I have recently switched into applications of mathematics to neuroscience.
Ramsey Theory and Brain Communication
We propose a principle of brain communication and compression that leverages the ideas of compressive sampling and sparse coding. This is joint work with Guy Isely and Friedrich Sommer (NIPS
2010 accepted oral spotlight submission pdf). | journal submission soon...
Surprisingly, the proof that the theory works depends on a theorem of Ramsey theory. Here is the statement in the simplest case:
Theorem: Color the integer points in the plane any way you choose with a finite number of colors. Then for any positive integer s, there is a s x s submatrix of points (see picture on right) all with the same color.
(with F. Sommer) Ramsey theory reveals the conditions when sparse coding on subsampled data is unique, preprint pdf.
Can you find a 5x5 submatrix of the same color?
Hint: The color is gray.
(rollover to see the answer)