Careers

We're looking for people who play with hard problems

We hire infrequently, but are always looking for people who would be fun to work with on hard problems.

How to apply

No resume required. No degree required.

Tell us who you are and what you want to build.

A few questions

Just for fun, to get to know you, and to help confirm you're human.

1. Who are you?

2. A regular tetrahedron

Consider the two vectors from the centroid of a regular tetrahedron to any two of its vertices. What is the angle between them?

Hint: Just as a 2D equilateral triangle can be represented in ℝ³ as the convex hull of the standard basis vectors e₁, e₂, e₃ (centroid at (⅓, ⅓, ⅓)), a 3D regular tetrahedron is the convex hull of e₁, e₂, e₃, e₄ in ℝ⁴, with centroid (¼, ¼, ¼, ¼). The two vectors from the centroid to a pair of vertices have simple components; cos θ falls out of their dot product and lengths.

The answer can be expressed as arccos(…) and is left as an exercise for the reader. You might note that 109.47° is also the methane bond angle from chemistry.

3. How do you feel?

Tell us honestly how you're feeling at this moment.

4. Taniyama's Problem 29 (optional)

While researching open conjectures, we came across Taniyama's Problem 29 — posed at the 1955 Tokyo–Nikko symposium on algebraic number theory. Unlike his celebrated Problems 12 and 13, which inspired the Modularity Theorem and the proof of Fermat's Last Theorem, Problem 29 received less attention: it did not spark a major theorem, and was never translated into English.

We present it here for the curious. The resolution is known, though not widely. It should be approachable with graduate-level mathematics, but the answer is not simple. This is left as an exercise for the reader — we're not asking for solutions, and we won't accept them.

Problem 29 (Taniyama, 1955) — Modern Reformulation

Consider the curve Cq : y² = 1 − xq for a prime q ≥ 5. Let p be a prime distinct from 2 and q, and let A be the Cartier–Manin matrix of Cq reduced modulo p.

Does the Jordan block structure of A depend only on the multiplicative order of p modulo q?

Original text of Taniyama Problem 29 (Sūgaku, 1956)
Original: Y. Taniyama, Sūgaku vol. 7 (1956), p. 269. The statement above is a reformulation in modern terminology.

If you feel like it, tell us how it felt to think about the problem, or point us to another mathematically or scientifically interesting problem that you feel has been unfairly forgotten.