Mathematics Problem Archive

Over $\mathsf{RCA}_0$, is Fraïssé's conjecture for countable linear orders equivalent to $\mathsf{ATR}_0$?...

Is there a subsystem weaker than $\mathsf{ATR}_0$ that proves that the three-element antichain is a better-quasi-order?...

Over $\mathsf{RCA}^*_0$, is Ramsey's theorem for triples equivalent to $\mathsf{ACA}_0$, as it is over $\mathsf{RCA}_0$?...

Do spin correlations in the classical Heisenberg model, and other $O(N)$ models with $N>2$, decay exponentially at every nonzero temperature, or do th...

(A) Prove long-range order for the quantum Heisenberg ferromagnet in dimension $D>2$ at temperature $T>0$. (B) Prove long-range order for spin $1/2$ i...

Establish, in some energy range, the existence of extended eigenstates or continuous spectrum for linear operators with extensive disorder, such as a ...

Determine the large-distance behavior of the one-particle reduced density matrix for a one-dimensional Fermi gas with spin and attractive interaction....

Give a fundamental and experimentally accessible definition of the entropy creation rate for general classical systems in stationary nonequilibrium st...

Give a mathematically precise meaning to determining the energy levels of the helium atom exactly, in the sense that hydrogen energy levels are exact,...

Let $V$ be a function on $\mathbb R^\nu$, $\nu\geq2$, satisfying $\int |x|^{-\nu+1}|V(x)|^2\,d^\nu x<\infty$. Prove that $-\Delta+V$ has absolutely co...

For the $N$-electron Coulomb Hamiltonian with nuclear charge $Z$, let $N_0(Z)$ be the least $N$ after which adding electrons no longer lowers the grou...

Give a mathematically rigorous justification of the techniques used to determine molecular configurations from first principles....

Prove that the integrated density of states $k(E)$ is continuous as a function of the energy $E$....

For spatial dimension $\nu=1$ and $1/2<\gamma<3/2$, determine the optimal constants $L_{\gamma,1}$ in the Lieb-Thirring inequality as predicted by the...

In many situations one is concerned with the asymptotic behavior of Riemann-Hilbert problems with exponentially varying data of the form $e^{in\phi(z)...

Random matrix theory was introduced into theoretical physics by Wigner in the 1950s in his study of neutron scattering resonances, but as a subject, R...

Much is now known, theoretically and also numerically, about the spectrum of self-adjoint operators. This is in great contrast to the situation regard...

The long-time behavior of the solution of the Cauchy problem for a great many integrable systems on the line is now well-understood, using, for exampl...

A number of so-called ``transportation'' problems have now been analyzed in terms of RMT. These include: the ``vicious'' walker problem of M. Fisher, ...

Let $J$ be a random tridiagonal matrix drawn from the tridiagonal Gaussian orthogonal ensemble and evolve it by the finite nonperiodic Toda flow, so i...

The bijective mapping properties of the scattering transform for a variety of integrable systems on the line between appropriate weighted Sobolev spac...

In the late 1990s Fokas introduced a new, more flexible approach to inverse scattering theory, and in recent years a number of researchers (Fokas, Its...

There has been considerable progress (Kuijlaars,...) in understanding basic statistics such as the correlation functions for the 2-matrix random matri...

Standard algorithms to compute the eigenvalues of a random matrix $H$ are completely integrable Hamiltonian systems (see ). The question raised in was...

Can one decide in deterministic polynomial time whether an integer $n$ is squarefree?...

Let $C_{10}$ find $x$ satisfying $x^e\equiv a\pmod n$ under $\gcd(e,\varphi(n))=\gcd(a,n)=1$. Is complete integer factorization randomized polynomial-...

Given a prime $p$ and degree $d$, can an irreducible polynomial of degree $d$ over $\mathbb{F}_p$ be constructed in deterministic polynomial time?...

Is complete integer factorization deterministically polynomial-time reducible to discrete logarithms modulo composites?...

Is complete integer factorization deterministically polynomial-time reducible to computing $\varphi(n)$?...

Given $a,b$ and a prime $p$ with nonsingular curve $y^2=x^3+ax+b$, can a point on the curve modulo $p$ be found in deterministic polynomial time?...

Does there exist a constant $c$ for which one can find in deterministic polynomial time a nonzero lattice vector of length at most $n^c$ times the min...

Given $f\in\mathbb{Q}[x]$, can the degree of its splitting field, equivalently the order of its Galois group, be computed in deterministic polynomial ...

Given $d\in\mathbb{N}$, can the class number $h(-d)$ of binary quadratic forms of discriminant $-d$ be computed in deterministic polynomial time?...

Can $\gcd(a,b)$ be computed in the parallel complexity class $NC$?...

Can two positive integers $a,b$ be multiplied using $O(\log(ab))$ bit operations?...

Determine the least positive integer $n_1$ and the greatest negative integer $n_2$ for which a $D(n_i)$-quintuple exists....

If $a$ and $b$ are two positive integers with $a<b$ and $R(a)=R(b)$ then there is a prime $p$ with $a< p< b$....

Let $\tau\in\mathbb{C}$ have positive imaginary part. Assume that $\tau$ is not quadratic. Set $q=e^{2i\pi\tau}$. Then at least $4$ of the $5$ numbers...

For each integer $n\ge 1$ there exists a positive constant $c(n)$ such that, for any algebraic subvariety $V$ of $\mathbb{G}_m^n$ which is defined ove...

There exists a positive constant $c$, depending only on $A$ and $\mathcal{L}$, such that for any $P\in A(\overline{\mathbb{Q}})$ which has infinite or...

Let $n \ge 2$. Denote by $\mathrm{ZH}_n$ the set of real numbers $\xi$ with the following property: there exists $c_1(\xi) >0$ and $c_2(\xi)>0$ such t...

Let $(n_i)_{i\ge 0}$ be an increasing sequence of positive integers. Assume there is a prime number $p$ such that the power series $$ \sum_{i\ge 0}z^{...

Given an elliptic curve $E/\mathbb{F}_q$ and $P\in E(\mathbb{F}_q)$, characterize maps or isogenies $\psi:E\to E'$ for which $\psi(P)$ can be evaluate...

Find bilinear structures that are useful for cryptographic constructions....

Find orders $\mathcal{O}$ whose Picard groups contain $\mathbb{Z}/\ell$, admit compact element representations, and allow group composition in $O(\log...

Make the proposed realization of existing finite-field discrete-logarithm systems inside class groups explicit for practical systems....

Determine whether Weil descent compromises the security of the proposed class-group realization of finite-field discrete-logarithm systems....

Explicitly construct global algebras or Brauer classes with prescribed local data, especially when the local splitting fields are dihedral....

Determine whether natural de Rham cup-product pairings can be used to improve zeta-function computations....

Remove the imaginary-hyperelliptic and $p\ne2$ restrictions from the complexity bound stated in the slides....