Mathematics Problem Archive

Prove that Yang–Mills theory exists and has a mass gap on $\mathbb{R}^4$, meaning the quantum particles have positive masses....

Develop a mathematical framework that axiomatizes physics, particularly mechanics, thermodynamics, and probability theory....

Apply quantum and statistical field theory methods to model and potentially control pathogen evolution....

Identify governing principles for biological systems, analogous to physical laws....

Extend understanding of symmetries and action principles in biology to include robustness, modularity, evolvability, and variability....

Connect the Langlands program to fundamental physics symmetries....

Give a mathematically precise meaning to an exact formula comparable to Onsager's formula for the two-dimensional Ising model, and prove or disprove t...

For the Edwards-Anderson Ising spin glass on $\mathbb{Z}^d$ with i.i.d. mean-zero finite-variance nearest-neighbor couplings, prove or disprove the ex...

For the two-dimensional square-lattice model of independent electrons at density $1/4$, does magnetic flux $\pi/2$ per plaquette minimize the ground-s...

Prove that Bose-Einstein condensation occurs in a continuum model of a weakly interacting Bose gas, or determine whether the long-held assertion fails...

Prove that the Anderson model has purely absolutely continuous spectrum in dimension $\nu\geq 3$, for suitable disorder width $b-a$, in some energy ra...

Prove that the spectrum of the Anderson model in dimension $\nu=2$ is dense pure point....

For the Anderson model in dimension $\nu\geq3$ and disorder strengths $|b-a|$ admitting absolutely continuous spectrum, prove that $\sum_{n\in\mathbb ...

For the $N$-electron Coulomb ground-state energy $E(N,Z)$, determine the asymptotics of the ionization energy $\delta E(Z)=E(Z,Z-1)-E(Z,Z)$ as $Z\to\i...

Give a mathematically rigorous formulation and justification of the nuclear shell model....

Prove that, as the number of nuclei tends to infinity, the ground state of some neutral system of nuclei and electrons approaches a periodic limit, es...

What I have in mind here is not a specific problem, but a project, a very large scale project. The six (nonlinear) Painlev\'e equations form the core ...

The fact that RMT, and the Tracy-Widom distributions, arise in so many problems in so many different areas leads one to the following question: how ca...

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...

Construct a set of at least four mutually unbiased bases in dimension six, or prove that there are no seven mutually unbiased bases in $\mathcal{H}_6$...

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