“Embarrassing” open problems in maths

There are many open problems in maths, most of which have the decency to sound appropriately daunting. For example, is it the case that “every Hodge class on a projective complex manifold X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X”? I have no idea what most of those words mean, so I’m certainly willing to believe that this is a Very Hard Problem. However, many open problems sound like they really should have been solved by now. In this post I’ll discuss a few “embarrassing” open problems in maths that are far harder than they might appear.

Four paradoxes involving infinity

I recently read an entertaining thought experiment claiming to show that energy isn’t conserved in Newtonian mechanics. It involves a sneaky trick using an infinite sequence of balls of decreasing size. In this post I’ll discuss this and some other superficially similar paradoxes and consider how they might be resolved, starting with Zeno’s famous paradox.

Some links 2

A webpage with no html

Parrot sings “Let the bodies hit the floor”

Mandatory minimums increase drug purity

Musical PVC pipes

The Alot is better than you at everything

Guessing polynomials

Consider the following game: Alice chooses a polynomial p with non-negative integer coefficients and challenges Bob to determine the polynomial. Bob is allowed to ask Alice for the value of p(n) for an n of his choice and upon receiving the answer can ask Alice for the value of p(m) for an m of his choice. Is there a strategy for Bob such that after receiving these two values he can always determine the coefficients of Alice’s polynomial?

A (fairly) interesting equivalent to the negation of the continuum hypothesis

I recently saw an interesting equivalent statement to the negation of the continuum hypothesis. The statement is essentially Freiling’s axiom of symmetry, presented in a very slightly different manner. It’s interesting in that it seems more geometric and less abstract than the continuum hypothesis itself. In this post, I briefly rehash the definition of the continuum hypothesis and prove the claimed equivalence.

Some links

Making Voodoo People

Brilliant acoustic covers

Medic discovers the trapezium rule, gets 75 citations

Dawkins hate mail

Inception (BWHARMMMMMMMMM!)