Into Darkness

Two things are infinite: the universe and human stupidity; and I’m not sure about the universe.
― Albert Einstein

Combinatorics:

  • (The linear arboricity conjecture) The linear arboricity of every \(d\)-regular graph is \(\lceil (d+1)/2 \rceil\).
  • (Packing squares) Let \(s(x)\) denote the maximum number of non-overlapping unit squares which can be packed into a large square of side length \(x\). Find the best asymptotics of the “wasted” area \(W(x)=x^2-s(x)\).
  • (Graham’s conjecture on sub-multiplicativity of pebbling number) \(\pi(G_1\times G_2) \le \pi(G_1)\pi(G_2)\).

Discrete Geometry

  • (Erdös-Szekeres problem) For any integer \(n\geq 3\), any set of at least \(2^{n-2}+1\) points in general position in the plane contains \(n\) points that are the vertices of a convex \(n\)-gon.
  • (Makai–Pach’s conjecture on translative slab covering) A sequence of slabs (region enclosed by two parallel hyperplanes) in \(\mathbb{R}^d\) with widths (distance between two hyperplanes) \(w_1, w_2, \dots\) permits a translative covering if \(\sum_i w_i = \infty\).

Computability

  • (Freeness of matrix semigroup) Is it decidable whether the multiplicative semigroup generated by two \(2\times 2\) matrices in the modular group \(SL_2(\mathbb{Z})\) is free?

Euclidean Geometry:

  • (Inequality of the Means) Is is possible to pack \(n^n\) rectangular \(n \)-dimensional boxes each of which has side lengths \(a_1,a_2,\ldots,a_n \) inside an \(n\)-dimensional cube with side length \(a_1 + a_2 + \ldots a_n\)?
  • (Michael Atiyah) Let \(x_1, \ldots, x_n\) be \(n\) distinct points in the unit ball. Let the oriented line \(x_ix_j\) meet the boundary 2-sphere in a point \(t_{ij}\) (regarded as a point of the complex Riemann sphere \(\mathbb{C}\cup\infty\)). For the complex polynomial \(p_i\) of degree \(n-1\), whose roots are \(t_{ij}\). For all \((x_1, \ldots, x_n)\) the \(n\) polynomials are linearly independent.

Number Theory:

  • (Lehmer’s conjecture) Is there an absolute constant \(\mu > 1\) such that for every monic polynomial \(p(x)\) with integer coefficients, the product of the roots of \(p(x)\) with length greater than \(1\) is greater than \(\mu\)?

Topology:

  • Is there a continuous mapping \(f: [0,1]\to\mathbb{R}^2\) such that for all \(\epsilon>0\), the image of \([0,\epsilon]\) under \(f\) is a convex set with three points not on a line?
42 (this post is made with love)

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