In this talk, intended for general mathematical audience, I will explain how complex analytic methods have been used to study the distribution of mathematical objects of number theoretic interest in the last couple of centuries, starting with Riemann's approach to proving the Prime Number Theorem. At the end I will sketch some recent progress.
Sobolev embeddings control the integrability of some power of a function by an
integral of the derivative of the function at a lower power. The limiting case
where the latter power is taken to be 1 due to Gagliardo and Nirenberg, is
inaccessible to classical methods of harmonic analysis and turns out to be a
functional version of the isoperimetric inequality. If one considers vector
fields instead of functions, one can hope that some redundancy in the
derivative would allow to obtain estimates with an integrand that does not
Abstract: In this talk, I will look at an old problem of very simple flow on a torus,
x' = wx + g(x,y)
y' = wy
In particular, I will be interested in the phase-locked states for this system. A common class of phase models has the form that
g(x,y) = a(y) sin(x) + b(y) cos(x)
I show that this system is equivalent to a simple map:
r' = m(r)