extended to a finite type invariant of long virtual knots. (Se e also Problem 3.9.) 2.8 Fini...

(K. Habiro [165], see also [153, “Theorem 5”]) Two m- strand string links L and L′ are Cd -equivalent if and only if v(L) = v(L′) for any A-valued fin...

(J. Roberts) What is graph cohomology the cohomology of?...

(R. Fenn, C. Rourke, B. Sanderson) H3(Rp)∼ = Z⊕ Z/pZ for p prime....

Is the Burau representation of B4 faithful?...

(3-move conjecture, Y. Nakanishi [305]) Any link can be related to a trivial link by a sequence of 3-moves....