Mathematics Problem Archive

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

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

Congruent number problem (a corollary to Birch and Swinnerton-Dyer conjecture, per Tunnell's theorem): determine precisely what rational numbers are c...

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

Let $X_t$ be $d$-dimensional Brownian motion starting at the origin, let $D$ be an open subset of $\mathbb{R}^d$ containing the origin, and let $\tau=...

Let $D\subset\mathbb{R}^d$, $d\ge2$, be bounded, connected, and open. Suppose there are coupled reflected Brownian motions $X_t,Y_t$ in $D$ and $\vare...

For the exclusion process on $\mathbb{Z}$ with $p(x,y)=p(y-x)$, assume $\sum_x|x|p(x)<\infty$, $\sum_xxp(x)>0$, and $\sum_{x<0}x^2p(x)=\infty$. Does t...

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...

For the exclusion process on $\mathbb{Z}^d$ with translation-invariant kernel $p(x,y)=p(y-x)$ and zero mean $\sum_xxp(x)=0$, prove that every stationa...

For nearest-neighbor asymmetric exclusion on $\mathbb{Z}$ with $p(1)=p>q=p(-1)$, start from the deterministic configuration $\cdots11110000\cdots$. Is...

For Aldous's self-similar random partition of the plane, determine its topological properties: in particular, do region boundaries have fractal dimens...

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

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 ...

Let $(X,\xi)$ consist of a random element and a diffuse random measure on a locally compact second countable Abelian group. Is mass-stationarity of $(...

Does the invariant-transport characterization of mass-stationarity remain valid if the bounded jointly invariant preserving kernels are restricted to ...

For every infinite vertex-transitive graph $G$, prove that $p_c(G)<p_u(G)$ if and only if $h(G)>0$....

Give general conditions implying $p_u(G)<1$. In particular, prove or disprove that every one-ended transitive graph has $p_u(G)<1$....

For nearest-neighbor independent bond percolation on $\mathbb{Z}^d$, $d\ge2$, let $p_c(d)$ be the critical edge-retention probability. Prove that at $...

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...

Is there any algorithm for recognition whether two graph-links are equivalent or not? Our conjecture is “no”. The idea behind that is that graph-links...

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?...

What is the smallest n = n(g) such that MCG (Sg) admits a properly dis- continuous action on Rn? on a contractible n-manifold? (The answers are expect...

(Fast word problem). Is there a sub-quadratic time algorithm to solve the word problem in Modg? One might guess that n logn is possible here, as there...

(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...

f Γ is an irreducible arithmetic group of rank ≥ 2, then every homomorphism Γ→ ModS has finite image. For many arithmetic groups Γ the conjecture can ...

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...

[Ehrenpreis Conjecture] Given two closed Riemann surfaces, there are finite un- branched covers with homeomorphic total spaces which are arbitrarily c...

Study, systematically and with the help of computers, the finite quotients of Mg which do not factor through Sp (2g, Z). We remark that Problem 3.1 wo...

Is there a faithful finite dimensional matrix representation of Mg,b,n for any value of the triplet (g,b,n ) other than (1, 0, 0), (1, 1, 0), (1, 0, 1...

Find a candidate for a faithful finite-dimensional matrix representation of Mg orMg,1,0....

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 linear? A locally compact group Γ is said to satisfy the Haagerup approximation property or is a-T- menable if there exists...

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...

(Orbit closures for moduli spaces). Determine the closures of the orbits for the GL+(2, R)-action on H(α) andQ(β). Are these closures always complex-a...

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...

Let M = H3/Γ be a finite volume hyperbolic 3-manifold, does Γ contain a surface subgroup. This was answered in [ 8] for non-compact but finite volume ...

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...

Forn> 3, does Aut(Fn) have property (T)? The corresponding question for mapping class groups is also open. If Aut( Fn) were to have Property (T), then...

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...