Math Matters, December 2006

Unfortunately, that is the only kind of math that I have given any thought to lately.

But here are some things to think about over break, (while I am finishing a referee job)

Geometric Topology: there is a way to form a wild knot as an intersection of nested solid tori, and I think that I can prove that if two knots from this class are equivalent, then the defining sequences must also be equivalent up to some sort of cofinality condition. If I can do this, then perhaps I can come up with some invariants that distinguish very ill behaved will knots (that is, knots that are so wild that they fail to pierce a disk anywhere)

College level mathematics: suppose one has the differential equation y”+y = f(t) and f(t) is both bounded and periodic of period 2pi as well as piecewise smooth. Then any solution to this differential equation is bounded on [0, infinity) if and only if the k = 1 term of the Fourier series expansion vanishes. Sufficiency is easy to show; necessity is where the work lies.
Of course, this is just a mild extension of the concept of resonance; for example the solution to y” + y = sin(t), y(0)=0, y’(0)= -1/2 is y(t) = -(1/2)tcos(t), which has unbounded oscillations as t goes to infinity, even though the driving function is bounded. For those who like to see examples, I suggest surfing to the interactive differential equations website. Click enter, and then go to the menu; there is tons of stuff there. You can use sliders to vary parameters and you can watch the phase plane as the systems evolve with time.

Ok, now I’ve said it. Now what am I going to do about it?