Unsolved Problems

Determine the maximum number and relative positions of limit cycles for polynomial vector fields of degree $n$, and investigate the topology of real a...

What is the densest packing of spheres in dimensions 4 through 23? More generally, what is the optimal sphere packing density in dimension $n$?...

If a compact set in $\mathbb{R}^d$ has Hausdorff dimension greater than $d/2$, must it determine a set of distances with positive Lebesgue measure?...

What is the relationship between curvature and Euler characteristic for even-dimensional Riemannian manifolds?...

What is the densest packing of unit spheres in dimensions other than 1, 2, 3, 8, and 24?...

Must a Kakeya set in $\mathbb{R}^n$ have Hausdorff and Minkowski dimension $n$?...

Does Yang-Mills theory exist mathematically and exhibit a mass gap in 4D?...

What are the relationships between curvature and Euler characteristic for higher-dimensional Riemannian manifolds?...

Is the first eigenvalue of the Laplace-Beltrami operator on an embedded minimal hypersurface of $S^{n+1}$ equal to $n$?...

What is the optimal sphere packing density in dimensions >3?...