Mathematics Problem Archive

In 2002, P. Zhou analyzed the behavior of solutions of the Cauchy problem for perturbations of the defocusing NLS equation i u_t + u_{xx}- 2|u|^2 u - ...

Give a rigorous explicit solution of the fixed-frequency scalar-wave diffraction problem for two finite slits in the plane, including asymptotics of t...

Determine whether there exist bound entangled bipartite quantum states with negative partial transpose....

Is discrete logarithm modulo a prime randomized polynomial-time reducible to computing $g^{xy}$ from $g,g^x,g^y$?...

Is there an absolute upper bound for the size of a rational Diophantine $m$-tuple, a set of nonzero rationals for which the product of every two disti...

Let $U$ be the set of integers $n\not\equiv2\pmod4$ for which there are at most two distinct $D(n)$-quadruples. Is $U$ finite?...

Are there infinitely many Diophantine triples that are also $D(n)$-triples for three distinct integers $n\ne1$?...

Let $P_n$ be the supremum of the sizes of nondegenerate polynomial $D(n)$-tuples over $\mathbb{Z}[X]$. Find an upper bound for $P_n$ depending only on...

For any $\varepsilon>0$, there is a constant $C(\varepsilon)>0$ such that, for any positive integers $x$, $y$, $p$, $q$ satisfying $x^p\not= y^q$, the...

If $x$ and $y$ are positive integers with $y^2\not=x^3$, then $$ |y^2-x^3|\ge C\max\{y^2,x^3\}^{1/6}. $$...

Let $\lambda_1,\ldots, \lambda_n$ be $\mathbb{Q}$-linearly independent complex numbers. Assume that the numbers $e^{\lambda_1},\ldots,e^{\lambda_n}$ a...

The numbers $\pi$, $\zeta(3),\zeta(5),\ldots,\zeta(2n+1),\ldots$ are algebraically independent over $\mathbb{Q}$....

There exists a positive absolute constant $c$ such that, for any nonzero algebraic number $\alpha$ which is not a root of unity, $$ \mathrm{M}(\alpha)...

Let $x_1,\ldots,x_n$ be $\mathbb{Q}$-linearly independent complex numbers. Assume that for any $\varepsilon>0$, there exists a positive number $H_0$ s...

Let $n$ be a positive integer. For almost all $n$-tuples $(x_1,\ldots,x_n)$, there are positive constants $c$ and $D_0$ (depending on $n$, $x_1,\ldots...

Is computation of a curve's zeta function polynomial simultaneously in the genus $g$ and in $\log q$?...

For a prime $p$, conjecturally $p$ does not divide the class number of the maximal real subfield $\mathbb{Q}(\zeta_p+\overline{\zeta_p})$ of the $p$th...

Let $\zeta_p:\mathbb{Z}_p\to\mathbb{Q}_p$ be the $p$-adic zeta function. Is $\zeta_p(k)\ne0$ for every even integer $k$?...

Given an integer $n$, determine whether there are rational numbers $x,y,z$ satisfying $x^2+y^2=z^2$ and $xy=2n$; equivalently, determine whether $n$ i...

Is every integer $n\equiv5,6,$ or $7\pmod 8$ a congruent number?...

For $E_n:y^2=x^3-n^2x$, is $n$ a congruent number if and only if $L(E_n,1)=0$?...

Find the two prime factors of the RSA-1024 challenge integer $1350664108659952233496032162788059699388814756056670275244851438515265106048595338339402...

Characterize all algebraic number fields that have some power basis....

Selberg's orthogonality conjecture: generalization of Mertens' theorem for functions in Selberg class....

Dubner's conjecture: every even number greater than $4208$ is the sum of two primes which both have a twin....

On $\mathbb{Z}^2$, take nearest-neighbor jump probabilities $p_1,q_1,p_2,q_2$ in directions $\pm e_1,\pm e_2$, with $p_1>q_1$ and $p_2>q_2$. If the an...

Wild Virtuals: Create the category of “wild virtual knots” and establish its axiomatics. In particular, one needs a theorem that states when a wild eq...

Can there be a degenerate Kleinian group that is not the fiber of a fibration and has algebraic trace field?...

Is there a singly degenerate Kleinian group for which all matrix entries of all group elements lie in one fixed number field?...

(Mod g is Kahler). Forg≥ 3, the group Modg is a Kahler group, i.e. it is isomorphic to the fundamental group of a compact Kahler manifold. It was show...

Ig is finitely presented for g≥ 4. One thing we do know is that, in contrast to Mod g, neither Ig norKg has a classifying space which is homotopy equi...

Compute L(Modg) explicitly for small g≥ 2. In principle L(Modg) can be computed for any given g. The point is that one can first bound the degree of L...

Every subgroup of finite index in ModS contains a congruence subgroup. V. Voevodsky had indicated (in a personal communication) a beautiful applicatio...

s it true that any normal subgroup is commensurable with such a subgroup? Recall that two subgroups Γ 1, Γ2 of a group G are commensurable if the inte...

Does ModS has the Kazhdan property (T)? A positive answer would imply the positive answer to the previous question, but this problems seems to be much...

Find special features of the monodromy of algebraic surfaces. There is some good motivation for this coming from at least three directions • The probl...

Is the mapping class group a-T-menable? 14. Geometric properties of the mapping class group 245...

(Billiards in general polygons). Does every billiard table have at least one regular periodic trajectory? If the answer is affirmative, does this trajec...

Forg≥ 2, does Γg contain a purely pseudo-Anosov surface subgroup? The paper is organized as follows. In §2, we discuss the existence of surface subgro...

LetH be a convex cocompact subgroup of Γg. Is Γg H-separable? Just focusing on surface subgroups, we can ask:...

Does there exist a closed hyperbolic 4-manifold X that is the total space of a smooth fiber bundle Σg→X→ Σh? We will call such an X a surface bundle o...

Let M be a closed hyperbolic 4-manifold. Then all the Seiberg-Witten invariants of M vanish. The relevance of this is given in the following propositi...

Forg,h≥ 2, does there exist a short exact sequence: 1→π1(Σg)→ Γ→π1(Σh)→ 1 for which Γ is a word hyperbolic group? Arguing as in the proof of Theorem 4...

Do mapping class groups or Out(Fn) satisfy the Baum-Connes conjecture? Does Out(Fn) satisfy the Novikov conjecture? An approach to proving these conje...

Determine whether the Torelli group Ig (g≥ 3) is finitely presentable or not (note that Ig (g≥ 3) is known to be finitely generated by Johnson [42])....

The generating hypothesis, which asserts that the stable homotopy functor is faithful on the category of finite spectra. That is, if f is a map of fin...

The Johnson question. This says that if X is a space, and x is in BP_n (X), then x is not v_n torsion. My guess is that one should consider this quest...